The adventure of physics - vol. IV : the quantum of change

Christoph Schiller

Motion Mountain

This book is written for anybody who is curious about nature and motion. Have you ever asked: Why do people, animals, things, images and space move? The answer leads to many adventures; this volume presents those due to the discovery that there is a smallest change value in nature. This smallest change value, the quantum of action, leads to what is called quantum physics. In the structure of modern physics, quantum physics covers three points; this volume covers the introduction to the point in the lower right: the foundations of quantum theory.

The present introduction to quantum physics arose from a threefold aim I have pursued since 1990: to present the basics of motion in a way that is simple, up to date and captivating.

In order to be simple, the text focuses on concepts, while keeping mathematics to the necessary minimum. Understanding the concepts of physics is given precedence over using formulae in calculations. The whole text is within the reach of an undergraduate.

In order to be up to date, the text is enriched by the many gems – both theoretical and empirical – that are scattered throughout the scientific literature.

In order to be captivating, the text tries to startle the reader as much as possible. Read- ing a book on general physics should be like going to a magic show. We watch, we are astonished, we do not believe our eyes, we think, and finally we understand the trick. When we look at nature, we often have the same experience. Indeed, every page presents at least one surprise or provocation for the reader to think about. Numerous interesting challenges are proposed.

The motto of the text, die Menschen stärken, die Sachen klären, a famous statement by Hartmut von Hentig on pedagogy, translates as: ‘To fortify people, to clarify things.' Clarifying things – and adhering only to the truth – requires courage, as changing the habits of thought produces fear, often hidden by anger. But by overcoming our fears we grow in strength. And we experience intense and beautiful emotions. All great adventures in life allow this, and exploring motion is one of them. Enjoy it!

Physics

English

9781447560371

https://open.umn.edu/opentextbooks/textbooks/the-adventure-of-physics-vol-iv-the-quantum-of-change

Christoph Schiller

MOTION MOUNTAIN
the adventure of physics – vol.iv
the quantum of change

www.motionmountain.net

Christoph Schiller

Motion Mountain
The Adventure of Physics
Volume IV

The Quantum of Change

Edition 29, available as free pdf
with films at www.motionmountain.net

Editio trigesima.
Proprietas scriptoris © Chrestophori Schiller
tertio anno Olympiadis trigesimae primae.
Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.

Thirtieth edition.
Copyright © 1990–2018 by Christoph Schiller,
from the third year of the 24th Olympiad
to the third year of the 31st Olympiad.

This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
creativecommons.org/licenses/by-nc-nd/3.0/de,
with the additional restriction that reproduction, distribution and use,
in whole or in part, in any product or service, be it
commercial or not, is not allowed without the written consent of
the copyright owner. The pdf file was and remains free for everybody
to read, store and print for personal use, and to distribute
electronically, but only in unmodified form and only at no charge.

To Britta, Esther and Justus Aaron

τῷ ἐμοὶ δαὶμονι

Die Menschen stärken, die Sachen klären.

PR E FAC E

Antiquity

”

Christoph Schiller
* ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motivating; both terms go back to the same Latin root.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

his book series is for anybody who is curious about motion in nature. How do
hings, people, animals, images and empty space move? The answer leads
o many adventures, and this volume presents those due to the discovery that
there is a smallest possible change value in nature. This smallest change value, the
quantum of action, leads to what is called quantum physics. In the structure of modern
physics, shown in Figure 1, quantum physics covers four of eight points. The present
volume introduces the foundations of quantum theory, explains the structure of atoms
and the appearance of probabilities, wave functions and colours.
The present introduction to quantum physics arose from a threefold aim I have pursued since 1990: to present the basics of quantum motion in a way that is simple, up to
date and captivating.
In order to be simple, the text focuses on concepts, while keeping mathematics to the
necessary minimum. Understanding the concepts of physics is given precedence over
using formulae in calculations. The whole text is within the reach of an undergraduate.
In order to be up to date, the text is enriched by the many gems – both theoretical and
empirical – that are scattered throughout the scientific literature.
In order to be captivating, the text tries to startle the reader as much as possible. Reading a book on general physics should be like going to a magic show. We watch, we are
astonished, we do not believe our eyes, we think, and finally we understand the trick.
When we look at nature, we often have the same experience. Indeed, every page presents
at least one surprise or provocation for the reader to think about. Numerous interesting
challenges are proposed.
The motto of the text, die Menschen stärken, die Sachen klären, a famous statement
on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and
adhering only to the truth – requires courage, as changing the habits of thought produces
fear, often hidden by anger. But by overcoming our fears we grow in strength. And we
experience intense and beautiful emotions. All great adventures in life allow this, and
exploring motion is one of them. Enjoy it.

Motion Mountain – The Adventure of Physics

T

“

Primum movere, deinde docere.*

8

preface

Final, unified description of motion
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).
PHYSICS:
Describing motion with precision,
i.e., using the least action principle.

Special relativity
Adventures: light,
magnetism, length
c contraction, time
limits dilation and
G
h, e, k
fast E0 = mc2
limits
limit
motion (vol. II).
uniform
tiny
motion
motion

Quantum field theory
(the ‘standard model’)
Adventures: building
accelerators, understanding quarks, stars,
bombs and the basis of
life, matter & radiation
(vol. V).

Quantum theory
Adventures: biology,
birth, love, death,
chemistry, evolution,
enjoying colours, art,
paradoxes, medicine
and high-tech business
(vol. IV and vol. V).

Galilean physics, heat and electricity
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).

F I G U R E 1 A complete map of physics, the science of motion, as first proposed by Matvei Bronshtein

(b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion,
and shows the connections to the fields of modern physics. Each connection increases the precision of
the description and is due to a limit to motion that is taken into account. The limits are given for
uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny
motion by the Planck constant h, the elementary charge e and the Boltzmann constant k.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Classical gravity
Adventures:
climbing, skiing,
space travel,
the wonders of
astronomy and
geology (vol. I).

Quantum theory
with classical gravity
Adventures: bouncing
neutrons, understanding tree
growth (vol. V).

Motion Mountain – The Adventure of Physics

General relativity
Adventures: the
night sky, measuring curved and
wobbling space,
exploring black
holes and the
universe, space
and time (vol. II).

An arrow indicates an
increase in precision by
adding a motion limit.

preface

9

Using this b o ok
Marginal notes refer to bibliographic references, to other pages or to challenge solutions.
In the colour edition, marginal notes, pointers to footnotes and links to websites are
typeset in green. Over time, links on the internet tend to disappear. Most links can be
recovered via www.archive.org, which keeps a copy of old internet pages. In the free
pdf edition of this book, available at www.motionmountain.net, all green pointers and
links are clickable. The pdf edition also contains all films; they can be watched directly
in Adobe Reader.
Solutions and hints for challenges are given in the appendix. Challenges are classified
as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for
which no solution has yet been included in the book are marked (ny).
Advice for learners

A teacher likes pupils and likes to lead them into exploring the field he or she chose. His
or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a
lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture,
feel and tell yourself how you will lead each of your pupils into enjoying that topic as
much as you do. Do this exercise consciously, every day. You will minimize trouble in
your class and maximize your teaching success.
This book is not written with exams in mind; it is written to make teachers and students understand and enjoy physics, the science of motion.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Advice for teachers

Motion Mountain – The Adventure of Physics

Learning allows us to discover what kind of person we can be. Learning widens knowledge, improves intelligence and provides a sense of achievement. Therefore, learning
from a book, especially one about nature, should be efficient and enjoyable. Avoid bad
learning methods like the plague! Do not use a marker or a pencil to highlight or underline text on paper. It is a waste of time, provides false comfort and makes the text
unreadable. And do not learn from a screen. In particular, never, ever, learn from the internet, from videos, from games or from a smartphone. Most of the internet, almost all
videos and all games are poisons and drugs for the brain. Smartphones are dispensers of
drugs that make people addicted and prevent learning. Nobody putting marks on paper
or looking at a screen is learning efficiently or is enjoying doing so.
In my experience as a pupil and teacher, one learning method never failed to transform unsuccessful pupils into successful ones: if you read a text for study, summarize
every section you read, in your own words and images, aloud. If you are unable to do
so, read the section again. Repeat this until you can clearly summarize what you read in
your own words and images, aloud. And enjoy the telling aloud! You can do this alone
or with friends, in a room or while walking. If you do this with everything you read, you
will reduce your learning and reading time significantly; you will enjoy learning from
good texts much more and hate bad texts much less. Masters of the method can use it
even while listening to a lecture, in a low voice, thus avoiding to ever take notes.

10

preface

Feedback
The latest pdf edition of this text is and will remain free to download from the internet.
I would be delighted to receive an email from you at fb@motionmountain.net, especially
on the following issues:
Challenge 1 s

— What was unclear and should be improved?
— What story, topic, riddle, picture or film did you miss?

Support

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Your donation to the charitable, tax-exempt non-profit organisation that produces, translates and publishes this book series is welcome. For details, see the web page www.
motionmountain.net/donation.html. The German tax office checks the proper use of
your donation. If you want, your name will be included in the sponsor list. Thank you in
advance for your help, on behalf of all readers across the world.
The paper edition of this book is available, either in colour or in black and white, from
www.amazon.com or www.createspace.com. And now, enjoy the reading.

Motion Mountain – The Adventure of Physics

Also help on the specific points listed on the www.motionmountain.net/help.html web
page is welcome. All feedback will be used to improve the next edition. You are welcome
to send feedback by mail or by sending in a pdf with added yellow notes, to provide
illustrations or photographs, or to contribute to the errata wiki on the website. If you
would like to translate a chapter of the book in your language, please let me know.
On behalf of all readers, thank you in advance for your input. For a particularly useful
contribution you will be mentioned – if you want – in the acknowledgements, receive a
reward, or both.

Contents
Minimum action – quantum theory for poets
The effects of the quantum of action on rest 18 • The consequences of the quantum
of action for objects 20 • Why ‘quantum’? 22 • The effect of the quantum of action
on motion 24 • The surprises of the quantum of action 26 • Transformation, life
and Democritus 28 • Randomness – a consequence of the quantum of action 31 •
Waves – a consequence of the quantum of action 33 • Particles – a consequence of
the quantum of action 34 • Quantum information 35 • Curiosities and fun challenges about the quantum of action 36 • The dangers of buying a can of beans 37
• A summary: quantum physics, the law and indoctrination 39

40

2

Light – the strange consequences of the quantum of action
How do faint lamps behave? 40 • Photons 44 • What is light? 46
• The size
of photons 47 • Are photons countable? – Squeezed light 47
• The positions
of photons 51 • Are photons necessary? 54 • Interference: how can a wave be
made up of particles? 56
• Interference of a single photon 59 • Reflection and
diffraction deduced from photon arrows 60 • Refraction and partial reflection
from photon arrows 62 • From photons to waves 63 • Can light move faster than
light? – Real and virtual photons 64 • Indeterminacy of electric fields 65 • How
can virtual photon exchange lead to attraction? 65 • Can two photons interfere? 66
• Curiosities and fun challenges about photons 67 • A summary on light: particle
and wave 69

72

3

Motion of matter – beyond classical physics
Wine glasses, pencils and atoms – no rest 72 • No infinite measurement precision 73 • Cool gas 73 • Flows and the quantization of matter 74
• Fluid flows
and quantons 74 • Knocking tables and quantized conductivity 74 • Matter quantons and their motion – matter waves 76 • Mass and acceleration of quantons 79
• Why are atoms not flat? Why do shapes exist? 79 • Rotation, quantization of
angular momentum, and the lack of north poles 81
• Rotation of quantons 82
• Silver, Stern and Gerlach – polarization of quantons 83 • Curiosities and fun
challenges about quantum matter 85 • First summary on the motion of quantum
particles 86

87

4

The quantum description of matter and its motion
States and measurements – the wave function 87 • Visualizing the wave function:
rotating arrows and probability clouds 89 • The state evolution – the Schrödinger
equation 91 • Self-interference of quantons 93 • The speed of quantons 94 • Dispersion of quantons 94 • Tunnelling and limits on memory – damping of quantons 95 • The quantum phase 97 • Can two electron beams interfere? Are there
coherent electron beams? 101 • The least action principle in quantum physics 102
• The motion of quantons with spin 104 • Relativistic wave equations 105 •
Bound motion, or composite vs. elementary quantons 107 • Curiosities and fun
challenges about quantum motion of matter 109
• A summary on motion of
matter quantons 111

112

5

Permutation of particles – are particles like gloves?
Distinguishing macroscopic objects 112 • Distinguishing atoms 113 • Why does
indistinguishability appear in nature? 115 • Can quantum particles be counted? 115

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

1

Motion Mountain – The Adventure of Physics

15

12

contents
• What is permutation symmetry? 116 • Indistinguishability and wave function
symmetry 117 • The behaviour of photons 118 • Bunching and antibunching 120
• The energy dependence of permutation symmetry 120 • Indistinguishability in
quantum field theory 121 • How accurately is permutation symmetry verified? 122
• Copies, clones and gloves 122 • Summary 124
Rotations and statistics – visualizing spin
Quantum particles and symmetry 125 • Types of quantum particles 127 • Spin
1/2 and tethered objects 130 • The extension of the belt trick 133 • Angels, Pauli’s
exclusion principle and the hardness of matter 135 • Is spin a rotation about an
axis? 137 • Rotation requires antiparticles 138 • Why is fencing with laser beams
impossible? 139 • Spin, statistics and composition 140 • The size and density of
matter 141 • A summary on spin and indistinguishability 141 • Limits and open
questions of quantum statistics 142

143

7

Superpositions and probabilities – quantum theory without
ideology
Why are people either dead or alive? 144 • Macroscopic superpositions, coherence and incoherence 144 • Decoherence is due to baths 146 • How baths lead to
decoherence – scattering 146 • How baths lead to decoherence – relaxation 148
• Summary on decoherence, life and death 150 • What is a system? What is an
object? 151 • Entanglement 151 • Is quantum theory non-local? A bit about the
Einstein–Podolsky–Rosen paradox 152 • Curiosities and fun challenges about
superpositions 155 • Why do probabilities and wave function collapse appear in
measurements? 157 • Why is ℏ necessary for probabilities? 162 • Hidden variables 163 • Summary on probabilities and determinism 165 • What is the difference between space and time? 167 • Are we good observers? 168 • What relates
information theory, cryptology and quantum theory? 168 • Is the universe a computer? 169 • Does the universe have a wave function? And initial conditions? 169

171

8

Colours and other interactions between light and matter
The causes of colour 171 • Using the rainbow to determine what stars are made
of 180 • What determines the colours of atoms? 181 • The shape of atoms 185 •
The size of atoms 186 • Relativistic hydrogen 188 • Relativistic wave equations –
again 189 • Getting a first feeling for the Dirac equation 191 • Antimatter 192 •
Virtual particles 193 • Curiosities and fun challenges about colour and atoms 194 •
Material properties 196 • A tough challenge: the strength of electromagnetism 196
• A summary on colours and materials 197

198

9

Quantum physics in a nutshell
Physical results of quantum theory 198 • Results on the motion of quantum
particles 199 • Achievements in accuracy and precision 201 • Is quantum theory
magic? 203 • Quantum theory is exact, but can do more 203

205

a Units, measurements and constants
SI units 205 • The meaning of measurement 208 • Planck’s natural units 208 •
Other unit systems 210 • Curiosities and fun challenges about units 211 • Precision and accuracy of measurements 212 • Limits to precision 214 • Physical
constants 214 • Useful numbers 222

223

b Numbers and vector spaces
Numbers as mathematical structures 223 • Complex numbers 225 • Quaternions 227 • Octonions 233 • Other types of numbers 234 • From vector spaces

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

6

Motion Mountain – The Adventure of Physics

125

to Hilbert spaces 235 • Mathematical curiosities and fun challenges 238
239

Challenge hints and solutions

247

Bibliography

263

Credits
Acknowledgements 263 • Film credits 264 • Image credits 264

267

Name index

274

Subject index

The Q uantum of Change

In our quest to understand how things move,
we discover that there is a smallest change value in nature,
implying that motion is fuzzy,
that boxes are never tight,
that matter is composed of elementary units,
and that light and interactions are streams of particles.
The smallest change value explains why antimatter exists,
why particles are unlike gloves,
why copying machines do not exist,
why probabilities are reasonable,
and how all colours in nature are formed.

Chapter 1

M I N I M UM AC T ION – QUA N T UM
T H E ORY F OR P OET S

”

limbing Motion Mountain up to this point, we completed three legs. We
ame across Galileo’s mechanics (the description of motion for kids), then
ontinued with Einstein’s relativity (the description of motion for science-fiction
enthusiasts), and finally explored Maxwell’s electrodynamics (the description of motion for business people). These three classical descriptions of motion are impressive,
beautiful and useful. However, they have a small problem: they are wrong. The reason is
simple: none of them describes life.
Whenever we observe a flower or a butterfly, such as those of Figure 2, we enjoy the
bright colours, the motion, the wild smell, the soft and delicate shape or the fine details
of their symmetries. However, we know:

Now, flowers and animals – but also many non-living systems – have characteristic sizes,
size ranges and proportions; and they have characteristic rhythms. And indeed, classical
physics cannot explain their origin, because
⊳ The classical constants of nature – the gravitational constant ?, the ideal gas
constant ?, the speed of light ?, the vacuum permittivity ?0 and the vacuum
permeability ?0 – do not allow defining length or time scales.
In fact, the classical constants do not even allow us to measure speed or force values, even
though these measurements are fractions of ? and ?4 /?; because in order to measure
fractions, we need to define them first; however, defining fractions also requires length
or time scales, which classical physics does not allow.
Without measurements, there are also no emotions! Indeed, our emotions are
triggered by our senses. And all the impressions and all the information that our senses
provide us are – among others – measurements. Since classical physics does not provide
measurement scales, we know:
Ref. 1

** ‘Nature [in its workings] makes no jumps.’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ Classical physics cannot explain any characteristic length or time scale observed in nature.

Motion Mountain – The Adventure of Physics

C

“

Natura [in operationibus suis] non facit
saltus.**
15th century

16

1 minimum action – quantum theory for poets

The reason for all these limitations is the following connection:
⊳ Classical physics alone cannot be used to build any measurement device.

Challenge 2 s

Every sense contains measurement devices. And every measurement device, like any pattern or rhythm, needs an internal scale, or, more generally, an internal measurement unit.
Because classical physics does not provide any scale, classical physics does not explain
how measurement devices work, not how senses work, and not how emotions appear.
To understand emotions and life, we need to go beyond classical physics. Take any example of a pleasant situation,* such as a beautiful evening sky, a waterfall, a happy child
or a caress. Classical physics is not able to explain any aspect of the situation: First, the
colours and their origin remain mysterious. Secondly, all shapes, sizes and proportions
remain mysterious. Thirdly, the timing and the duration of the involved processes cannot be understood. Fourthly, all the sensations and emotions produced by the situation
remain mysterious. To understand and explain these aspects, we need quantum theory.

* The photograph on page 14 shows a female glow worm, Lampyris noctiluca, as commonly found in the
United Kingdom (© John Tyler, www.johntyler.co.uk/gwfacts.htm).

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ Classical physics does not allow understanding senses or emotions.

Motion Mountain – The Adventure of Physics

F I G U R E 2 Examples of quantum machines (© Linda de Volder).

1 minimum action – quantum theory for poets

Vol. I, page 410

Vol. III, page 230
Vol. III, page 246

Ref. 2

In fact, we will find out that both life and every type of pleasure are examples of quantum
motion. Emotions are quantum processes.
In the early days of physics, the impossibility to describe life and pleasure was not
seen as a shortcoming, because neither senses nor material properties nor scales were
thought to be related to motion. And pleasure was not considered a serious subject of
investigation for a respectable researcher anyway. Today, the situation is different. In our
adventure we have learned that our senses of time, hearing, touch, smell and sight are
primarily detectors of motion. Without motion, there would be no senses. Furthermore,
all detectors are made of matter. During the exploration on electromagnetism we began
to understand that all properties of matter are due to motions of charged constituents.
Density, stiffness, colour and all other material properties result from the electromagnetic behaviour of the Lego bricks of matter: namely, the molecules, the atoms and the
electrons. Thus, the properties of matter are also consequences of motion. Moreover,
we saw that these tiny constituents are not correctly described by classical electrodynamics. We even found that light itself does not behave classically. Therefore the inability
of classical physics to describe matter, light and the senses is indeed due to its intrinsic
limitations.
In fact, every failure of classical physics can be traced back to a single, fundamental
discovery made in 1899 by Max Planck:*
⊳ In nature, action values smaller than ℏ = 1.06 ⋅ 10−34 Js are not observed.

Ref. 3, Ref. 4

* Max Planck (1858–1947), professor of physics in Berlin, was a central figure in thermostatics and modern physics. He discovered and named the Boltzmann constant ? and the quantum of action ℎ, often called
Planck’s constant. His introduction of the quantum hypothesis gave birth to quantum theory. He also made
the works of Einstein known in the physical community, and later organized a job for him in Berlin. He
received the Nobel Prize for physics in 1918. He was an important figure in the German scientific establishment; he also was one of the very few who had the courage to tell Adolf Hitler face to face that it was
a bad idea to fire Jewish professors. (He got an outburst of anger as answer.) Famously modest, with many
tragedies in his personal life, he was esteemed by everybody who knew him.
** In fact, this story is a slight simplification: the constant originally introduced by Planck was the (unreduced) constant ℎ = 2πℏ. The factor 2π leading to the final quantum principle was added somewhat later,
by other researchers.
This somewhat unconventional, but didactically useful, approach to quantum theory is due to Niels Bohr.
Nowadays, it is hardly ever encountered in the literature, despite its simplicity.
Niels Bohr (b. 1885 Copenhagen, d. 1962 Copenhagen) was one of the great figures of modern physics.
A daring thinker and a polite man, he made Copenhagen University into the new centre of development of
quantum theory, overshadowing Göttingen. He developed the description of the atom in terms of quantum
theory, for which he received the 1922 Nobel Prize in Physics. He had to flee Denmark in 1943 after the
German invasion, because of his Jewish background, but returned there after the war, continuing to attract
the best physicists across the world.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 3 s

All attempts to observe physical action values smaller than this fail.** In other words, in
nature – as in a good cinema film – there is always some action. The existence of a smallest action value – the so-called quantum principle – is in complete contrast with classical physics. (Why?) Despite this contrast, the quantum principle has passed an enormous number of experimental tests, many of which we will encounter in this part of our
mountain ascent. Above all, the quantum principle has never failed even a single test.

Motion Mountain – The Adventure of Physics

Vol. III, page 150

17

18

1 minimum action – quantum theory for poets

F I G U R E 3 Max Planck (1858–1947)

F I G U R E 4 Niels Bohr

(1885–1962)

⊳ In nature, a change smaller than ℏ = 1.06 ⋅ 10−34 Js cannot be observed.
Therefore, a smallest action implies that there is a smallest change value in nature. If we
compare two observations, there will always be change between them. Thus the quantum
of action would perhaps be better named the quantum of change.
Can a minimum change really exist in nature? To accept the idea, we need to explore
three points, detailed in Table 1. We need to show that a smaller change is never observed
in nature, show that smaller change values can never be observed, and finally, show that
all consequences of this smallest change, however weird they may be, apply to nature. In
fact, this exploration constitutes all of quantum physics. Therefore, these checks are all
we do in the remaining of this part of our adventure. But before we explore some of the
experiments that confirm the existence of a smallest change, we directly present some of
its more surprising consequences.
The effects of the quantum of action on rest
Since action is a measure of change, a minimum observable action means that two successive observations of the same system always differ by at least ℏ. In every system, there
is always something happening. As a consequence we find:

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 247

Motion Mountain – The Adventure of Physics

Vol. III, page 150

The fundamental constant ℏ, which is pronounced ‘aitch-bar’, is called the quantum of
action, or alternatively Planck’s constant. Planck discovered the quantum principle when
studying the properties of incandescent light, i.e., of light emanating from hot bodies. But
the quantum principle also applies to motion of matter, and even, as we will see later, to
motion of empty space, such as gravitational waves.
The quantum principle states that no experiment can measure an action smaller than
ℏ. For a long time, Einstein tried to devise experiments to overcome this limit. But he
failed in all his attempts: nature does not allow it, as Bohr showed again and again. The
same occurred to many other researchers.
We recall that in physics – as in the theatre – action is a measure for the change occurring in a system. The quantum principle can thus rephrased as

1 minimum action – quantum theory for poets

19

TA B L E 1 How to convince yourself and others that there is a smallest
action, or smallest change ℏ in nature. Compare this table with the two
tables in volume II, that about maximum speed on page 26, and that
about maximum force on page 109.

S tat e m e n t

Te s t

The smallest action value ℏ is
observer-invariant.
Local change or action values < ℏ
are not observed.

Check all observations.

Local change or action values < ℏ
cannot be produced.
Local change or action values < ℏ
cannot even be imagined.

Check all attempts.

The smallest local change or action
value ℏ is a principle of nature.

Deduce quantum theory
from it.
Show that all consequences,
however weird, are
confirmed by observation.

Check all observations.

Solve all paradoxes.

Page 15

⊳ There is motion inside matter.
Since there is a minimum action for all observers, and since there is no rest, we deduce:
⊳ In nature there is no perfectly straight or perfectly uniform motion.

Challenge 4 s

Forget all you have learnt so far: Inertial motion is an approximation! An object can
move in straight, uniform motion only approximately, and only when observed over long
distances or long times. We will see later that the more massive the object is, the better
the approximation is. (Can you confirm this?) So macroscopic observers can still talk
about space-time symmetries; and special relativity can thus be reconciled with quantum
theory.
Also free fall, or motion along a geodesic, exists only as a long-time average. So
general relativity, which is based on the existence of freely-falling observers, cannot be
* ‘Nature makes jumps.’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Everything moves, all the time, at least a little bit. Natura facit saltus.* True, these jumps
are tiny, as ℏ is too small to be observable by any of our senses. Nevertheless, rest can
be observed only macroscopically, and only as a long-time or many-particle average. For
example, the quantum of action implies that in a mountain – an archetypal ‘system at
rest’ – all the atoms and electrons are continually buzzing around. In short,

Motion Mountain – The Adventure of Physics

⊳ In nature there is no rest.

20

1 minimum action – quantum theory for poets

correct when actions of the order of ℏ are involved. Indeed, the reconciliation of the
quantum principle with general relativity – and thus with curved space – is a big challenge. (The solution is simple only for weak, everyday fields.) The issues involved are so
mind-shattering that they form a separate, final, part of this adventure. We thus explore
situations without gravity first.
The consequences of the quantum of action for objects

Vol. I, page 437

⊳ All measurements are quantum effects.

⊳ In nature, all times and all frequencies are due to the quantum of action.
All processes that take time are quantum processes. If you prefer, waiting is a quantum
effect! In particular, without the quantum of action, oscillations and waves could not
exist:
⊳ Every colour is a quantum effect.
But this is not all. Planck also realized that the quantum of action allows us to understand
the size of all things.
⊳ Every size is a quantum effect.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 5

When Planck saw that the quantum of action allowed defining all units in nature, he was
as happy as a child; he knew straight away that he had made a fundamental discovery,
even though (in 1899) quantum theory did not yet exist. He even told his seven-year-old
son Erwin about it, while walking with him through the woods around Berlin. Planck
explained to his son that he had made a discovery as important as universal gravity.
Indeed, Planck knew that he had found the key to understanding many of the effects
that were then unexplained.

Motion Mountain – The Adventure of Physics

Challenge 5 s

Have you ever wondered why leaves are green? You probably know that they are green
because they absorb blue (short-wavelength) and red (long-wavelength) light, while allowing green (medium-wavelength) light to be reflected. How can a system filter out the
small and the large, and let the middle pass through? To do so, leaves must somehow
measure the frequency. But we have seen that classical physics does not allow measurement of time (or length) intervals, as any measurement requires a measurement unit,
and classical physics does not allow such units to be defined. On the other hand, it
takes only a few lines to confirm that with the help of the quantum of action ℏ (and
the Boltzmann constant ?, both of which Planck discovered), fundamental units for all
measurable quantities can be defined, including time and therefore frequency. (Can you
find a combination of the speed of light ?, the gravitational constant ? and the quantum
of action ℏ that gives a time? It will only take a few minutes.)
In short, measurements are only possible at all because of the existence of the quantum
of action.

1 minimum action – quantum theory for poets

21

O

H
H
F I G U R E 5 An artist’s impression of a water molecule made of

two hydrogen (H) and one oxygen (O) atom.

Challenge 6 e

⊳ Atom sizes are quantum effects.

Page 186

At the time, Haas was widely ridiculed.** Nowadays, his formula for the size of atoms is
found in all textbooks, including this one. In determining the size of atoms, the quantum
of action has another important consequence:
⊳ Gulliver’s travels are impossible.

Challenge 9 s

Challenge 7 s

There are no tiny people and no giant ones. Classically, nothing speaks against the idea;
but the quantum of action prevents it. Can you supply the detailed argument?
* In fact, it is also possible to define all measurement units in terms of the speed of light ?, the gravitational
constant ? and the electron charge ?. Why is this not fully satisfactory?
** Before the discovery of ℏ, the only simple length scale for the electron was the combination
?2 /(4π?0 ?e ?2 ) ≈ 3 fm; this is ten thousand times smaller than an atom. We stress that any length scale
containing ? is a quantum effect, and not a classical length scale, because ? is the quantum of electric charge.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 8 s

Motion Mountain – The Adventure of Physics

Vol. I, page 337

Can you find the combination of ?, ? and ℏ that yields a length? With the quantum of
action, it was finally possible to determine the maximum size of mountains, of trees and
of humans. Planck knew that the quantum of action confirmed what Galileo had already
deduced long before him: that sizes are due to fundamental, smallest scales in nature.
Max Planck also understood that the quantum of action ℏ was the last missing constant of nature. With ℏ, it becomes possible to define a natural unit for every observable
property in nature. Together, ?, ? and ℏ allow to define units that are independent of
culture or civilization – even extraterrestrials would understand them.* In short, ℏ allows understanding all observables. Therefore, with ℏ it is possible to draw the diagram
shown in Figure 1 that encompasses all motion in nature, and thus all of physics.
In our environment, the size of all objects is related and due to the size of atoms. In
turn, the size of atoms is a direct consequence of the quantum of action. Can you derive
an approximation for the size of atoms, knowing that it is given by the motion of electrons
of mass ?e and charge ?, constrained by the quantum of action? This connection, a simple
formula, was discovered in 1910 by Arthur Erich Haas, 15 years before quantum theory
was formulated.

22

1 minimum action – quantum theory for poets

F I G U R E 6 Max Born (1882 –1970)

All shapes in everyday life are due to molecular shapes, or to their generalizations.
The mass of an object is also a consequence of the quantum of action, as we will see
later on. Since all material properties – such as density, colour, stiffness or polarizability
– are defined as combinations of length, time and mass units, we find:
⊳ All material properties arise from the quantum of action.
In short, the quantum of action determines the size, shape, colour, mass, and all other
properties of objects, from stones to whipped cream.
Why ‘ quantum ’ ?
Quantum effects surround us on all sides. However, since the quantum of action is so
small, its effects on motion appear mostly, but not exclusively, in microscopic systems.
The study of such systems was called quantum mechanics by Max Born, one of the major
contributors to the field.* Later, the term quantum theory became more popular.
* Max Born (b. 1882 Breslau, d. 1970 Göttingen) first studied mathematics, then turned to physics. A professor at Göttingen University, he made the city one of the world centres of physics. He developed quantum
mechanics with his assistants Werner Heisenberg and Pascual Jordan, and then applied it to scattering,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ All shapes are quantum effects.

Motion Mountain – The Adventure of Physics

But if rest does not exist, how can shapes exist? Any shape of everyday life, including
that of a flower, is the result of body parts remaining at rest with respect to each other.
Now, all shapes result from interactions between the constituents of matter, as shown
most clearly in the shapes of molecules. But how can a molecule, such as the water molecule H2 O, shown in Figure 5, have a shape? In fact, a molecule does not have a fixed
shape, but its shape fluctuates, as would be expected from the quantum of action. Despite the fluctuations, every molecule does have an average shape, because different angles
and distances correspond to different energies. Again, these average length and angle values only exist because the quantum of action yields fundamental length scales in nature.
Without the quantum of action, there would be no shapes in nature.

1 minimum action – quantum theory for poets

23

TA B L E 2 Some small systems in motion and the observed action values for their changes.

System and change

1ℏ
2ℏ
c. 5 ℏ
1ℏ
c. 3 ℏ
c. 1017 ℏ

quantum
quantum
quantum
quantum
quantum
classical

c. 1–2 ℏ
c. 1–2 ℏ
c. 2–6 ℏ
c. 5 ℏ
c. 1038 ℏ

quantum
quantum
quantum
quantum
classical

c. 1–2 ℏ
c. 1035 ℏ
1ℏ
1ℏ
1ℏ

quantum
classical
quantum
quantum
quantum

1ℏ
c. 1 − 5 ℏ
c. 1 − 5 ℏ
c. 2 − 6 ℏ
c. 300 ℏ
c. 10 ℏ
c. 1037 ℏ

quantum
quantum
quantum
quantum
quantum
quantum
classical

c. 2 ℏ

quantum

Challenge 10 ny

c. 100 ℏ
c. 1014 ℏ
c. 5 ℏ
c. 1015 ℏ
c. 1019 ℏ
c. 1024 ℏ
c. 2 ⋅ 1036 ℏ

quantum
classical
quantum
classical
classical
classical
classical

c. 1 − 5 ℏ
c. 1080 ℏ

quantum
classical

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Motion

Motion Mountain – The Adventure of Physics

Light
Smallest amount of light absorbed by a coloured surface
Smallest impact when light reflects from mirror
Smallest consciously visible amount of light
Smallest amount of light absorbed in flower petal
Blackening of photographic film
Photographic flash
Electricity
Electron ejected from atom or molecule
Electron extracted from metal
Electron motion inside microprocessor
Signal transport in nerves, from one molecule to the next
Current flow in lightning bolt
Materials
Tearing apart two neighbouring iron atoms
Breaking a steel bar
Basic process in superconductivity
Basic process in transistors
Basic magnetization process
Chemistry
Atom collision in liquid at room temperature
Shape oscillation of water molecule
Shape change of molecule, e.g. in chemical reaction
Single chemical reaction curling a hair
Tearing apart two mozzarella molecules
Smelling one molecule
Burning fuel in a cylinder in an average car engine explosion
Life
Air molecule hitting eardrum
Smallest sound signal detectable by the ear
Single DNA duplication step during cell division
Ovule fertilization
Smallest step in molecular motor
Sperm motion by one cell length
Cell division
Fruit fly’s wing beat
Person walking one body length
Nuclei and stars
Nuclear fusion reaction in star
Explosion of gamma-ray burster

Action

24

Vol. I, page 334

1 minimum action – quantum theory for poets

There is another way to characterize the difference between a microscopic, or quantum,
system and a macroscopic, or classical, one. A smallest action implies that the difference
between the action values ? of two successive observations of the same system, a time Δ?
apart, cannot vanish. We have
|?(? + Δ?) − ?(?)| = |(? ± Δ?)(? + Δ?) − ??| = |?Δ? ± ?Δ? ± Δ?Δ?| ⩾

ℏ
.
2

(1)

The factor 1/2 arises because a smallest action ℏ automatically implies an action indeterminacy of half its value. Now the values of the energy ? and time ? – but not of (the
positive) Δ? or Δ? – can be set to zero if we choose a suitable observer. Thus, the existence of a quantum of action implies that in any system the evolution is constrained

Ref. 6

solid-state physics, optics and liquids. He was the first to understand that the wave function, or state function, describes a probability amplitude. Later, Born and Emil Wolf wrote what is still the main textbook on
optics. Many of Born’s books were classics and read all over the world.
Born attracted to Göttingen the most brilliant talents of the time, receiving as visitors Hund, Pauli, Nordheim, Oppenheimer, Goeppert-Mayer, Condon, Pauling, Fock, Frenkel, Tamm, Dirac, Mott, Klein, Heitler,
London, von Neumann, Teller, Wigner, and dozens of others. Being Jewish, Born lost his job in 1933, when
criminals took over the German government. He emigrated, and became professor in Edinburgh, where he
stayed for 20 years. Physics at Göttingen never recovered from this loss. For his elucidation of the meaning
of the wave function he received the 1954 Nobel Prize in Physics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The effect of the quantum of action on motion

Motion Mountain – The Adventure of Physics

Quantum theory arises from the existence of smallest measurable values in nature,
generalizing the idea that Galileo had in the seventeenth century. As discussed in detail earlier on, it was Galileo’s insistence on ‘piccolissimi quanti’ – smallest quanta –
of matter that got him into trouble. We will soon discover that the idea of a smallest
change is necessary for a precise and accurate description of matter and of nature as a
whole. Therefore Born adopted Galileo’s term for the new branch of physics and called it
‘Quantentheorie’ or ‘theory of quanta’. The English language adopted the Latin singular
‘quantum’ instead of the plural used in most other languages.
Note that the term ‘quantum’ does not imply that all measurement values are multiples
of a smallest one: this is so only in a few cases.
Quantum theory is the description of microscopic motion. Quantum theory is necessary whenever a process produces an action value of the order of the quantum of action.
Table 2 shows that all processes on atomic and molecular scales, including biological
and chemical processes, are quantum processes. So are processes of light emission and
absorption. These phenomena can only be described with quantum theory.
Table 2 also shows that the term ‘microscopic’ has a different meaning for a physicist
and for a biologist. For a biologist, a system is ‘microscopic’ if it requires a microscope
for its observation. For a physicist, a system is microscopic if its characteristic action is of
the order of the quantum of action. In other words, for a physicist a system is usually microscopic if it is not even visible in a (light) microscope. To increase the confusion, some
quantum physicists nowadays call their own class of microscopic systems ‘mesoscopic’,
while others call their systems ‘nanoscopic’. Both terms were introduced only to attract
attention and funding: they are conceptually useless.

1 minimum action – quantum theory for poets

25

F I G U R E 7 Werner Heisenberg (1901–1976)

by

Challenge 11 e

ℏ
,
2

(2)

where ? is the energy of the system and ? is its age, so that Δ? is the change of energy
and Δ? is the time between two successive observations.
By a similar reasoning, we find that for any physical system the position and momentum are constrained by
ℏ
(3)
Δ?Δ? ⩾ ,
2

Ref. 7

* It is often said that the indeterminacy relation for energy and time has a different weight from that for
momentum and position. This is a wrong idea, propagated by the older generation of physicists, which has
survived through many textbooks for over 70 years. Just forget it. It is essential to remember that all four
quantities appearing in the inequalities describe the internal properties of the system. In particular, ? is a
time variable deduced from changes observed inside the system, and not the time coordinate measured by
an outside clock; similarly, the position ? is not the external space coordinate, but the position characterizing the system.
Werner Heisenberg (1901–1976) was an important theoretical physicist and an excellent table-tennis
and tennis player. In 1925, as a young man, he developed, with some help from Max Born and Pascual
Jordan, the first version of quantum theory; from it he deduced the indeterminacy relations. For these
achievements he received the Nobel Prize in Physics in 1932. He also worked on nuclear physics and on
turbulence. During the Second World War, he worked on the nuclear-fission programme. After the war, he
published several successful books on philosophical questions in physics, slowly turned into a crank, and
tried unsuccessfully – with some half-hearted help from Wolfgang Pauli – to find a unified description of
nature based on quantum theory, the ‘world formula’.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

where Δ? is the indeterminacy in position and Δ? is the indeterminacy in momentum.
These two famous relations were called indeterminacy relations by their discoverer,
Werner Heisenberg.* In English they are often called ‘uncertainty relations’; however,
this term is incorrect. The quantities are not uncertain, but undetermined. Because of the
quantum of action, system observables have no definite value. There is no way to ascribe
a precise value to momentum, position, or any other observable of a quantum system.
We will use the term ‘indeterminacy relation’ throughout. The habit to call the relation
a ‘principle’ is even more mistaken.
Any system whose indeterminacy is of the order of ℏ is a quantum system; if the

Motion Mountain – The Adventure of Physics

Δ?Δ? ⩾

26

1 minimum action – quantum theory for poets

indeterminacy product is much larger, the system is classical, and then classical physics
is sufficient for its description. So even though classical physics assumes that there are no
measurement indeterminacies in nature, a system is classical only if its indeterminacies
are large compared to the minimum possible ones!
In other terms, quantum theory is necessary whenever we try to measure some quantity as precisely as possible. In fact, every measurement is itself a quantum process. And
the indeterminacy relation implies that measurement precision is limited. The quantum
of action shows that
⊳ Motion cannot be observed to infinite precision.

The surprises of the quantum of action

⊳ Perfect clocks do not exist.
The deep implications of this statement will become clear step by step.
It is also impossible to avoid that an object makes small displacement sideways. In
fact, quantum theory implies that, strictly speaking,
⊳ Neither uniform nor one-dimensional motion exists.
Also this statement harbours many additional surprises.
Quantum limitations apply also to metre rules. It is impossible to ensure that the rule
is completely at rest with respect to the object being measured. Thus the quantum of
action implies again, on the one hand, that measurements are possible, and on the other
hand:

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 237

The quantum of action ℏ implies a fuzziness of all motion. This fuzziness also implies
the existence of short-time deviations from energy, momentum and angular-momentum
conservation in microscopic systems. For general assurance it must be stressed that for
long observation times – surely for all times longer than a microsecond – conservation
holds. But in the first part of our adventure, we realized that any type of non-conservation
implies the existence of surprises in nature. Well, here are some of them.
Since precisely uniform motion does not exist, a system moving in one dimension
only – such as the hand of a clock – always has the possibility of moving a bit in the
opposite direction, thus leading to incorrect readings. Indeed, quantum theory predicts
that clocks have essential limitations:

Motion Mountain – The Adventure of Physics

In other words, the microscopic world is fuzzy. This fact has many important consequences and many strange ones. For example, if motion cannot be observed with infinite precision, the very concept of motion needs to be handled with great care, as it cannot
be applied in certain situations. In a sense, the rest of our quest is just an exploration of
the implications of this result.
In fact, as long as space-time is flat, it turns out that we can retain the concept of
motion to describe observations, provided we remain aware of the limitations implied
by the quantum principle.

1 minimum action – quantum theory for poets

27

⊳ Measurement accuracy is limited.
It also follows from the quantum of action that any inertial or freely-falling observer
must be large, as only large systems approximate inertial motion.
⊳ An observer cannot be microscopic.
If humans were not macroscopic, they could neither observe nor study motion.
Because of the finite accuracy with which microscopic motion can be observed, we
discover that
⊳ Faster-than-light motion is possible in the microscopic domain.

⊳ Motion backwards in time is possible over microscopic times and distances.

Challenge 12 s

⊳ The quantum of action implies that there is no permanence in nature.
The quantum of action also implies:
⊳ The vacuum is not empty.

Challenge 13 e

If we look at empty space twice, the two observations being separated by a tiny time interval, some energy will be observed the second time. If the time interval is short enough,
the quantum of action will lead to the observation of radiation or matter particles. Indeed, particles can appear anywhere from nowhere, and disappear just afterwards: the
action limit requires it. In summary, nature exhibits short-term appearance and disappearance of matter and radiation. In other words, the classical idea of an empty vacuum
is correct only when the vacuum is observed over a long time.
The quantum of action implies that compass needles cannot work. If we look twice in
quick succession at a compass needle, or even at a house, we usually observe that it stays
oriented in the same direction. But since physical action has the same dimensions as
angular momentum, a minimum value for action implies a minimum value for angular
momentum. Even a macroscopic object has a minimum value for its rotation. In other
words, quantum theory predicts

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In short, a quantum of action implies the existence of microscopic time travel. However,
this remains impossible in the macroscopic domain, such as everyday life.
But there is more. Imagine a moving car suddenly disappearing for good. In such
a situation, neither momentum nor energy would be conserved. The action change for
such a disappearance is large compared to ℏ, so that its observation would contradict
even classical physics – as you may wish to check. However, the quantum of action allows a microscopic particle, such as an electron, to disappear for a short time, provided it
reappears afterwards.

Motion Mountain – The Adventure of Physics

Quantum theory thus predicts tachyons, at least over short time intervals. For the same
reason,

28

1 minimum action – quantum theory for poets

E
m

p
Δx

0

F I G U R E 8 Hills are never high

enough.

⊳ Everything rotates.

Transformation, life and Demo critus
At the beginning of our adventure, we mentioned that the Greeks distinguished three
types of changes: transport, growth, and transformation. We also mentioned that Democritus had deduced that all these types of changes – including life and death – were in
fact the same, and due to the motion of atoms. The quantum of action makes exactly this
point.
First of all, a minimum action implies that cages in zoos are dangerous and banks are
not safe. A cage is a feature that needs a lot of energy to overcome. Physically speaking,
the wall of a cage is an energy hill, resembling the real hill shown in Figure 8. Imagine
that a particle with momentum ? approaches one side of the hill, which is assumed to
have width Δ?.
In everyday life – and thus in classical physics – the particle will never be observed
on the other side of the hill if its kinetic energy ?2 /2? is less than the height ? of the
hill. But imagine that the missing momentum to overcome the hill, Δ? = √2?? − ?,
satisfies Δ?Δ? ⩽ ℏ/2. The particle will have the possibility to overcome the hill, despite
its insufficient energy. The quantum of action thus implies that a hill of width
Δ? ⩽

ℏ/2
√2?? − ?

(4)

is not an obstacle to a particle of mass ?. But this is not all. Since the value of the particle
momentum ? is itself undetermined, a particle can overcome the hill even if the hill is
wider than the value (4) – although the broader it is, the lower the probability will be.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 20

Motion Mountain – The Adventure of Physics

An object can be non-rotating only approximately, when observations are separated by
long time intervals.
For microscopic systems, the quantum limits on rotation have specific effects. If the
rotation angle can be observed – as for molecules – the system behaves like a macroscopic
object: its position and orientation are fuzzy. But for a system whose rotation angle cannot be observed, the quantum of action limits the angular momentum to multiples of
ℏ/2. In particular, all microscopic bound systems – such as molecules, atoms, or nuclei
– contain rotational motion and rotating components.

1 minimum action – quantum theory for poets

29

E1
m

E2

F I G U R E 9 Leaving enclosures.

Page 89

⊳ Matter is not impenetrable.

Challenge 14 s

⊳ Decay is part of nature.

Ref. 8

Challenge 15 s

Note that decay often appears in everyday life, under a different name: breaking. In fact,
all breakages require the quantum of action for their description. Obviously, the cause
of breaking is often classical, but the mechanism of breaking is always quantum. Only
objects that obey quantum theory can break. In short, there are no stable excited systems
in nature. For the same reason, by the way, no memory can be perfect. (Can you confirm
this?)
Taking a more general view, ageing and death also result from the quantum of action.
Death, like ageing, is a composition of breaking processes. When dying, the mechanisms
in a living being break. Breaking is a form of decay, and is due to tunnelling. Death is

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The penetrability of all matter is in contrast to everyday, classical observation. Can you
explain why lion cages work despite the quantum of action?
By the way, the quantum of action also implies that a particle with a kinetic energy
greater than the energy height of a hill can be reflected by the hill. Also this effect is
impossible in classical physics.
The minimum-action principle also implies that bookshelves are dangerous. Why?
Shelves are obstacles to motion. A book on a shelf is in the same situation as the mass in
Figure 9: the mass is surrounded by energy hills hindering its escape to the outer, lowerenergy world. But thanks to the tunnelling effect, escape is always possible. The same
picture applies to a branch of a tree, a nail in a wall, or anything attached to anything
else. Things can never be permanently fixed together. In particular, we will discover that
every example of light emission – even radioactivity – results from this effect.
In summary, the quantum of action thus implies that

Motion Mountain – The Adventure of Physics

So any particle can overcome any obstacle. This is called the tunnelling effect, for obvious
reasons. Classically, tunnelling is impossible. In quantum theory, the feat is possible, because the wave function does not vanish at the location of the hill; sloppily speaking, the
wave function is non-zero inside the hill. It thus will be also non-zero behind the hill. As
a result, quantum systems can penetrate or ‘tunnel’ through hills.
In short, the minimum-action principle implies that there are no tight boxes in nature.
Thanks to the tunnelling effect,

30

1 minimum action – quantum theory for poets

m

m
F I G U R E 10 Identical objects with
crossing paths.

Challenge 17 s

⊳ All flows and all waves are made of microscopic particles.

Challenge 18 s

The term ‘microscopic’ (or ‘quantum’) is essential, as such particles do not behave like
little stones. We have already encountered several differences, and we will encounter others shortly. For these reasons, there should be a special name for microscopic particles;
but so far all proposals, of which quanton is the most popular, have failed to catch on.
The quantum of action has several strange consequences for microscopic particles.
Take two such particles with the same mass and composition. Imagine that their paths
cross, and that at the crossing they approach each other very closely, as shown in Figure 10. A minimum action implies that in such a situation, if the distance becomes small
enough, the two particles can switch roles, without anybody being able to avoid, or notice, it. Thus, in a volume of gas it is impossible – thanks to the quantum of action – to
follow particles moving around and to say which particle is which. Can you confirm this
deduction, and specify the conditions, using the indeterminacy relations? In summary
⊳ In nature it is impossible to distinguish between identical particles.

Challenge 19 s

Can you guess what happens in the case of light?
But matter deserves still more attention. Imagine again two particles – even two different ones – approaching each other very closely, as shown in Figure 11. We know that if

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

thus a quantum process. Classically, death does not exist. Might this be the reason why
so many people believe in immortality or eternal youth?
We will also discover that the quantum of action is the reason for the importance of
the action observable in classical physics. In fact, the existence of a smallest action is the
reason for the least-action principle of classical physics.
A minimum action also implies that matter cannot be continuous, but must be composed of smallest entities. Indeed, any flow of a truly continuous material would contradict the quantum principle. Can you give the precise argument? Of course, at this point
in our adventure, the non-continuity of matter is no longer a surprise. But the quantum
of action implies that even radiation cannot be continuous. As Albert Einstein was the
first to state clearly, light is made of quantum particles.
Even more generally, the quantum of action implies that in nature

Motion Mountain – The Adventure of Physics

Challenge 16 s

1 minimum action – quantum theory for poets

31

M
m1

m2
m

m3

F I G U R E 11 Transformation through

reaction.

⊳ The quantum of action allows transformation of matter.

⊳ All growth processes in nature are quantum processes.
Above all, as mentioned already, the quantum of action explains life. Only the quantum
of action makes reproduction and heredity possible. Birth, sexuality and death are consequences of the quantum of action.
So Democritus was both right and wrong. He was right in deducing fundamental
constituents for matter and radiation. He was right in unifying all change in nature –
from transport to transformation and growth – as motion of particles. But he was wrong
in assuming that the small particles behave like stones. As we will show in the following,
the smallest particles behave like quantons: they behave randomly, and they behave partly
as waves and partly as particles.
R and omness – a consequence of the quantum of action
What happens if we try to measure a change smaller than the quantum of action? Nature
has a simple answer: we get random results. If we build an experiment that tries to produce a change or action of the size of a quarter of the quantum of action, the experiment
will produce, for example, a change of one quantum of action in a quarter of the cases,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

One also says that the quantum of action allows particle reactions. In fact, we will discover that all kinds of reactions in nature, including breathing, digestion, and all other
chemical and nuclear reactions, are due just to the existence of the quantum of action.
One type of process that is especially dear to us is growth. The quantum of action
implies that all growth happens in small steps. Indeed,

Motion Mountain – The Adventure of Physics

Challenge 20 e

the approach distance gets small, things get fuzzy. Now, the minimum-action principle
makes it possible for something to happen in that small domain as long as resulting outgoing products have the same total linear momentum, angular momentum and energy as
the incoming ones. Indeed, ruling out such processes would imply that arbitrarily small
actions could be observed, thus eliminating nature’s fuzziness, as you may wish to check
for yourself. In short,

32

1 minimum action – quantum theory for poets

F I G U R E 12 A famous quantum effect: how do train windows manage to show two superimposed
images? (Photo © Greta Mansour)

⊳ Attempts to measure actions below ℏ lead to random results.

⊳ Microscopic systems behave randomly.
Obviously, there will be some average outcome; but in all cases, microscopic observations
are probabilistic. Many find this conclusion of quantum theory the most difficult to swallow. But fact is: the quantum of action implies that the behaviour of quantum systems is
strikingly different from that of classical systems. The conclusion is unavoidable:
⊳ Nature behaves randomly.
Can we observe randomness in everyday life? Yes. Every window proves that nature behaves randomly on a microscopic scale. Everybody knows that we can use a train window
either to look at the outside landscape or, by concentrating on the reflected image, to ob* In this context, ’no change’ means ’no change’ in the physical variable to be measured; generally speaking,
there is always some change, but not necessarily in the variable being measured.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

If you want to condense quantum physics in one key statement, this is it.
The quantum of action leads to randomness at microscopic level. This connection
can be seen also in the following way. Because of the indeterminacy relations, it is impossible to obtain definite values for both the momentum and the position of a particle.
Obviously, definite values are also impossible for the individual components of an experimental set-up or an observer. Therefore, initial conditions – both for a system and
for an experimental set-up – cannot be exactly duplicated. The quantum of action thus
implies that whenever an experiment on a microscopic system is performed twice, the
outcomes will (usually) be different. The outcomes could only be the same if both the
system and the observer were in exactly the same configuration each time. However, because of the second principle of thermodynamics, and because of the quantum of action,
reproducing a configuration is impossible. Therefore,

Motion Mountain – The Adventure of Physics

and no change in three quarters of the cases,* thus giving an average of one quarter of ℏ.

1 minimum action – quantum theory for poets

33

F I G U R E 13 A particle and a screen with two nearby slits.

⊳ Quantons move randomly.

Waves – a consequence of the quantum of action

Vol. I, page 313

The quantum of action implies an important result about the paths of particles. If a
particle travels from one point to another, there is no way to say which path it has taken
in between. Indeed, in order to distinguish between two possible, but slightly different, paths, actions smaller than ℏ would have to be measured reliably. In particular, if
a particle is sent through a screen with two sufficiently close slits, as illustrated in Figure 13, it is impossible to say which slit the particle passed through. This impossibility is
fundamental.
We already know phenomena of motion for which it is not possible to say with precision how something moves or which path is taken behind two slits: waves behave in this
way. All waves are subject to the indeterminacy relations
Δ?Δ? ⩾

1
2

and

Δ?Δ? ⩾

1
.
2

(5)

A wave is a type of motion described by a phase that changes over space and time. This
turns out to hold for all motion. In particular, this holds for matter.
We saw above that quantum systems are subject to
Δ?Δ? ⩾

ℏ
2

and

Δ?Δ? ⩾

ℏ
.
2

(6)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Without the quantum of action, train journeys would be much more boring.

Motion Mountain – The Adventure of Physics

serve some interesting person inside the carriage. In other words, observations like that
of Figure 12 show that glass reflects some of the light particles and lets some others pass
through. More precisely, glass reflects a random selection of light particles; yet the average proportion is constant. In these properties, partial reflection is similar to the tunnelling effect. Indeed, the partial reflection of photons in glass is a result of the quantum of
action. Again, the situation can be described by classical physics, but the precise amount
of reflection cannot be explained without quantum theory. We retain:

34

1 minimum action – quantum theory for poets

We are thus led to ascribe a frequency and a wavelength to a quantum system:
? = ℏ? and

Page 76

? = ℏ? = ℏ

2π
.
?

(7)

The energy–frequency relation for light and the equivalent momentum–wavelength relation were deduced by Max Planck in 1899. In the years from 1905 onwards, Albert Einstein confirmed that the relations are valid for all examples of emission and absorption
of light. In 1923 and 1924, Louis de Broglie* predicted that the relation should hold also
for all quantum matter particles. The experimental confirmation came a few years later.
(This is thus another example of a discovery that was made about 20 years too late.) In
short, the quantum of action implies:
⊳ Matter particles behave like waves.

Particles – a consequence of the quantum of action
Vol. I, page 353

⊳ All matter has particle aspects.
In the same way, the quantum of action, the smallest change, implies that light cannot
be arbitrarily faint. There is a smallest illumination in nature; it is called a photon or a
light quantum. Now, light is a wave, and the argument can be made for any other wave
as well. In short, the quantum of action thus implies:
⊳ All waves have particle aspects.
* Louis de Broglie (b. 1892 Dieppe, d. 1987 Paris), physicist and professor at the Sorbonne. The energy–
frequency relation for light had earned Max Planck and Albert Einstein the Nobel Prize in Physics, in 1918
and 1921. De Broglie expanded the relation to predict the wave nature of the electron (and of all other
quantum matter particles): this was the essence of his doctoral thesis. The prediction was first confirmed
experimentally a few years later, in 1927. For the prediction of the wave nature of matter, de Broglie received
the Nobel Prize in Physics in 1929. Being an aristocrat, he did no more research after that. For example, it
was Schrödinger who then wrote down the wave equation, even though de Broglie could equally have done
so.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The quantum of action, the smallest change, implies that flows cannot be arbitrarily weak.
This applies to all flows: in particular, it applies to rivers, solid matter flows, gas flows,
light beams, energy flows, entropy flows, momentum flows, angular momentum flows,
probability flows, signals of all kind, electrical charge flows, colour charge flows and weak
charge flows.
Water flows in rivers, like any other matter flow, cannot be arbitrary small: the
quantum of action implies that there is a smallest matter flow in nature. Depending on
the situation, the smallest matter flow is a molecule, an atom or a smaller particle. Indeed, the quantum of action is also at the origin of the observation of a smallest charge
in electric current. Since all matter can flow, the quantum of action implies:

Motion Mountain – The Adventure of Physics

In particular, the quantum of action implies the existence of interference for streams of
matter.

1 minimum action – quantum theory for poets

35

This has been proved for light waves, water waves, X-rays, sound waves, plasma waves,
fluid whirls and any other wave type that has ever been observed. There is one exception:
gravitational waves have finally been observed in 2016, many decades after their prediction; it is expected that their particle-like aspects, the gravitons, also exist, though this
might take a long time to prove by experiment.
In summary, the quantum of action states:
⊳ If something moves, it is made of quantum particles, or quantons.
Later on we will explore and specify the exact differences between a quantum particle and
a small stone or a grain of sand. We will discover that matter quantons move differently,
behave differently under rotation, and behave differently under exchange.
Q uantum information

The quantum of action makes it impossible to gather and use all information in a way
that allows production of a perfect copy.
The exploration of copying machines will remind us again that the precise order
in which measurements are performed in an experiment matters. When the order
of measurements can be reversed without affecting the net result, physicists speak of
‘commutation’. The quantum of action implies:
⊳ Physical observables do not commute.

Page 152

We will also find that the quantum of action implies that systems are not always
independent, but can be entangled. This term, introduced by Erwin Schrödinger, de-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ Perfect copying machines do not exist.

Motion Mountain – The Adventure of Physics

In computer science, the smallest unit of change is called a ‘bit change’. The existence of
a smallest change in nature implies that computer science – or information science – can
be used to describe nature, and in particular quantum theory. This analogy has attracted
much research in the past decades, and explored many interesting questions: Is unlimited
information storage possible? Can information be read out and copied completely? Can
information be transmitted while keeping it secret? Can information transmission and
storage be performed independently of noise? Can quantum physics be used to make
new types of computers? So far, the answer to all these questions is negative; but the
hope to change the situation is not dead yet.
The analogy between quantum theory and information science is limited: information
science can describe only the ‘software’ side of devices. For a physicist, the ‘hardware’
side of nature is central. The hardware of nature enters the description whenever the
actual value ℏ of the quantum of action must be introduced.
As we explore the similarities and differences between nature and information science, we will discover that the quantum of action implies that macroscopic physical systems cannot be copied – or ‘cloned’, as quantum theorists like to say. Nature does not
allow copies of macroscopic objects. In other words:

36

1 minimum action – quantum theory for poets

scribes one of the most absurd consequences of quantum theory. Entanglement makes
everything in nature connected to everything else. Entanglement produces effects that
seem (but are not) faster than light.
⊳ Entanglement produces a (fake) form of non-locality.
Ref. 9

Page 157

Entanglement implies that trustworthy communication cannot exist.
We will also discover that decoherence is an ubiquitous process in nature that influences all quantum systems. For example, it allows measurements on the one hand and
makes quantum computers impossible on the other.
Curiosities and fun challenges ab ou t the quantum of action

∗∗
Challenge 21 s

Where is the quantum scale in a pendulum clock?
∗∗

∗∗
Challenge 22 s

Adult plants stop growing in the dark. Without light, the reactions necessary for growth
cease. Can you show that this is a quantum effect, not explainable by classical physics?
∗∗

Challenge 23 s

Most quantum processes in everyday life are electromagnetic. Can you show that the
quantum of action must also hold for nuclear processes, i.e., for processes that are not
electromagnetic?
∗∗

Challenge 24 s

Is the quantum of action independent of the observer, even near the speed of light? This
question was the reason why Planck contacted the young Einstein, inviting him to Berlin,
thus introducing him to the international physics community.
∗∗
The quantum of action implies that tiny people, such as Tom Thumb, cannot exist. The

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. III, page 86

When electromagnetic fields come into play, the value of the action (usually) depends
on the choice of the vector potential, and thus on the choice of gauge. We saw in the part
on electrodynamics that a suitable choice of gauge can change the value of the action
by adding or subtracting any desired amount. Nevertheless, there is a smallest action in
nature. This is possible, because in quantum theory, physical gauge changes cannot add
or subtract any amount, but only multiples of twice the minimum value. Thus they do
not allow us to go below the minimum action.

Motion Mountain – The Adventure of Physics

Even if we accept that no experiment performed so far contradicts the minimum action,
we still have to check that the minimum action does not contradict reason. In particular,
the minimum action must also be consistent with all imagined experiments. This is not
self-evident.

1 minimum action – quantum theory for poets

Challenge 25 s

37

quantum of action implies that fractals cannot exist in nature. The quantum of action
implies that ‘Moore’s law’ of semiconductor electronics, which states that the number of
transistors on a chip doubles every two years, cannot be valid for ever. Why not?
∗∗
Take a horseshoe. The distance between the two ends is not fixed, since otherwise their
position and velocity would be known at the same time, contradicting the indeterminacy
relation. Of course, this reasoning is also valid for any other solid object. In short, both
quantum mechanics and special relativity show that rigid bodies do not exist, albeit for
different reasons.
∗∗

∗∗
Challenge 27 s

Could we have started the whole discussion of quantum theory by stating that there is a
minimum angular momentum instead of a minimum action?
∗∗

The dangers of buying a can of beans

Ref. 10

Another way to show the absurd consequences of quantum theory is given by the ultimate product warning, which according to certain well-informed lawyers should be
printed on every can of beans and on every product package. It shows in detail how
deeply our human condition fools us.
Warning: care should be taken when looking at this product:
It emits heat radiation.
Bright light has the effect to compress this product.
Warning: care should be taken when touching this product:
Part of it could heat up while another part cools down, causing severe burns.
Warning: care should be taken when handling this product:
This product consists of at least 99.999 999 999 999 % empty space.
This product contains particles moving with speeds higher than one million kilometres per hour.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 28 s

Niels Bohr, besides propagating the idea of a minimum action, was also an enthusiast of
the so-called complementarity principle. This is the idea that certain pairs of observables
of a system – such as position and momentum – have linked precision: if one of the pair
is known to high precision, the other is necessarily known with low precision. Can you
deduce this principle from the minimum action?

Motion Mountain – The Adventure of Physics

Challenge 26 s

Angular momentum has the same dimensions as action. A smallest action implies that
there is a smallest angular momentum in nature. How can this be, given that some
particles have spin zero, i.e., have no angular momentum?

38

1 minimum action – quantum theory for poets

Every kilogram of this product contains the same amount of energy as liberated by
about one hundred nuclear bombs.*
In case this product is brought in contact with antimatter, a catastrophic explosion
will occur.
In case this product is rotated, it will emit gravitational radiation.
Warning: care should be taken when transporting this product:
The force needed depends on its velocity, as does its weight.
This product will emit additional radiation when accelerated.
This product attracts, with a force that increases with decreasing distance, every
other object around, including its purchaser’s kids.
Warning: care should be taken when storing this product:

Except when stored underground at a depth of several kilometres, over time cosmic
radiation will render this product radioactive.
This product may disintegrate in the next 1035 years.
It could cool down and lift itself into the air.
This product warps space and time in its vicinity, including the storage container.

This product can disappear from its present location and reappear at any random
place in the universe, including your neighbour’s garage.
Warning: care should be taken when travelling away from this product:
It will arrive at the expiration date before the purchaser does so.
Warning: care should be taken when using this product:
Any use whatsoever will increase the entropy of the universe.
The constituents of this product are exactly the same as those of any other object
in the universe, including those of rotten fish.
All these statements are correct. The impression of a certain paranoid side to quantum
physics is purely coincidental.

Ref. 11

* A standard nuclear warhead has an explosive yield of about 0.2 megatons (implied is the standard explosive
trinitrotoluene or TNT), about thirteen times the yield of the Hiroshima bomb, which was 15 kilotonne. A
megatonne is defined as 1 Pcal=4.2 PJ, even though TNT delivers about 5 % slightly less energy than this
value. In other words, a megaton is the energy content of about 47 g of matter. That is less than a handful
for most solids or liquids.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Even if stored in a closed container, this product is influenced and influences all
other objects in the universe, including your parents in law.

Motion Mountain – The Adventure of Physics

It is impossible to keep this product in a specific place and at rest at the same time.

1 minimum action – quantum theory for poets

39

A summary: quantum physics, the law and ind o ctrination

Page 167

Ref. 12

”

* ‘Never and nowhere has matter existed, nor can it exist, without motion.’ Friedrich Engels (1820–1895)
was one of the theoreticians of Marxism.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

“

Nie und nirgends hat es Materie ohne
Bewegung gegeben, oder kann es sie geben.
Friedrich Engels, Anti-Dühring.*

Motion Mountain – The Adventure of Physics

Challenge 29 d

The mere existence of a quantum of action, a quantum of change, has many deep consequences: randomness, wave-particle duality, matter transformation, death, and, above
all, new thinking habits.
Don’t all the deductions from the quantum of action presented so far look wrong, or
at least crazy? In fact, if you or your lawyer made some of the statements on quantum
physics in court, maybe even under oath, you might end up in prison! However, all the
above statements are correct: they are all confirmed by experiment. And there are many
more surprises to come. You may have noticed that, in the preceding examples, we have
made no explicit reference to electricity, to the nuclear interactions or to gravity. In these
domains the surprises are even more astonishing. Observation of antimatter, electric current without resistance, the motion inside muscles, vacuum energy, nuclear reactions in
stars, and – maybe one day – the boiling of empty space, will fascinate you as much as
they have fascinated, and still fascinate, thousands of researchers.
In particular, the consequences of the quantum of action for the early universe are
mind-boggling. Just try to explore for yourself its consequences for the big bang. Together, all these topics will lead us a long way towards the aim of our adventure. The
consequences of the quantum of action are so strange, so incredible, and so numerous,
that quantum physics can rightly be called the description of motion for crazy scientists.
In a sense, this generalizes our previous definition of quantum physics as the description
of motion related to pleasure.
Unfortunately, it is sometimes claimed that ‘nobody understands quantum theory’.
This is wrong. In fact, it is worse than wrong: it is indoctrination and disinformation.
Indoctrination and disinformation are methods that prevent people from making up
their own mind and from enjoying life. In reality, the consequences of the quantum of
action can be understood and enjoyed by everybody. In order to do so, our first task on
our way towards completing our adventure will be to use the quantum of action to study
our classical standard of motion: the motion of light.

Chapter 2

L IG H T – T H E ST R A NG E
C ON SE QU E NC E S OF T H E QUA N T UM
OF AC T ION

S

”

ince all the colours of materials are quantum effects, it becomes mandatory to
tudy the properties of light itself. If a smallest change really exists, then there
hould also be a smallest illumination in nature. This conclusion was already drawn
in ancient Greece, for example by Epicurus (341–271 b ce), who stated that light is a
stream of little particles. The smallest possible illumination would then be that due to
a single light particle. Today, the particles are called light quanta or photons. Incredibly,
Epicurus himself could have checked his prediction with an experiment.
How d o faint lamps behave?

Around 1930, Brumberg and Vavilov found a beautiful way to check the existence of
photons using the naked eye and a lamp. Our eyes do not allow us to consciously detect
single photons, but Brumberg and Vavilov found a way to circumvent this limitation.
In fact, the experiment is so simple that it could have been performed many centuries
earlier; but nobody had had a sufficiently daring imagination to try it.
Brumberg and Vavilov constructed a mechanical shutter that could be opened for
time intervals of 0.1 s. From the other side, in a completely dark room, they illuminated
the opening with extremely weak green light: about 200 aW at 505 nm, as shown in Figure 14. At that intensity, whenever the shutter opens, on average about 50 photons can
pass. This is just the sensitivity threshold of the eye. To perform the experiment, they
repeatedly looked into the open shutter. The result was simple but surprising. Sometimes they observed light, and sometimes they did not. Whether they did or did not was
completely random. Brumberg and Vavilov gave the simple explanation that at low lamp
powers, because of fluctuations, the number of photons is above the eye threshold half the
time, and below it the other half. The fluctuations are random, and so the conscious detection of light is as well. This would not happen if light were a continuous stream: in that
case, the eye would detect light at each and every opening of the shutter. (At higher light
intensities, the percentage of non-observations quickly decreases, in accordance with the
explanation given.)
In short, a simple experiment proves:
** ‘From light all beings live, each fair-created thing.’ Friedrich Schiller (b. 1759 Marbach, d. 1805 Weimar),
poet, playwright and historian.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 14

Motion Mountain – The Adventure of Physics

Ref. 13

“

Alle Wesen leben vom Lichte,
jedes glückliche Geschöpfe.
Friedrich Schiller, Wilhelm Tell.**

2 light – and the quantum of action

lamp

head, after
45 minutes
in complete
darkness

strong shutter
filter

F I G U R E 14 How to experience single
photon effects (see text).

photographic
film

white

red
green
violet

⊳ Light is made of photons.
Nobody knows how the theory of light would have developed if this simple experiment
had been performed 100 or even 2500 years earlier.
The reality of photons becomes more convincing if we use devices to help us. A simple
way is to start with a screen behind a prism illuminated with white light, as shown in
Figure 15. The light is split into colours. As the screen is placed further and further away,
the illumination intensity cannot become arbitrarily small, as that would contradict the
quantum of action. To check this prediction, we only need some black-and-white photographic film. Film is blackened by daylight of any colour; it becomes dark grey at medium
intensities and light grey at lower intensities. Looking at an extremely light grey film under the microscope, we discover that, even under uniform illumination, the grey shade is
actually composed of black spots, arranged more or less densely. All these spots have the
same size, as shown in Figure 16. This regular size suggests that a photographic film reacts
to single photons. Detailed research confirms this conjecture; in the twentieth century,
the producers of photographic films have elucidated the underlying atomic mechanism
in all its details.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 15 How does a
white-light spectrum appear at
extremely long screen distances?
(The short-screen-distance
spectrum shown, © Andrew
Young, is optimized for CRT
display, not for colour printing, as
explained on mintaka.sdsu.edu/
GF/explain/optics/rendering.
html.)

Motion Mountain – The Adventure of Physics

glass

41

42

2 light – and the quantum of action

F I G U R E 16 Exposed photographic film at increasing magnification (© Rich Evans).

Motion Mountain – The Adventure of Physics

Ref. 15

Single photons can be detected most elegantly with electronic devices. Such devices
can be photomultipliers, photodiodes, multichannel plates or rod cells in the eye; a selection is shown in Figure 17. Also these detectors show that low-intensity light does not
produce a homogeneous colour: on the contrary, low-intensity produces a random pattern of equal spots, even when observing typical wave phenomena such as interference
patterns, as shown in Figure 18. Today, recording and counting individual photons is a
standard experimental procedure. Photon counters are part of many spectroscopy setups, such as those used to measure tiny concentrations of materials. For example, they
are used to detect drugs in human hair.
All experiments thus show the same result: whenever sensitive light detectors are constructed with the aim of ‘seeing’ as accurately as possible – and thus in environments as

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 17 Detectors that allow photon counting: photomultiplier tubes (left), an avalanche
photodiode (top right, c. 1 cm) and a multichannel plate (bottom right, c. 10 cm) (© Hamamatsu
Photonics).

2 light – and the quantum of action

43

light detectors

F I G U R E 19 An atom radiating one
photon triggers only one detector and
recoils in only one direction.

dark as possible – one finds that light manifests as a stream of light quanta. Nowadays
they are usually called photons, a term that appeared in 1926. Light of low or high intensity corresponds to a stream with a small or large number of photons.
A particularly interesting example of a low-intensity source of light is a single atom.
Atoms are tiny spheres. When atoms radiate light or X-rays, the radiation should be emitted as a spherical wave. But in all experiments – see Figure 19 for a typical set-up – the
light emitted by an atom is never found to form a spherical wave, in contrast to what we
might expect from everyday physics. Whenever a radiating atom is surrounded by many
detectors, only a single detector is triggered. Only the average over many emissions and
detections yields a spherical shape. The experiments shows clearly that partial photons
cannot be detected.
All experiments in dim light thus show that the continuum description of light is

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

radiating
atom

Motion Mountain – The Adventure of Physics

F I G U R E 18 Light waves are made of particles: observation of photons – black spots in these negatives
– in a low intensity double slit experiment, with exposure times of 1, 2 and 5 s, using an image
intensifier (© Delft University of Technology).

44

Vol. III, page 87

incorrect. All such experiments thus prove directly that light is a stream of particles, as
Epicurus had proposed in ancient Greece. More precise measurements confirm the role
of the quantum of action: every photon leads to the same amount of change. All photons
of the same frequency blacken a film or trigger a scintillation screen in the same way. In
short, the amount of change induced by a single photon is indeed the smallest amount
of change that light can produce.
If there were no smallest action value, light could be packaged into arbitrarily small
amounts. But nature is different. In simple terms: the classical description of light by a
continuous vector potential ?(?, ?), or electromagnetic field ?(?, ?), whose evolution is
described by a principle of least action, is wrong. Continuous functions do not describe
the observed particle effects. A modified description is required. The modification has to
be significant only at low light intensities, since at high, everyday intensities the classical
Lagrangian describes all experimental observations with sufficient accuracy.*
At which intensities does light cease to behave as a continuous wave? Human eyesight
does not allow us to consciously distinguish single photons, although experiments show
that the hardware of the eye is in principle able to do so. The faintest stars that can be
seen at night produce a light intensity of about 0.6 nW/m2 . Since the pupil of the eye is
small, and we are not able to see individual photons, photons must have energies smaller
than 100 aJ. Brumberg and Vavilov’s experiment yields an upper limit of around 20 aJ.
An exact value for the quantum of action found in light must be deduced from laboratory experiment. Some examples are given in the following.
Photons

Vol. III, page 150
Page 214

This relation was first deduced by Max Planck in 1899. He found that for light, the smallest measurable action is given by the quantum of action ℏ. In short, colour is a property
of photons. A coloured light beam is a hailstorm of corresponding photons.
The value of Planck’s constant can be determined from measurements of black bodies
or other light sources. All such measurements coincide and yield
ℏ = 1.054 571 726(47) ⋅ 10−34 Js ,

Challenge 30 e

Ref. 16
Challenge 31 ny

(9)

a value so small that we can understand why photons go unnoticed by humans. For example, a green photon with a wavelength of 555 nm has an energy of 0.37 aJ. Indeed, in
normal light conditions the photons are so numerous that the continuum approximation
for the electromagnetic field is highly accurate. In the dark, the insensitivity of the signal
processing of the human eye – in particular the slowness of the light receptors – makes
photon counting impossible. However, the eye is not far from the maximum possible
sensitivity. From the numbers given above about dim stars, we can estimate that humans
* The transition from the classical case to the quantum case used to be called quantization. This concept,
and the ideas behind it, are only of historical interest today.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In general, all experiments show that a beam of light of frequency ? or angular frequency
?, which determines its colour, is accurately described as a stream of photons, each with
the same energy ? given by
? = ℏ 2π? = ℏ ? .
(8)

Motion Mountain – The Adventure of Physics

Ref. 16

2 light – and the quantum of action

2 light – and the quantum of action

Challenge 32 s
Ref. 17

Challenge 33 e

are able to see consciously, under ideal conditions, flashes of about half a dozen photons;
in normal conditions, the numbers are about ten times higher.
Let us explore the other properties of photons. Above all, photons have no measurable
(rest) mass and no measurable electric charge. Can you confirm this? In fact, experiments
can only provide an upper limit for both quantities. The present experimental upper limit
for the (rest) mass of a photon is 10−52 kg, and for the charge is 5⋅10−30 times the electron
charge. These limits are so small that we can safely say that both the mass and the charge
of the photon vanish.
We know that intense light can push objects. Since the energy, the lack of mass and
the speed of photons are known, we deduce that the photon momentum is given by
?=

Vol. III, page 124

Vol. III, page 124

or ? = ℏ ? .

(10)

In other words, if light is made of particles, we should be able to play billiard with them.
This is indeed possible, as Arthur Compton showed in a famous experiment in 1923.
He directed X-rays, which are high-energy photons, onto graphite, a material in which
electrons move almost freely. He found that whenever the electrons in the material are
hit by the X-ray photons, the deflected X-rays change colour. His experiment is shown
in Figure 20. As expected, the strength of the hit is related to the deflection angle of the
photon. From the colour change and the reflection angle, Compton confirmed that the
photon momentum indeed satisfies the expression ? = ℏ ?.
All other experiments agree that photons have momentum. For example, when an
atom emits light, the atom feels a recoil. The momentum again turns out to be given by
the expression ? = ℏ ?. In short, the quantum of action determines the momentum of
the photon.
The value of a photon’s momentum respects the indeterminacy relation. Just as it is
impossible to measure exactly both the wavelength of a wave and the position of its crest,
so it is impossible to measure both the momentum and the position of a photon. Can you
confirm this? In other words, the value of the photon momentum is a direct consequence
of the quantum of action.
From our study of classical physics, we know that light has a property beyond its colour: light can be polarized. That is only a complicated way to say that light can turn
the objects that it shines on. In other words, light has an angular momentum oriented
(mainly) along the axis of propagation. What about photons? Measurements consistently
find that each light quantum carries an angular momentum given by ? = ℏ. It is called
its helicity. The quantity is similar to one found for massive particles: one therefore also
speaks of the spin of a photon. In short, photons somehow ‘turn’ – in a direction either
parallel or antiparallel to their direction of motion. Again, the magnitude of the photon
helicity, or spin, is no surprise; it confirms the classical relation ? = ?/? between energy
and angular momentum that we found in the section on classical electrodynamics. Note
that, counterintuitively, the angular momentum of a single photon is fixed, and thus independent of its energy. Even the most energetic photons have ? = ℏ. Of course, the
value of the helicity also respects the limit given by the quantum of action. The many
consequences of the helicity (spin) value ℏ will become clear in the following.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 34 s

2π
?
=ℏ
?
?

Motion Mountain – The Adventure of Physics

Ref. 18

45

46

2 light – and the quantum of action

photon with
wavelength λ
X-ray
source
X-ray
source

sample

X-ray
detector

deflection
angle

collision
in
sample

X-ray
detector
deflected
photon after
the collision,
with wavelength λ+Δλ
electron
after the
collision

F I G U R E 20 A modern version of Compton’s experiment fits on a table. The experiment shows that
photons have momentum: X-rays – and thus the photons they consist of – change frequency when
they hit the electrons in matter in exactly the same way as predicted from colliding particles (© Helene
Hoffmann).

“

La lumière est un mouvement luminaire de
corps lumineux.
Blaise Pascal*

”

* ‘Light is the luminary movement of luminous bodies.’ Blaise Pascal (b. 1623 Clermont, d. 1662 Paris),
important mathematician and physicist up to the age of 26, after which he became a theologian and philosopher.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In the seventeenth century, Blaise Pascal used the above statement about light to make
fun of certain physicists, ridiculing the blatant use of a circular definition. Of course, he
was right: in his time, the definition was indeed circular, as no meaning could be given to
any of the terms. But whenever physicists study an observation with care, philosophers
lose out. All those originally undefined terms now have a definite meaning and the circular definition is resolved. Light is indeed a type of motion; this motion can rightly be
called ‘luminary’ because, in contrast to the motion of material bodies, it has the unique
property ? = ?; the luminous bodies, called light quanta or photons, are characterized,
and differentiated from all other particles, by their dispersion relation ? = ??, their energy ? = ℏ?, their spin ? = ℏ, the vanishing of all other quantum numbers, and the
property of being the quanta of the electromagnetic field.
In short, light is a stream of photons. It is indeed a ‘luminary movement of luminous
bodies’. Photons provide our first example of a general property of the world on small
scales: all waves and all flows in nature are made of quantum particles. Large numbers
of (coherent) quantum particles – or quantons – behave and form as waves. We will see
shortly that this is the case even for matter. Quantons are the fundamental constituents of
all waves and all flows, without exception. Thus, the everyday continuum description of
light is similar in many respects to the description of water as a continuous fluid: photons
are the atoms of light, and continuity is an approximation valid for large numbers of
particles. Single quantons often behave like classical particles.
Physics books used to discuss at length a so-called wave–particle duality. Let us be
clear from the start: quantons, or quantum particles, are neither classical waves nor clas-

Motion Mountain – The Adventure of Physics

What is light?

2 light – and the quantum of action

47

sical particles. In the microscopic world, quantons are the fundamental objects.
However, there is much that is still unclear. Where, inside matter, do these monochromatic photons come from? Even more interestingly, if light is made of quantons, all
electromagnetic fields, even static ones, must be made of photons as well. However, in
static fields nothing is flowing. How is this apparent contradiction resolved? And what
implications does the particle aspect have for these static fields? What is the difference
between quantons and classical particles? The properties of photons require more careful
study.
The size of photons

“

Vol. II, page 180

Also gibt es sie doch.

”

Max Planck*

We saw above that the simplest way to count photons is to distribute them across a large
screen and then to absorb them. But this method is not entirely satisfactory, as it destroys
the photons. How can we count photons without destroying them?
One way is to reflect photons in a mirror and measure the recoil of the mirror. It
seems almost unbelievable, but nowadays this effect is becoming measurable even for
small numbers of photons. For example, it has to be taken into account in relation to the
laser mirrors used in gravitational wave detectors, whose position has to be measured
with high precision.
Another way of counting photons without destroying them involves the use of special
high-quality laser cavities. It is possible to count photons by the effect they have on atoms
cleverly placed inside such a cavity.
In other words, light intensity can indeed be measured without absorption. These
measurement show an important issue: even the best light beams, from the most sophist* ‘Thus they do exist after all.’ Max Planck, in his later years, said this after standing silently, for a long time,
in front of an apparatus that counted single photons by producing a click for each photon it detected. For
a large part of his life, Planck was sceptical of the photon concept, even though his own experiments and
conclusions were the starting point for its introduction.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Are photons countable? – S queezed light

Motion Mountain – The Adventure of Physics

Challenge 35 s

First of all, we might ask: what are these photons made of? All experiments so far, performed down to the present limit of about 10−20 m, give the same answer: ‘we can’t find
anything’. This is consistent with both a vanishing mass and a vanishing size of photons.
Indeed, we would intuitively expect a body with a finite size to have a finite mass. Thus,
although experiments can give only an upper limit, it is consistent to claim that a photon
has zero size.
A particle with zero size cannot have any constituents. Thus a photon cannot be divided into smaller entities: photons are not composite. For this reason, they are called
elementary particles. We will soon give some further strong arguments for this result.
(Can you find one?) Nevertheless, the conclusion is strange. How can a photon have
vanishing size, have no constituents, and still be something? This is a hard question; the
answer will appear only in the last volume of our adventure. At the moment we simply
have to accept the situation as it is. We therefore turn to an easier question.

48

2 light – and the quantum of action

icated lasers, fluctuate in intensity. There are no steady beams. This comes as no surprise:
if a light beam did not fluctuate, observing it twice would yield a vanishing value for the
action. However, there is a minimum action in nature, namely ℏ. Thus any beam and any
flow in nature must fluctuate. But there is more.
A light beam is described, in a cross section, by its intensity and phase. The change –
or action – that occurs while a beam propagates is given by the product of intensity and
phase. Experiments confirm the obvious deduction: the intensity and phase of a beam
behave like the momentum and position of a particle in that they obey an indeterminacy
relation. You can deduce it yourself, in the same way as we deduced Heisenberg’s relations. Using as characteristic intensity ? = ?/?, the beam energy divided by the angular
frequency, and calling the phase ?, we get*
Δ? Δ? ⩾

ℏ
.
2

(12)

* A large photon number is assumed in the expression. This is obvious, as Δ? cannot grow beyond all
bounds. The exact relations are
ℏ
|⟨sin ?⟩|
2
ℏ
Δ? Δ sin ? ⩾ |⟨cos ?⟩|
2

Δ? Δ cos ? ⩾

(11)

where ⟨?⟩ denotes the expectation value of the observable ?.
** Coherent light is light for which the photon number probability distribution is Poissonian; in particular,
the variance is equal to the mean photon number. Coherent light is best described as composed of photons
in coherent quantum states. Such a (canonical) coherent state, or Glauber state, is formally a state with Δ? →
1/? and Δ? → ?.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 19

For light emitted from an ordinary lamp, so-called thermal light, the indeterminacy
product on the left-hand side of the above inequality is a large number. Equivalently, the
indeterminacy product for the action (12) is a large multiple of the quantum of action.
For laser beams, i.e., beams of coherent light,** the indeterminacy product is close to
1/2. An illustration of coherent light is given in Figure 22.
Today it is possible to produce light for which the product of the two indeterminacies
in equation (13) is near 1/2, but whose two values differ (in the units of the so-called
phasor space illustrated in Figure 21). Such light is called non-classical or squeezed. The
photon statistics is either hyper- or sub-Poissonian. Such light beams require involved
laboratory set-ups for their production and are used in many modern research applications. Non-classical light has to be treated extremely carefully, as the smallest disturbances transforms it back into ordinary coherent (or even thermal light), in which Poisson (or even Bose-Einstein) statistics hold again. A general overview of the main types
of light beams is given in Figure 21, together with their intensity and phase behaviour.
(Several properties shown in the figure are defined for a single phase space cell only.)

Motion Mountain – The Adventure of Physics

Equivalently, the indeterminacy product for the average photon number ? = ?/ℏ = ?/ℏ?
and the phase ? obeys:
1
Δ? Δ? ⩾ .
(13)
2

2 light – and the quantum of action

Thermal equilibrium light

Coherent laser light

Photon clicks show bunching Weak bunching

49

Non-classical,
phase- squeezed light

Non-classical,
intensity-squeezed light

Strong bunching

Anti-bunching

time
Intensity I(t)

time
Photon number probability
Bose-Einstein
(super-Poisson)

Poisson

hyperPoisson

subPoisson

Intensity correlation g2(t)
bunching

2
1

2
1

coherence time
Phasor
diagram

weak bunching

2
1

strong bunching

2
1

anti-bunching

delay

Im
ω

ω

ω

Re

F I G U R E 21 Four types of light and their photon properties: thermal light, laser light, and two extreme
types of non-classical, squeezed light.

One extreme of non-classical light is phase-squeezed light. Since a phase-squeezed
light beam has an (almost) determined phase, the photon number in such a beam fluctuates from zero to (almost) infinity. In other words, in order to produce coherent laser
light that approximates a pure sine wave as perfectly as possible, we must accept that the
photon number is as undetermined as possible. Such a beam has extremely small phase
fluctuations that provide high precision in interferometry; the phase noise is as low as
possible.
The other extreme of non-classical light is a beam with a given, fixed number of
photons, and thus with an extremely high phase indeterminacy. In such an amplitudesqueezed light beam, the phase fluctuates erratically.* This sort of squeezed, non-classical
* The most appropriate quantum states to describe such light are called number states, sometimes Fock states.
These states are stationary, thus eigenstates of the Hamiltonian, and contain a fixed number of photons.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

ω

Motion Mountain – The Adventure of Physics

n

<n>

50

2 light – and the quantum of action

8
6

electric field (a. u.)

4
2
0
-2
-4
-6
0

2

4

6

8

10

12

14

16

18

20

time (a. u.)
F I G U R E 22 A simple way to illustrate the indeterminacy of a light beam’s intensity and phase: the
measured electric field of a coherent electromagnetic wave with low intensity, consisting of about a
dozen photons. The cloudy sine wave corresponds to the phasor diagram at the bottom of the second
column in the previous overview. For large number of photons, the relative noise amplitude is
negligible. (© Rüdiger Paschotta)

Challenge 37 ny

Challenge 38 s

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 36 s

light is ideal for precision intensity measurements as it provides the lowest intensity noise
available. This kind of light shows anti-bunching of photons. To gain more insight, sketch
the graphs corresponding to Figure 22 for phase-squeezed and for amplitude-squeezed
light.
In contrast, the coherent light that is emitted by laser pointers and other lasers lies
between the two extreme types of squeezed light: the phase and photon number indeterminacies are of similar magnitude.
The observations about thermal light, coherent laser light and non-classical light highlight an important property of nature: the number of photons in a light beam is not a
well-defined quantity. In general, it is undetermined, and it fluctuates. Photons, unlike
stones, cannot be counted precisely – as long as they are propagating and not absorbed.
In flight, it is only possible to determine an approximate, average photon number, within
the limits set by indeterminacy. Is it correct to claim that the number of photons at the
beginning of a beam is not necessarily the same as the number at the end of the beam?
The fluctuations in the number of photons are of most importance at optical frequencies. At radio frequencies, the photon number fluctuations are usually negligible, due to
the low photon energies and the usually high photon numbers involved. Conversely, at
gamma-ray energies, wave effects play little role. For example, we saw that in deep, dark
intergalactic space, far from any star, there are about 400 photons per cubic centimetre;
they form the cosmic background radiation. This photon density number, like the number of photons in a light beam, also has a measurement indeterminacy. Can you estimate
it?

Motion Mountain – The Adventure of Physics

-8

2 light – and the quantum of action

51

The Mach-Zehnder interferometer
source

detectors

mirrors
beam
splitter

beam
splitter
two identical
photons

possible
light
paths

Motion Mountain – The Adventure of Physics

In short, unlike pebbles, photons are countable, but their number is not fixed. And this
is not the only difference between photons and pebbles.
The positions of photons
Where is a photon when it moves in a beam of light? Quantum theory gives a simple
answer: nowhere in particular. This is proved most spectacularly by experiments with
interferometers, such as the basic interferometer shown in Figure 23. Interferometers
show that even a beam made of a single photon can be split, led along two different paths,
and then recombined. The resulting interference shows that the single photon cannot be

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 23 The Mach–Zehnder interferometer and a practical realization, about 0.5 m in size (© Félix
Dieu and Gaël Osowiecki).

52

Challenge 39 e

* We cannot avoid this conclusion by saying that photons are split at the beam splitter: if we place a detector
in each arm, we find that they never detect a photon at the same time. Photons cannot be divided.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 20

said to have taken either of the two paths. If one of the two paths is blocked, the pattern
on the screen changes. In other words, somehow the photon must have taken both paths
at the same time. Photons cannot be localized: they have no position.*
We come to the conclusion that macroscopic light pulses have paths, but the individual photons in it do not. Photons have neither position nor paths. Only large numbers
of photons can have positions and paths, and then only approximately.
The impossibility of localizing photons can be quantified. Interference shows that it
is impossible to localize photons in the direction transverse to the motion. It might seem
less difficult to localize photons along the direction of motion, when it is part of a light
pulse, but this is a mistake. The quantum of action implies that the indeterminacy in the
longitudinal position is given at least by the wavelength of the light. Can you confirm
this? It turns out that photons can only be localized within a coherence length. In fact, the
transversal and the longitudinal coherence length differ in the general case. The longitudinal coherence length (divided by ?) is also called temporal coherence, or simply, the
coherence time. It is also indicated in Figure 21. The impossibility of localizing photons is
a consequence of the quantum of action. For example, the transverse coherence length is
due to the indeterminacy of the transverse momentum; the action values for paths leading to points separated by less than a coherence length differ by less than the quantum of
action ℏ. Whenever a photon is detected somewhere, e.g., by absorption, a precise statement on its direction or its origin cannot be made. Sometimes, in special cases, there can
be a high probability for a certain direction or source, though.
Lack of localisation means that photons cannot be simply visualized as short wave
trains. For example, we can increase the coherence length by sending light through a narrow filter. Photons are truly unlocalizable entities, specific to the quantum world. Photons
are neither little stones nor little wave packets. Conversely, ‘light path’, ‘light pulse position’ and ‘coherence’ are properties of a photon ensemble, and do not apply to a single
photon.
Whenever photons can almost be localized along their direction of motion, as in coherent light, we can ask how photons are lined up, one after the other, in a light beam. Of
course, we have just seen that it does not make sense to speak of their precise position.
But do photons in a perfect beam arrive at almost-regular intervals?
To the shame of physicists, the study of photon correlations was initiated by two astronomers, Robert Hanbury Brown and Richard Twiss, in 1956, and met with several years
of disbelief. They varied the transversal distance of the two detectors shown in Figure 24
– from a few to 188 m – and measured the intensity correlations between them. Hanbury
Brown and Twiss found that the intensity fluctuations within the volume of coherence
are correlated. Thus the photons themselves are correlated. With this experiment, they
were able to measure the diameter of numerous distant stars.
Inspired by the success of Hanbury Brown and Twiss, researchers developed a simple
method to measure the probability that a second photon in a light beam arrives at a given
time after the first one. They simply split the beam, put one detector in the first branch,
and varied the position of a second detector in the other branch. The set-up is sketched
in Figure 25. Such an experiment is nowadays called a Hanbury Brown Twiss experiment.

Motion Mountain – The Adventure of Physics

Page 49

2 light – and the quantum of action

2 light – and the quantum of action

53

The Hanbury Brown–Twiss experiment
light detector
D1

adjustable
position

light detector
D2

Page 63

coincidence
counter

F I G U R E 25 How
to measure
photon statistics
with an
electronic
intensity
correlator or
coincidence
counter, the
variation being
measured by
varying the
position of a
detector.

One finds that, for coherent light within the volume of coherence, the clicks in the two
counters – and thus the photons themselves – are correlated. To be more precise, such
experiments show that whenever the first photon hits, the second photon is most likely
to hit just afterwards. Thus, photons in light beams are bunched. Bunching is one of the
many results showing that photons are quantons, that they are indeed necessary to describe light, and that they are unlocalizable entities. As we will see below, the result also
implies that photons are bosons.
Every light beam has an upper time limit for bunching: the coherence time. For times
longer than the coherence time, the probability for bunching is low, and independent of
the time interval, as shown in Figure 25. The coherence time characterizes every light

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

incoming
light
beam

Motion Mountain – The Adventure of Physics

F I G U R E 24 The original experimental set-up with which Hanbury Brown and Twiss measured stellar
diameters at Narrabri in Australia. The distance between the two light collectors could be changed by
moving them on rails. The light detectors are at the end of the poles and each of them, as they wrote,
‘collected light as rain in a bucket.’ (© John Davis).

54

2 light – and the quantum of action

kinetic energy of
emitted electrons

Ekin
lamp
electrons

Ekin=h (ω−ωt)
threshold
metal plate
in vacuum

Page 49

ω

Are photons necessary?
In light of the results uncovered so far, the answer to the above question is obvious. But
the issue is tricky. In textbooks, the photoelectric effect is usually cited as the first and
most obvious experimental proof of the existence of photons. In 1887, Heinrich Hertz
observed that for certain metals, such as lithium or caesium, incident ultraviolet light
leads to charging of the metal. Later studies of the effect showed that the light causes
emission of electrons, and that the energy of the ejected electrons does not depend on
the intensity of the light, but only on the difference between ℏ times its frequency and
a material-dependent threshold energy. Figure 26 summarizes the experiment and the
measurements.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

beam. In fact, it is often easier to think in terms of the coherence length of a light beam.
For thermal light, the coherence length is only a few micrometres: a small multiple of
the wavelength. The largest coherence lengths, of over 300 000 km, are obtained with
research lasers that have an extremely narrow laser bandwith of just 1 Hz. Interestingly,
coherent light is even found in nature: several special stars have been found to emit it.
Although the intensity of a good laser beam is almost constant, the photons do not
arrive at regular intervals. Even the best laser light shows bunching, though with different statistics and to a lesser degree than lamp light, as illustrated in Figure 21. Light
whose photons arrive regularly, thus exhibiting so-called (photon) anti-bunching, is obviously non-classical in the sense defined above; such light can be produced only by special
experimental arrangements. Extreme examples of this phenomenon are being investigated at present by several research groups aiming to construct light sources that emit one
photon at a time, at regular time intervals, as reliably as possible. In short, we can state
that the precise photon statistics in a light beam depends on the mechanism of the light
source.
In summary, experiments force us to conclude that light is made of photons, but also
that photons cannot be localized in light beams. It makes no sense to talk about the
position of a photon in general; the idea makes sense only in some special situations,
and then only approximately and as a statistical average.

Motion Mountain – The Adventure of Physics

Ref. 21

frequency of lamp light

F I G U R E 26 The kinetic
energy of electrons
emitted in the
photoelectric effect.

2 light – and the quantum of action

Ref. 22

Ref. 23

Ref. 26

Page 43

Page 49
Ref. 27

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 25

In classical physics, the photoelectric effect is difficult to explain. But in 1905, Albert
Einstein deduced the measurements from the assumption that light is made of photons
of energy ? = ℏ?. He imagined that this energy is used partly to take the electron over
the threshold, and partly to give it kinetic energy. More photons only lead to more electrons, not to faster ones. In 1921, Einstein received the Nobel Prize for the explanation
of the photoelectric effect. But Einstein was a genius: he deduced the correct result by a
somewhat incorrect reasoning. The (small) mistake was the assumption that a classical,
continuous light beam would produce a different effect. In fact, it is easy to see that a
classical, continuous electromagnetic field interacting with discrete matter, made of discrete atoms containing discrete electrons, would lead to exactly the same result, as long as
the motion of electrons is described by quantum theory. Several researchers confirmed
this early in the twentieth century. The photoelectric effect by itself does not imply the
existence of photons.
Indeed, many researchers in the past were unconvinced that the photoelectric effect
shows the existence of photons. Historically, the most important argument for the necessity of light quanta was given by Henri Poincaré. In 1911 and 1912, aged 57 and only a few
months before his death, he published two influential papers proving that the radiation
law of black bodies – in which the quantum of action had been discovered by Max Planck
– requires the existence of photons. He also showed that the amount of radiation emitted
by a hot body is finite only because of the quantum nature of the processes leading to light
emission. A description of these processes in terms of classical electrodynamics would
lead to (almost) infinite amounts of radiated energy. Poincaré’s two influential papers
convinced most physicists that it was worthwhile to study quantum phenomena in more
detail. Poincaré did not know about the action limit ? ⩾ ℏ; yet his argument is based on
the observation that light of a given frequency has a minimum intensity, namely a single
photon. Such a one-photon beam may be split into two beams, for example by using a
half-silvered mirror. However, taken together, those two beams never contain more than
a single photon.
Another interesting experiment that requires photons is the observation of ‘molecules
of photons’. In 1995, Jacobson et al. predicted that the de Broglie wavelength of a packet of
photons could be observed. According to quantum theory, the packet wavelength is given
by the wavelength of a single photon divided by the number of photons in the packet.
The team argued that the packet wavelength could be observable if such a packet could
be split and recombined without destroying the cohesion within it. In 1999, this effect was
indeed observed by de Pádua and his research group in Brazil. They used a careful set-up
with a nonlinear crystal to create what they call a biphoton, and observed its interference
properties, finding a reduction in the effective wavelength by the predicted factor of two.
Since then, packages with three and even four entangled photons have been created and
observed.
Yet another argument for the necessity of photons is the above-mentioned recoil felt
by atoms emitting light. The recoil measured in these cases is best explained by the emission of a photon in a particular direction. In contrast, classical electrodynamics predicts
the emission of a spherical wave, with no preferred direction.
Obviously, the observation of non-classical light, also called squeezed light, also argues
for the existence of photons, as squeezed light proves that photons are indeed an intrinsic
aspect of light, necessary even when interactions with matter play no role. The same is

Motion Mountain – The Adventure of Physics

Ref. 24

55

56

2 light – and the quantum of action

lasers or other
coherent light source

pocket lamps

F I G U R E 27 Two situations in which light crosses light: different light sources lead to different results.

“

Die ganzen fünfzig Jahre bewusster Grübelei
haben mich der Antwort auf die Frage ‘Was
sind Lichtquanten?’ nicht näher gebracht.
Heute glaubt zwar jeder Lump er wisse es, aber
er täuscht sich.
Albert Einstein, 1951 *

”

If a light wave is made of particles, we must be able to explain each and every wave
property in terms of photons. The experiments mentioned above already hint that this is
possible only because photons are quantum particles. Let us take a more detailed look at
this connection.
Light can cross other light undisturbed, for example when the light beams from two
pocket lamps shine through each other. This observation is not hard to explain with

Ref. 28

* ‘Fifty years of conscious brooding have not brought me nearer to the answer to the question ‘What are
light quanta?’ Nowadays every bounder thinks he knows it, but he is wrong.’ Einstein wrote this a few years
before his death in a letter to Michele Besso.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Interference: how can a wave be made up of particles?

Motion Mountain – The Adventure of Physics

Challenge 40 s

true for the Hanbury Brown–Twiss effect.
Finally, the spontaneous decay of excited atomic states also requires the existence of
photons. This cannot be explained by a continuum description of light.
In summary, the concept of a photon is indeed necessary for a precise description
of light; but the details are often subtle, as the properties of photons are unusual and
require a change in our habits of thought. To avoid these issues, most textbooks stop
discussing photons after coming to the photoelectric effect. This is a pity, as it is only then
that things get interesting. Ponder the following. Obviously, all electromagnetic fields are
made of photons. At present, photons can be counted for gamma rays, X-rays, ultraviolet
light, visible light and infrared light. However, for lower frequencies, such as radio waves,
photons have not yet been detected. Can you imagine what would be necessary to count
the photons emitted from a radio station? This issue leads directly to the most important
question of all:

2 light – and the quantum of action

57

Vol. V, page 130

⊳ Photons behave like moving little arrows.
Some further thought leads to the following description:
—
—
—
—
—
—

The arrow is always perpendicular to the direction of motion.
The arrow’s direction stays fixed in space when the photons move.
The length of an arrow shrinks with the square of the distance travelled.
The probability of a photon arriving somewhere is given by the square of an arrow.
The final arrow is the sum of all the arrows arriving there by all possible paths.
Photons emitted by single-coloured sources are emitted with arrows of constant
length pointing in the direction ??; in other words, such sources spit out photons
with a rotating mouth.
— Photons emitted by incoherent sources – e.g., thermal sources, such as pocket lamps
– are emitted with arrows of constant length pointing in random directions.
Challenge 41 s

* If lasers are used, fringes can only be observed if the two beams are derived from a single beam by splitting,
or if two expensive high-precision lasers are used. (Why?)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

photons; since photons do not interact with each other, and are point-like, they ‘never’
hit each other. In fact, there is an extremely small positive probability for their interaction, as we will find out later, but this effect is not observable in everyday life.
But if two coherent light beams, i.e., two light beams of identical frequency and fixed
phase relation cross, we observe alternating bright and dark regions: so-called interference fringes. The schematic set-up is shown in Figure 27. Examples of actual interference
effects are given in Figure 28 and Figure 29. How do these interference fringes appear?*
How can it be that photons are not detected in the dark regions? We already know the
only possible answer: the brightness at a given place corresponds to the probability that
a photon will arrive there. The fringes imply:

Motion Mountain – The Adventure of Physics

F I G U R E 28 Examples of interference patterns that appear when coherent light beams cross: the
interference produced by a self-made parabolic telescope mirror of 27 cm diameter, and a speckle laser
pattern on a rough surface (© Mel Bartels, Epzcaw).

58

2 light – and the quantum of action

Motion Mountain – The Adventure of Physics

With this simple model* we can explain the wave behaviour of light. In particular, we
can describe the interference stripes seen in laser experiments, as shown schematically
in Figure 30. You can check that in some regions the two arrows travelling through the
two slits add up to zero for all times. No photons are detected there: those regions are
black. In other regions, the arrows always add up to the maximal value. These regions
are always bright. Regions in between have intermediate shades. Obviously, in the case
of usual pocket lamps, shown in the left-hand diagram of Figure 27, the brightness in the
common region also behaves as expected: the averages simply add up.
Obviously, the photon model implies that an interference pattern is built up as the sum
of a large number of single-photon hits. Using low-intensity beams, we should therefore
be able to see how these little spots slowly build up an interference pattern by accumulating in the bright regions and never hitting the dark regions. This is indeed the case, as
* The model gives a correct description of light except that it neglects polarization. To add polarization, it
is necessary to combine arrows that rotate in both senses around the direction of motion.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 29 Top: calculated interference patterns – and indistinguishable from observed ones under
ideal, “textbook” conditions – produced by two parallel narrow slits illuminated with green light and
with white light. Bottom: two Gaussian beams interfering at an angle (© Dietrich Zawischa, Rüdiger
Paschotta).

2 light – and the quantum of action

two lasers or
point sources

59

screen

S1
s

d
S2

the arrow model:
t1

t3
F I G U R E 30 Interference and the description of
light with arrows (at three instants of time).

we have seen earlier on. All experiments confirm this description.
In other words, interference is the superposition of coherent light fields or, more generally, of coherent electromagnetic fields. Coherent light fields have specific, more regular photon behaviour, than incoherent light fields. We will explore the details of photon
statistics in more detail shortly.
In summary, photons are quantum particles. Quantum particles can produce interference patterns – and all other wave effects – when they appear in large numbers, because
they are described by an arrow whose length squared gives the probability for its detection.
Interference of a single photon

Ref. 29

It is important to point out that interference between two light beams is not the result of
two different photons cancelling each other out or being added together. Such cancellation would contradict conservation of energy and momentum. Interference is an effect
applicable to each photon separately – as shown in the previous section – because each
photon is spread out over the whole set-up: each photon takes all possible paths. As Paul
Dirac stressed:
⊳ Each photon interferes only with itself.
Interference of a photon with itself only occurs because photons are quantons, and not
classical particles.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 43

Motion Mountain – The Adventure of Physics

t2

60

2 light – and the quantum of action

screen
source

image

mirror

Page 51

⊳ A photon interferes only within its volume of coherence. And in that
volume, it is impossible to distinguish photons.

Page 66

In the coherence volume formed by the longitudinal and transversal coherence length –
sometimes also called a phase space cell – we cannot completely say that light is a flow
of photons, because a flow cannot be defined in it. Despite regular claims to the contrary, Dirac’s statement is correct, as we will see below. It is a strange consequence of the
quantum of action.
R eflection and diffraction deduced from photon arrows
Waves also show diffraction. Diffraction is the change of propagation direction of light
or any other wave near edges. To understand this phenomenon with photons, let us start
with a simple mirror, and study reflection first. Photons (like all quantum particles) move
from source to detector by all possible paths. As Richard Feynman,* who discovered this
explanation, liked to stress, the term ‘all’ has to be taken literally. This is not a big deal in
* Richard (‘Dick’) Phillips Feynman (b. 1918 New York City, d. 1988 Los Angeles), physicist, was one
of the founders of quantum electrodynamics. He also discovered the ‘sum-over-histories’ reformulation of quantum theory, made important contributions to the theory of the weak interaction and to

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Dirac’s oft-quoted statement leads to a famous paradox: if a photon can interfere only
with itself, how can two laser beams from two different lasers interfere with each other?
The answer given by quantum physics is simple but strange: in the region where the
beams interfere – as mentioned above – it is impossible to say from which source a
photon has come. The photons in the crossing region cannot be said to come from a
specific source. Photons, also in the interference region, are quantons, and they indeed
interfere only with themselves.
Another description of the situation is the following:

Motion Mountain – The Adventure of Physics

arrow sum

F I G U R E 31 Light reflected by a
mirror, and the corresponding
arrows (at an instant of time).

2 light – and the quantum of action

source

61

point

arrow sum
at point
vanishes

usual
mirror

screen
source

arrow sum
at image
F I G U R E 32 The light reflected
by a badly-placed mirror and by
a grating.

quantum gravity, and co-authored a famous textbook, the Feynman Lectures on Physics, now online at www.
feynmanlectures.info. He is one of those theoretical physicists who made his career mainly by performing
complex calculations – but he backtracked with age, most successfully in his teachings and physics books,
which are all worth reading. He was deeply dedicated to physics and to enlarging knowledge, and was a
collector of surprising physical explanations. He helped building the nuclear bomb, wrote papers in topless bars, avoided to take any professional responsibility, and was famously arrogant and disrespectful of
authority. He wrote several popular books on the events of his life. Though he tried to surpass the genius of
Wolfgang Pauli throughout his life, he failed in this endeavour. He shared the 1965 Nobel Prize in Physics
for his work on quantum electrodynamics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

the explanation of interference. But in order to understand a mirror, we have to include
all possibilities, however crazy they seem, as shown in Figure 31.
As stated above, a light source emits rotating arrows. To determine the probability
that light arrives at a certain location within the image, we have to add up all the arrows
arriving at the same time at that location. For each path, the arrow orientation at the
image is shown – for convenience only – below the corresponding segment of the mirror.
The angle and length of the arriving arrow depends on the path. Note that the sum of all
the arrows does not vanish: light does indeed arrive at the image. Moreover, the largest
contribution comes from the paths near to the middle. If we were to perform the same
calculation for another image location, (almost) no light would get there.
In short, the rule that reflection occurs with the incoming angle equal to the outgoing
angle is an approximation, following from the arrow model of light. In fact, a detailed

Motion Mountain – The Adventure of Physics

striped
mirror

image

62

2 light – and the quantum of action

light beam

air
water

F I G U R E 33 If light were made of little stones, they would
move faster in water.

Challenge 43 e
Challenge 44 e
Challenge 45 e

All waves have a signal velocity. The signal velocity also depends on the medium in which
they propagate. As a consequence, waves show refraction when they move from one medium into another with different signal velocity. Interestingly, the naive particle picture
of photons as little stones would imply that light is faster in materials with high refractive
indices: the so-called dense materials. (See Figure 33.) Can you confirm this? However,
experiments show that light in dense materials moves slowly. The wave picture has no
difficulty explaining this observation. (Can you confirm this?) Historically, this was one
of the arguments against the particle theory of light. In contrast, the arrow model of light
presented above is able to explain refraction properly. It is not difficult: try it.
Waves also reflect partially from materials such as glass. This is one of the most difficult wave properties to explain with photons. But it is one of the few effects that is not
explained by a classical wave theory of light. However, it is explained by the arrow model,
as we will find out. Partial reflection confirms the first two rules of the arrow model. Par-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

R efraction and partial reflection from photon arrows

Motion Mountain – The Adventure of Physics

Challenge 42 s

calculation, with more arrows, shows that the approximation is quite precise: the errors
are much smaller than the wavelength of the light.
The proof that light does indeed take all these strange paths is given by a more specialized mirror. As show in Figure 32, we can repeat the experiment with a mirror that
reflects only along certain stripes. In this case, the stripes have been carefully chosen so
that the corresponding path lengths lead to arrows with a bias in one direction, namely
to the left. The arrow addition now shows that such a specialized mirror – usually called
a grating – allows light to be reflected in unusual directions. Indeed, this behaviour is
standard for waves: it is called diffraction. In short, the arrow model for photons allows
us to describe this wave property of light, provided that photons follow the ‘crazy’ probability scheme. Do not get upset! As was said above, quantum theory is the theory for
crazy people.
You may wish to check that the arrow model, with the approximations it generates
by summing over all possible paths, automatically ensures that the quantum of action is
indeed the smallest action that can be observed.

2 light – and the quantum of action

Page 57

63

tial reflection shows that photons indeed behave randomly: some are reflected and other
are not, without any selection criterion. The distinction is purely statistical. More about
this issue shortly.
From photons to waves

Ref. 30

Page 112

—
—
—
—
—
—
—
—
—

are not precisely countable – never with a precision better than √? ,
are not localizable – never with a precision better than the coherence length,
have no size, no charge and no (rest) mass,
show a phase that increases as ??, i.e., as the product of frequency and time,
carry spin 1,
of the same frequency are indistinguishable bosons – within a coherence volume,
can take any path whatsoever – as long as allowed by the boundary conditions,
have no discernable origin, and
have a detection probability given by the square of the sum of amplitudes* for all
allowed paths leading to the point of detection.

* The amplitude of a photon field, however, cannot and should not be identified with the wave function of
any massive spin 1 particle.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 31

Motion Mountain – The Adventure of Physics

Page 52

In waves, the fields oscillate in time and space. One way to show how waves can be made
of particles is to show how to build up a sine wave using a large number of photons. A
sine wave is a coherent state of light. The way to build them up was explained in detail by
Roy Glauber. In fact, to build a pure sine wave, we need a superposition of a beam with
one photon, a beam with two photons, a beam with three photons, and so on. Together,
they give a perfect sine wave. As expected, its photon number fluctuates to the highest
possible degree.
If we repeat the calculation for non-ideal beams, we find that the indeterminacy relation for energy and time is respected: every emitted beam will possess a certain spectral
width. Purely monochromatic light does not exist. Similarly, no system that emits a wave
at random can produce a monochromatic wave. All experiments confirm these results.
In addition, waves can be polarized. So far, we have disregarded this property. In the
photon picture, polarization is the result of carefully superposing beams of photons spinning clockwise and anticlockwise. Indeed, we know that linear polarization can be seen
as a result of superposing circularly-polarized light of both signs, using the proper phase.
What seemed a curiosity in classical optics turns out to be a fundamental justification for
quantum theory.
Finally, photons are indistinguishable. When two photons of the same colour cross,
there is no way to say afterwards which of the two is which. The quantum of action makes
this impossible. The indistinguishability of photons has an interesting consequence. It is
impossible to say which emitted photon corresponds to which arriving photon. In other
words, there is no way to follow the path of a photon, as we are used to following the
path of a billiard ball. Photons are indeed indistinguishable. In addition, the experiment
by Hanbury Brown and Twiss implies that photons are bosons. We will discover more
details about the specific indistinguishability of bosons later on.
In summary, we find that light waves can indeed be described as being built of
particles. However, this is only correct with the proviso that photons

64

2 light – and the quantum of action

In other words, light can be described as made of particles only if these particles have
special, quantum properties. These quantum properties differ from everyday particles
and allow photons to behave like waves whenever they are present in large numbers.
C an light move faster than light? – R eal and virtual photons

Challenge 46 ny

Challenge 47 ny

Challenge 48 s

Δ?Δ? ⩽ ℏ/2 .

Page 193

(14)

Rather than obeying the usual indeterminacy relation, they obey the opposite relation,
which expresses their very brief appearance. Despite their intrinsically short life, and despite the impossibility of detecting them directly, virtual particles have important effects.
We will explore them in detail shortly.
In fact, the vector potential ? allows four polarizations, corresponding to the four
coordinates (?, ?, ?, ?). It turns out that for the photons one usually talks about – the
free or real photons – the polarizations in the ? and ? directions cancel out, so that one

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 153

Motion Mountain – The Adventure of Physics

Ref. 32

In a vacuum, light can move faster than ?, as well as slower than ?. The quantum principle
provides the details. As long as this principle is obeyed, the speed of a short light flash
can differ – though only by a tiny amount – from the ‘official’ value. Can you estimate
the allowable difference in arrival time for a light flash coming from the dawn of time?
The arrow description for photons gives the same result. If we take into account the
crazy possibility that photons can move with any speed, we find that all speeds very different from ? cancel out. The only variation that remains, translated into distances, is the
indeterminacy of about one wavelength in the longitudinal direction, which we mentioned above.
In short, light, or real photons, can indeed move faster than light, though only by an
amount allowed by the quantum of action. For everyday situations, i.e., for high values of
the action, all quantum effects average out, including light and photon velocities different
from ?.
Not only the position, but also the energy of a single photon can be undefined. For
example, certain materials split one photon of energy ℏ? into two photons, whose two
energies add up to the original one. Quantum mechanics implies that the energy partitioning is known only when the energy of one of the two photons is measured. Only at
that very instant is the energy of the second photon known. Before the measurement,
both photons have undefined energies. The process of energy fixing takes place instantaneously, even if the second photon is far away. We will explain below the background
to this and similar strange effects, which seem to be faster than light. In fact, despite the
appearance, these observations do not involve faster-than-light transmission of energy
or information.
More bizarre consequences of the quantum of action appear when we study static electric fields, such as the field around a charged metal sphere. Obviously, such a field must
also be made of photons. How do they move? It turns out that static electric fields are
made of virtual photons. Virtual photons are photons that do not appear as free particles:
they only appear for an extremely short time before they disappear again. In the case of
a static electric field, they are longitudinally polarized, and do not carry energy away.
Virtual photons, like other virtual particles, are ‘shadows’ of particles that obey

2 light – and the quantum of action

Vol. V, page 122
Vol. II, page 72

65

Indeterminacy of electric fields

?? = ? (? + ? × ?) .

Challenge 49 ny

(15)

Since it is impossible to measure both the momentum and the position of a particle, they
deduced an indeterminacy for the electrical field, given by
Δ? =

ℏ
,
? Δ? ?

(16)

where ? is the measurement time and Δ? is the position indeterminacy. Thus every value
of an electric field, and similarly of a magnetic field, possesses an indeterminacy. The
state of the electromagnetic field behaves like the state of matter in this respect: both
follow an indeterminacy relation.
How can virtual photon exchange lead to at traction?
Exchange of real photons always leads to recoil. But exchange of virtual photons can lead
either to attraction or repulsion, depending on the signs of the two charges involved. This
is worth looking at.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 33

We have seen that the quantum of action implies an indeterminacy for light intensity.
Since light is an electromagnetic wave, this indeterminacy implies similar, separate limits
for electric and magnetic fields at a given point in space. This conclusion was first drawn
in 1933 by Bohr and Rosenfeld. They started from the effects of the fields on a test particle
of mass ? and charge ?, which are described by:

Motion Mountain – The Adventure of Physics

observes only the ? and ? polarizations in actual experiments.
For bound or virtual photons, the situation is different. All four polarizations are possible. Indeed, the z and t polarizations of virtual photons – which do not appear for real
photons, i.e., for free photons – are the ones that can be said to be the building blocks of
static electric and magnetic fields.
In other words, static electric and magnetic fields are continuous flows of virtual
photons. In contrast to real photons, virtual photons can have mass, can have spin directions not pointing along the path of motion, and can have momentum opposite to
their direction of motion. Exchange of virtual photons leads to the attraction of bodies
of different charge. In fact, virtual photons necessarily appear in any description of electromagnetic interactions. Later on we will discuss their effects further – including the
famous attraction of neutral bodies.
We have seen already early on that virtual photons, for example those that are needed
to describe collisions of charges, must be able to move with speeds higher than that of
light. This description is required in order to ensure that the speed of light remains a
limit in all experiments.
In summary, it might be intriguing to note that virtual photons, in contrast to real
photons, are not bound by the speed of light; but it is also fair to say that virtual photons
move faster-than-light only in a formal sense.

66

2 light – and the quantum of action

Page 59

Ref. 29

Ref. 34

In 1930, Paul Dirac made the famous statement already mentioned earlier on:
⊳ Each photon interferes only with itself. Interference between two different
photons never occurs.
Often this statement is misinterpreted as implying that light from two separate photon
sources cannot interfere. Unfortunately, this false interpretation has spread through a part
of the literature. Everybody can check that this statement is incorrect with a radio: signals
from two distant radio stations transmitting on the same frequency lead to beats in amplitude, i.e., to wave interference. (This should not to be confused with the more common
radio interference, which usually is simply a superposition of intensities.) Radio transmitters are coherent sources of photons, and any radio receiver shows that signals form
two different sources can indeed interfere.
In 1949, interference of fieds emitted from two different photon sources has been

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

C an t wo photons interfere?

Motion Mountain – The Adventure of Physics

We start with two localized charges of same sign, located both on the ?-axis, and want
to determine the momentum transferred from the charge on the right side via a virtual
photon to the charge on the left side.
For the virtual photon, the important part of its state in momentum space is its imaginary part, which, if emitted by a negative charge, has a positive peak (delta function shape)
at the negative of its momentum value and a negative peak at its positive momentum
value.
When the virtual photon hits the other charged particle, on the left, it can push it
either to the left or to the right. The probability amplitude for each process is given by
the particle charge times the photon momentum state value times ? times time. Both
amplitudes need to be added.
In the case that the second particle has the same charge as the first, the effect of the
virtual photon absorption in momentum space is to add a wave function that originally
was antisymmetric and positively valued on the positive axis, and that is then shifted
to the left, to a second wave function which originally was the negative of the first, but
is then shifted to the right. The result for this one-photon absorption process is a realvalued, antisymmetric function in momentum space, with positive values for negative
momenta, and negative values for positive momenta.
To understand repulsion, we need to add the wave function for this one-photon process to the zero-photon (thus unmodified) function of the second particle, and then
square the sum. This unmodified function was positive in the case of same charges. The
squaring process of the sum yields a probability distribution in momentum space whose
maximum is at a negative momentum value; thus the second particle has been repelled
from the first.
If the charges had different signs, the maximum of the sum would be at a positive momentum value, and the second particle would be attracted to the first. In short, attraction
or repulsion is determined by the interference between the wave function for one-photon
absorption (more precisely, for odd-photon-number absorption) and the wave function
for zero-photon absorption (more precisely, for even-photon-number absorption).

2 light – and the quantum of action

Ref. 35

67

demonstrated also with microwave beams. From the nineteen fifties onwards, numerous
experiments with two lasers and even with two thermal light sources have shown light
interference. For example, in 1963, Magyar and Mandel used two ruby lasers emitting
light pulses and a rapid shutter camera to produce spatial interference fringes.
However, all these experimental results with two interfering sources do not contradict
the statement by Dirac. Indeed, two photons cannot interfere for several reasons.

⊳ A photon interferes only within its volume of coherence, i.e., within its own
cell of phase space. Outside, there is no interference. And inside that volume
or cell, it is impossible to distinguish photons, states or histories.
The concept of ‘photon’ remains deep even today. The quantum particle model of coherence and light remains fascinating to this day. Summarizing, we can say: Two different
electromagnetic beams can interfere, but two different photons cannot.
Curiosities and fun challenges ab ou t photons
Can one explain refraction with photons? Newton was not able to do so, but today we
can. In refraction by a horizontal surface, as shown in Figure 34, the situation is translationally invariant along the horizontal direction. Therefore, the momentum component
along this direction is conserved: ?1 sin ?1 = ?2 sin ?2 . The photon energy ? = ?1 = ?2
is obviously conserved. The index of refraction ? is defined in terms of momentum and

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The statement of Dirac thus depends on the definition of the term ‘photon’. A better
choice of words is to say that interference is always between two (indistinguishable) histories, but never between two quantum particles. Or, as expressed above:

Motion Mountain – The Adventure of Physics

— Interference is a result of the space-time propagation of waves; photons appear only
when the energy–momentum picture is used, mainly when interaction with matter
takes place. The description of space-time propagation and the particle picture are
mutually exclusive – this is one aspect of the complementary principle. Why does
Dirac seem to mix the two in his statement? Dirac employs the term ‘photon’ in a
very general sense, as quantized state of the electromagnetic field. When two coherent beams are superposed, the quantized entities, the photons, cannot be ascribed to
either of the sources. Interference results from superposition of two coherent states,
not of two particles.
— Interference is only possible if one cannot know where the detected photon comes
from. The quantum mechanical description of the field in a situation of interference
never allows ascribing photons of the superposed field to one of the sources. In other
words, if it is possible to say from which source a detected photon comes from, interference cannot be observed.
— Interference between two coherent beams requires a correlated or fixed phase
between them, i.e., an undetermined particle number; in other words, interference
is possible if and only if the photon number for each of the two beams is unknown.
And a beam has an unknown photon number when the number indeterminacy is of
similar size as the average number.

68

2 light – and the quantum of action

p1

α1

air

water
p2
α2
F I G U R E 34 Refraction and photons.

energy as

Challenge 50 e

??
.
?

(17)

The ‘law’ of refraction follows:
sin ?1
=?.
sin ?2

(18)

∗∗
If an electromagnetic wave has amplitude ?, the photon density ? is
?=
Challenge 51 ny

?2
.
ℏ?

(19)

Can you show this?
∗∗

Challenge 52 e

Show that for a laser pulse in vacuum, the coherence volume increases during propagation, whereas the volume occupied in phase space remains constant. Its entropy is constant, as its path is reversible.
∗∗

Challenge 53 s

A typical effect of the quantum ‘laws’ is the yellow colour of the lamps used for street
illumination in most cities. They emit pure yellow light of (almost) a single frequency;
that is why no other colours can be distinguished in their light. According to classical
electrodynamics, harmonics of that light frequency should also be emitted. Experiments
show, however, that this is not the case; classical electrodynamics is thus wrong. Is this
argument correct?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The relation is known since the middle ages.
There is an important issue here. In a material, the velocity of a photon ? = ??/??
in a light ray differs from the phase velocity ? = ?/? that enters into the calculation. In
summary, inside matter, the concept of photon must be used with extreme care.

Motion Mountain – The Adventure of Physics

?=

2 light – and the quantum of action

69

∗∗

∗∗

Page 171
Challenge 55 e

What happens to photons that hit an object but are not absorbed or transmitted? Generally speaking, they are scattered. Scattering is the name for any process that changes the
motion of light (or that of any other wave). The details of the scattering process depend
on the object; some scattering processes only change the direction of motion, others also
change the frequency. Table 3 gives an overview of processes that scatter light.
All scattering properties depend on the material that produces the deflection of light.
Among others, the study of scattering processes explains many colours of transparent
materials, as we will see below.
We note that the bending of light due to gravity is not called scattering. Why not?
A summary on light : particle and wave

Ref. 36

In summary, light is a stream of light quanta or photons. A single photon is the smallest
possible light intensity of a given colour. Photons, like all quantons, are quite different
from everyday particles. In fact, we can argue that the only (classical) particle aspects of
photons are their quantized energy, momentum and spin. In all other respects, photons
are not like little stones. Photons move with the speed of light. Photons cannot be localized in light beams. Photons are indistinguishable. Photons are bosons. Photons have no
mass, no charge and no size. It is more accurate to say that photons are calculating devices
to precisely describe observations about light.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 54 ny

How can you check whether a single-photon-triggered bomb is functional without exploding it? This famous puzzle, posed by Avshalom Elitzur and Lev Vaidman, requires
interference for its solution. Can you find a way?

Motion Mountain – The Adventure of Physics

F I G U R E 35 The blue shades of the sky and the colours
of clouds are due to various degrees of Rayleigh, Mie
and Tyndall scattering (© Giorgio di Iorio).

70

2 light – and the quantum of action

TA B L E 3 Types of light scattering.

S c at t e r i n g
type

S c at t e r e r

D e ta i l s

Examples

Rayleigh scattering

atoms, molecules

blue sky, red evening
sky, blue cigarette
smoke

Mie scattering

transparent objects,
droplets

elastic, intensity
changes as 1/?4 ,
scatterers smaller
than ?/10
elastic, intensity
changes as 1/?0.5 to
1/?2 , scatterer size
around ?
elastic, scatterer size
larger than ?

Geometric scattering edges

Smekal–Raman
scattering
Inverse Raman
scattering
Thomson scattering

Brillouin scattering

Von Laue or X-ray
scattering

The strange properties of photons are the reason why earlier attempts to describe
light as a stream of (classical) particles, such as the attempt of Newton, failed miserably,
and were rightly ridiculed by other scientists. Indeed, Newton upheld his theory against
all experimental evidence – especially with regard to light’s wave properties – which is
something that a physicist should never do. Only after people had accepted that light is
a wave, and then discovered and understood that quantum particles are fundamentally
different from classical particles, was the quanton description successful.
The quantum of action implies that all waves are streams of quantons. In fact, all waves

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Compton scattering

better called
diffraction, used in
interference
non-transparent objects
elastic, angle weakly smog, white clouds,
or not wavelengthfog, white cigarette
dependent
smoke
excited atoms, molecules inelastic, light gains used in lidar
energy
investigations of the
atmosphere
atoms, molecules
inelastic, light loses used in material
energy
research
electrons
elastic
used for electron
density
determination
electrons
inelastic, X-ray lose proves particle
energy
nature of light (see
page 46)
acoustic phonons, density inelastic, frequency used to study
variations in solids/fluids shift of a few GHz
phonons and to
diagnose optical
fibres
crystalline solids
elastic, due to
used to determine
interference at
crystal structures;
crystal planes
also called Bragg
diffraction

Motion Mountain – The Adventure of Physics

Tyndall scattering

blue sky, red
evenings, blue
distant mountains

2 light – and the quantum of action

71

are correlated streams of quantons. This is true for light, for any other form of radiation,
and for all forms of matter waves.
The indeterminacy relations show that even a single quanton can be regarded as a
wave; however, whenever it interacts with the rest of the world, it behaves as a particle. In
fact, it is essential that all waves be made of quantons: if they were not, then interactions
would be non-local, and objects could not be localized at all, contrary to experience.
To decide whether the wave or the particle description is more appropriate, we can
use the following criterion. Whenever matter and light interact, it is more appropriate to
describe electromagnetic radiation as a wave if the wavelength ? satisfies
?≫

(20)

where ? = 1.4 ⋅ 10−23 J/K is Boltzmann’s constant and ? is the temperature of the
particle. If the wavelength is much smaller than the quantity on the right-hand side, the
particle description is most appropriate. If the two sides are of the same order of magnitude, both descriptions play a role. Can you explain the criterion?

Motion Mountain – The Adventure of Physics

Challenge 56 e

ℏ?
,
??

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Chapter 3

MOT ION OF M AT T E R – BEYON D
C L A S SIC A L PH YSIC S

”

he existence of a smallest action has numerous important consequences for
he motion of matter. We start with a few experimental results that show
hat the quantum of action is indeed the smallest measurable action value, also in
the case of matter. Then we show that the quantum of action implies the existence of a
phase and thus of the wave properties of matter. Finally, from the quantum of action, we
deduce for the motion of matter the same description that we already found for light:
matter particles behave like rotating arrows.
Wine glasses, pencils and atoms – no rest
Cicero, De oratore.

”

If the quantum of action is the smallest observable change in a physical system, then two
observations of the same system must always differ. Thus there cannot be perfect rest in
nature. Is that true? Experiments show that this is indeed the case.
A simple consequence of the lack of perfect rest is the impossibility of completely
filling a glass of wine. If we call a glass at maximum capacity (including surface tension
effects, to make the argument precise) ‘full’, we immediately see that the situation requires the liquid’s surface to be completely at rest. This is never observed. Indeed, a completely quiet surface would admit two successive observations that differ by less than ℏ.
We could try to reduce all motions by reducing the temperature of the system. To achieve
absolute rest we would need to reach absolute zero temperature. Experiments show that
this is impossible. (Indeed, this impossibility, the so-called third ‘law’ of thermodynamics, is equivalent to the existence of a minimum action.) All experiments confirm: There
is no rest in nature. In other words, the quantum of action proves the old truth that a
glass of wine is always partially empty and partially full.
The absence of microscopic rest, predicted by the quantum of action, is confirmed
in many experiments. For example, a pencil standing on its tip cannot remain vertical,
as shown in Figure 36, even if it is isolated from all disturbances, such as vibrations, air
molecules and thermal motion. This – admittedly very academic – conclusion follows
** ‘Rest with dignity.’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

“

Otium cum dignitate.**

Motion Mountain – The Adventure of Physics

T

“

All great things begin as blasphemies.
George Bernard Shaw

3 motion of matter – beyond classical physics

73

?

axis

Page 79

from the indeterminacy relation. In fact, it is even possible to calculate the time after
which a pencil must have fallen over. In practice however, pencils fall over much earlier,
because in usual conditions, external disturbances are much larger than the effects of the
quantum of action.
But the most important consequence of the absence of rest is another. The absence
of rest for the electrons inside atoms prevents them from falling into the nuclei, despite
their mutual attraction. In other words, the existence and the size of atoms, and thus of
all matter, is a direct consequence of the absence of microscopic rest! We will explore
this consequence in more detail below. Since we are made of atoms, we can say: we only
exist and live because of the quantum of action.

Not only does the quantum of action prevent the existence of rest; the quantum of action also prevents the observation or measurement of rest. In order to check whether
an object is at rest, we need to observe its position with high precision. Because of the
wave properties of light, we need a high-energy photon: only a high-energy photon has
a small wavelength and thus allows a precise position measurement. As a result of this
high energy, however, the object is disturbed. Worse, the disturbance itself is not precisely measurable; so there is no way to determine the original position even by taking
the disturbance into account. In short, perfect rest cannot be observed – even if it existed.
Indeed, all experiments in which systems have been observed with highest precision
confirm that perfect rest does not exist. The absence of rest has been confirmed for electrons, neutrons, protons, ions, atoms, molecules, atomic condensates and crystals. The
absence of rest has been even confirmed for objects with a mass of about a tonne, as
used in certain gravitational wave detectors. No object is ever at rest.
The same argument on measurement limitations also shows that no measurement, of
any observable, can ever be performed to infinite precision. This is another of the farreaching consequences of the quantum of action.
C o ol gas
The quantum of action implies that rest is impossible in nature. In fact, even at extremely
low temperatures, all particles inside matter are in motion. This fundamental lack of rest

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

No infinite measurement precision

Motion Mountain – The Adventure of Physics

Challenge 57 d

F I G U R E 36 A falling pencil.

74

Challenge 58 s
Ref. 37
Challenge 59 s

3 motion of matter – beyond classical physics

is said to be due to the so-called zero-point fluctuations. A good example is provided by
the recent measurements of Bose–Einstein condensates. They are trapped gases, with a
small number of atoms (between ten and a few million), cooled to extremely low temperatures (around 1 nK). The traps allow to keep the atoms suspended in mid-vacuum.
These cool and trapped gases can be observed with high precision. Using elaborate experimental techniques, Bose–Einstein condensates can be put into states for which Δ?Δ?
is almost exactly equal to ℏ/2 – though never lower than this value. These experiments
confirm directly that there is no observable rest, but a fundamental fuzziness in nature.
And the fuzziness is described by the quantum of action.
This leads to an interesting puzzle. In a normal object, the distance between the atoms
is much larger than their de Broglie wavelength. (Can you confirm this?) But today it is
possible to cool objects to extremely low temperatures. At sufficiently low temperatures,
less than 1 nK, the wavelength of the atoms may be larger than their separation. Can you
imagine what happens in such cases?

“

”

Die Bewegung ist die Daseinsform der Materie.
Friedrich Engels, Anti-Dühring.*

Fluid flows and quantons
Vol. I, page 390

The flow of matter also exhibits smallest units. We mentioned early on in our adventure
that a consequence of the particulate structure of liquids is that oil or any other smooth
liquid produces noise when it flows through even the smoothest of pipes. We mentioned
that the noise we hear in our ears in situations of absolute silence – for example, in a
snowy and windless landscape in the mountains or in an anechoic chamber – is partly
due to the granularity of blood flow in the veins. All experiments confirm that all flows
of matter produce vibrations. This is a consequence of the quantum of action, and of the
resulting granularity of matter. In fact, the quantum of action can be determined from
noise measurements in fluids.
Kno cking tables and quantized conductivit y

Ref. 38, Ref. 39
Ref. 12

If electrical current were a continuous flow, it would be possible to observe action values
as small as desired. The simplest counter-example was discovered in 1996, by José CostaKrämer and his colleagues. They put two metal wires on top of each other on a kitchen
* ‘Motion is matter’s way of being.’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Not only does the quantum of action make rest impossible, it also makes impossible any
situation that does not change in time. The most important examples of (apparently) stationary situations are flows. The quantum of action implies that no flow can be stationary.
More precisely, a smallest action implies that no flow can be continuous. All flows fluctuate. In nature, all flows are made of smallest entities: all flows are made of quantum
particles. We saw above that this is valid for light; it also applies to matter flows. Two
simple kinds of flow from our everyday experience directly confirm this consequence
from the quantum of action: flows of fluids and flows of electricity.

Motion Mountain – The Adventure of Physics

Flows and the quantization of mat ter

3 motion of matter – beyond classical physics

F I G U R E 37 Steps in the flow of electricity in metal wire crossings: the set-up, the nanowires at the
basis of the effect, and three measurement results (© José Costa-Krämer, AAPT from Ref. 39).

Motion Mountain – The Adventure of Physics

table and attached a battery, a current-voltage converter – or simply a resistor – and a
storage oscilloscope to them. Then they measured the electrical current while knocking
on the table. That is all.
Knocking the table breaks the contact between the two wires. In the last millisecond
before the wires detach, the conductivity and thus the electrical current diminishes in
regular steps of about 7 μA, as can easily be seen on the oscilloscope. Figure 37 shows
such a measurement. This simple experiment could have beaten, if it had been performed
a few years earlier, a number of other, enormously expensive experiments which discovered this same quantization at costs of several million euro each, using complex setups at extremely low temperatures.
In fact, the quantization of conductivity appears in any electrical contact with a small
cross-section. In such situations the quantum of action implies that the conductivity can
only be a multiple of 2?2 /ℏ ≈ (12 906 Ω)−1 . Can you confirm this result? Note that electrical conductivity can be as small as required; only the quantized electrical conductivity
has the minimum value of 2?2 /ℏ.
Many more elaborate experiments confirm the observation of conductance steps.
They force us to conclude that there is a smallest electric charge in nature. This smallest
charge has the same value as the charge of an electron. Indeed, electrons turn out to be
part of every atom, in a construction to be explained shortly. In metals, a large number
of electrons can move freely: that is why metals conduct electricity so well and work as

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 60 e

75

76

3 motion of matter – beyond classical physics

F I G U R E 38 Electrons beams
diffract and interfere at multiple
slits (© Claus Jönsson).

Ref. 40

In 1923 and 1924, the influential physicist Louis de Broglie pondered the consequences
of the quantum of action for matter particles. He knew that in the case of light, the
quantum of action connects wave behaviour to particle behaviour. He reasoned that the
same should apply to matter. It dawned to him that streams of matter particles with the
same momentum should behave as waves, just as streams of light quanta do. He thus predicted that like for light, coherent matter flows should have a wavelength ? and angular
frequency ? given by
?
2π ℏ
and ? = ,
(21)
?=
?
ℏ
where ? and ? are the momentum and the energy, respectively, of the single particles.
Equivalently, we can write the relations as
? = ℏ?

and ? = ℏ? .

(22)

All these relations state that matter quantons also behave as waves. For everyday objects,
the predicted wavelength is unmeasurably small – though not for microscopic particles.
Soon after de Broglie’s prediction, experiments began to confirm it. Matter streams

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Mat ter quantons and their motion – mat ter waves

Motion Mountain – The Adventure of Physics

mirrors.
In short, matter and electricity flow in smallest units. Depending on the flowing
material, the smallest flowing units of matter may be ‘molecules’, ‘atoms’, ‘ions’, or
‘electrons’. All of them are quantum particles, or quantons. In short, the quantum of
action implies that matter is made of quantons. Matter quantons share some properties
with ordinary stones, but also differ from them in many ways. A stone has position and
momentum, mass and acceleration, size, shape, structure, orientation and angular momentum, and colour. We now explore each of these properties for quantons, and see how
they are related to the quantum of action.

3 motion of matter – beyond classical physics

77

Vol. V, page 142
Ref. 42
Vol. III, page 102

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 41

were observed to diffract, refract and interfere; and all observations matched the values
predicted by de Broglie. Because of the smallness of the wavelength of quantons, careful
experiments are needed to detect these effects. But one by one, all experimental confirmations of the wave properties of light were repeated for matter beams. For example, just as
light is diffracted when it passes around an edge or through a slit, matter is also diffracted in these situations. This is true even for electrons, the simplest particles of everyday
matter, as shown in Figure 38. In fact, the experiment with electrons is quite difficult. It
was first performed by Claus Jönsson in Tübingen in 1961; in the year 2002 it was voted
the most beautiful experiment in all of physics. Many years after Jönsson, the experiment
was repeated with a modified electron microscope, as shown in Figure 39.
Inspired by light interferometers, researchers began to build matter interferometers.
Matter interferometers have been used in many beautiful experiments, as we will find
out. Today, matter interferometers work with beams of electrons, nucleons, nuclei, atoms,
or even large molecules. Just as observations of light interference prove the wave character of light, so the interference patterns observed with matter beams prove the wave
character of matter. They also confirm the value of ℏ.
Like light, matter is made of particles; like light, matter behaves as a wave when large
numbers of particles with the same momentum are involved. But although beams of large

Motion Mountain – The Adventure of Physics

F I G U R E 39 Formation over time of the interference pattern of electrons, here in a low-intensity
double-slit experiment: (a) 8 electrons, (b) 270 electrons, (c) 2000 electrons, (d) 6000 electrons, after 20
minutes of exposure. The last image corresponds to the situation shown in the previous figure.
(© Tonomura Akira/Hitachi).

78

3 motion of matter – beyond classical physics

molecules behave as waves, everyday objects – such as cars on a motorway – do not.
There are several reasons for this. First, for cars on a motorway the relevant wavelength
is extremely small. Secondly, the speeds of the cars vary too much. Thirdly, cars can be
counted. In summary, streams of cars with the same speed cannot be made coherent.
If matter behaves like a wave, we can draw a strange conclusion. For any wave, the
position and the wavelength cannot both be sharply defined simultaneously: the indeterminacies of the wave number ? = 2π/? and of the position ? obey the relation
Δ?Δ? ≥

1
.
2

(23)

Similarly, for every wave the angular frequency ? = 2π? and the instant ? of its peak
amplitude cannot both be sharply defined. Their indeterminacies are related by
1
.
2

(24)

Using de Broglie’s wave properties of matter (22), we get
Δ?Δ? ⩾

Page 193

and

Δ?Δ? ⩾

ℏ
.
2

(25)

These famous relations are called Heisenberg’s indeterminacy relations. They were discovered by Werner Heisenberg in 1925. They are valid for all quantum particles, be they
matter or radiation. The indeterminacy relations state that there is no way to simultaneously ascribe a precise momentum and position to a quantum system, nor to simultaneously ascribe a precise energy and age. The more accurately one quantity is known, the
less accurately the other is.* As a result, matter quantons – rather like stones – can always
be localized, but always only approximately. On the other hand, we saw that photons often cannot be localized.
Both indeterminacy relations have been checked experimentally in great detail. All
experiments confirm them. In fact, every experiment proving that matter behaves like a
wave is a confirmation of the indeterminacy relation – and vice versa.
When two variables are linked by indeterminacy relations, one says that they are complementary to each other. Niels Bohr systematically explored all possible such pairs. You
can also do that for yourself. Bohr was deeply fascinated by the existence of a complementarity principle, and he later extended it in philosophical directions. In a wellknown scene, somebody asked him what was the quantity complementary to precision.
He answered: ‘clarity’.
We remark that the usual, real, matter quantons always move more slowly than light.
Due to the inherent fuzziness of quantum motion, it should not come to a surprise that
exceptions exist. Indeed, in some extremely special cases, the quantum of action allows
the existence of particles that move faster than light – so-called virtual particles – which
we will meet later on.
* A policeman stopped the car being driven by Werner Heisenberg. ‘Do you know how fast you were driving?’ ‘No, but I know exactly where I was!’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 61 s

ℏ
2

Motion Mountain – The Adventure of Physics

Δ?Δ? ≥

3 motion of matter – beyond classical physics

79

In summary, the quantum of action means that matter quantons do not behave like
point-like stones, but as waves. In particular, like for waves, the values of position and
momentum cannot both be exactly defined for quantons. The values are fuzzy – position
and momentum are undetermined. The more precisely one of the two is known, the less
precisely the other is known.
Mass and acceleration of quantons

Vol. III, page 30
Challenge 62 s

Matter quantons, like stones, have mass. Indeed, hits by single electrons, atoms or molecules can be detected, if sensitive measurement set-ups are used. Quantons can also
be slowed down or accelerated. We have already explored some of these experiments in
the section on electrodynamics. However, quantons differ from pebbles. Using the time–
energy indeterminacy relation, you can deduce that

Ref. 43

2??3
.
ℏ

(26)

Thus there is a maximum acceleration for quantons.* Indeed, no particle has ever been
observed with a higher acceleration than this value. In fact, no particle has ever been
observed with an acceleration anywhere near this value. The quantum of action thus
prevents rest but also limits acceleration.
Why are atoms not flat? Why d o shapes exist?

Ref. 45
Page 181

Challenge 64 e

* We note that this acceleration limit is different from the acceleration limit due to general relativity:
?⩽

Challenge 63 e

?4
.
4??

(27)

In particular, the quantum limit (26) applies to microscopic particles, whereas the general-relativistic limit
applies to macroscopic systems. Can you confirm that in each domain the relevant limit is the smaller of
the two?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 44

The quantum of action determines all sizes in nature. In particular, it determines all
shapes. Let us start to explore this topic.
Experiments show that all composed quantons, such as atoms or molecules, have
structures of finite size and often with complex shape. The size and the shape of every
composed quanton are due to the motion of their constituents. The motion of the constituents is due to the quantum of action; but how do they move?
In 1901, Jean Perrin and independently, in 1904, Nagaoka Hantaro proposed that
atoms are small ‘solar systems’. In 1913, Niels Bohr used this idea, combining it with
the quantum of action, and found that he could predict the size and the colour of hydrogen atoms, two properties that had not until then been understood. We will perform
the calculations below. Even Bohr knew that the calculations were not completely understood, because they seemed to assume that hydrogen atoms were flat, like the solar
system is. But first of all, atoms are observed to be spherical. Secondly, a flat shape would
contradict the quantum of action. Indeed, the quantum of action implies that the motion of quantum constituents is fuzzy. Therefore, all composed quantons, such as atoms
or molecules, must be made of clouds of constituents.

Motion Mountain – The Adventure of Physics

?⩽

80

3 motion of matter – beyond classical physics

In short, the quantum of action predicts:
⊳ Atoms are spherical clouds.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Experiment and theory confirm that the shape of any atom is due to the cloud, or probability distribution, of its lightest components, the electrons. The quantum of action thus
states that atoms or molecules are not hard balls, as Democritus or Dalton believed, but
that they are clouds. Matter is made of clouds.
Atomic electron clouds are not infinitely hard, but can to a certain degree interpenetrate and be deformed. The region where this deformation occurs is called a chemical
bond. Bonds lead to molecules. Molecules, being composed of atoms, are composed of
(deformed) spherical clouds. Bonds also lead to liquids, solids, flowers and people. A detailed exploration confirms that all shapes, from the simplest molecules to the shape of
people, are due to the interactions between electrons and nuclei of the constituent atoms.
Nowadays, molecular shapes can be calculated to high precision. Small molecules, like
water, have shapes that are fairly rigid, though endowed with a certain degree of elasticity. Large molecules, such as polymers or peptides, have flexible shapes. These shape
changes are essential for their effects inside cells and thus for our survival. A large body
of biophysical and biochemical research is exploring molecular shape effects.
In summary, the quantum of action implies that shapes exist – and that they fluctuate.
For example, if a long molecule is held fixed at its two ends, it cannot remain at rest in
between. Such experiments are easy to perform nowadays, for example with DNA. In
fact, all experiments confirm that the quantum of action prevents rest, produces sizes
and shapes, and enables chemistry and life.
In nature, all sizes and shapes are due to the quantum of action. Now, every macroscopic object and every quantum object with a non-spherical shape is able to rotate. We
therefore explore what the quantum of action can say about rotation.

Motion Mountain – The Adventure of Physics

F I G U R E 40 Probability clouds: a hydrogen atom in its spherical ground state (left) and in a
non-spherical excited state (right) as seen by an observer travelling around it (QuickTime film produced
with Dean Dauger’s software package ‘Atom in a Box’, available at daugerresearch.com).

3 motion of matter – beyond classical physics

81

?

?

source

?

Rotation, quantization of angular momentum, and the lack of
north poles

“

Challenge 65 e

”

In everyday life, rotation is a frequent type of motion. Wheels are all around us. It turns
out that the quantum of action has important consequences for rotational motion. First
of all, we note that action and angular momentum have the same physical dimension:
both are measured in Js or Nms. It only takes a little thought to show that if matter or
radiation has a momentum and wavelength related by the quantum of action, then angular momentum is fixed in multiples of the quantum of action. This beautiful argument
is due to Dicke and Wittke.
Imagine a circular fence, made of ? vertical steel bars spaced apart at a distance
? = 2π?/?, as shown in Figure 41. At the centre of the fence, imagine a source of matter
or radiation that can emit particles towards the fence in any chosen direction. The linear
momentum of such a particle is ? = ℏ? = 2πℏ/?. At the fence slits, the wave will interfere. Outside the fence, the direction of the motion of the particle is determined by the
condition of positive interference. In other words, the angle ?, describing the direction of
motion outside the fence, is given by ? sin ? = ??, where ? is an integer. Through the
deflection due to the interference process, the fence receives a linear momentum ? sin ?,
or an angular momentum ? = ?? sin ?. Combining all these expressions, we find that
the angular momentum transferred to the fence is
? = ??ℏ .

(28)

* ‘Sad is that disciple who does not surpass his master.’ This statement from one of his notebooks, the
Codice Forster III, is sculpted in large letters in the chemistry aula of the University of Rome La Sapienza.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 46

Tristo è quel discepolo che non avanza il suo
maestro.
Leonardo da Vinci*

Motion Mountain – The Adventure of Physics

F I G U R E 41 The quantization of
angular momentum.

82

3 motion of matter – beyond classical physics

In other words, the angular momentum of the fence is an integer multiple of ℏ. Fences
can only have integer intrinsic angular momenta (in units of ℏ). The generalization of
the argument to all bodies is also correct. (Of course, this latter statement is only a hint,
not a proof.)
⊳ The measured intrinsic angular momentum of bodies is always a multiple of
ℏ.

Ref. 47
Ref. 48

Δ? Δ? ⩾

ℏ
.
2

(29)

Rotation of quantons
The effects of the quantum of action on the rotation of microscopic particles, such as
atoms, molecules or nuclei, are especially interesting. We note again that action and angular momentum have the same units. The precision with which angular momentum
* An exact formulation of the indeterminacy relation for angular momentum is
Δ? Δ? ⩾

Ref. 49

ℏ
|1 − 2π?(π)| ,
2

(30)

where ?(π) is the normalized probability that the angular position has the value π. For an angularmomentum eigenstate, one has Δ? = π/√3 and ?(π) = 1/2π. This exact expression has been tested and
confirmed by experiments.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

This is obviously an approximation: the relation is only valid for large angular momenta.
In any case, the expression tells us that rotation behaves similarly to translation. The expression cannot be valid for small angular momentum values, as Δ? by definition cannot
grow beyond 2π. In particular, angular-momentum eigenstates have Δ? = 0.*
The indeterminacy of angular momentum appears for all macroscopic bodies. We can
say that the indeterminacy appears for all cases when the angular phase of the system can
be measured.
The quantization and indeterminacy of angular momentum have important consequences. Classically speaking, the poles of the Earth are the places that do not move
when observed by a non-rotating observer. Therefore, at those places matter would have
a defined position and a defined momentum. However, the quantum of action forbids
this. There cannot be a North Pole on Earth. More precisely, the idea of a fixed rotational axis is an approximation, not valid in general. This applies in particular to rotating
quantum particles.

Motion Mountain – The Adventure of Physics

Page 48

Quantum theory thus states that every object’s angular momentum increases in steps.
Angular momentum is quantized. This result is confirmed by all experiments.
But rotation has more interesting aspects. Thanks to the quantum of action, just as
linear momentum is usually fuzzy, so is angular momentum. There is an indeterminacy
relation for angular momentum ?. The complementary variable is the phase angle ? of
the rotation. The indeterminacy relation can be expressed in several ways. The simplest
approximation – and thus not the exact expression – is

3 motion of matter – beyond classical physics

83

Ref. 50

After a year of hard work, in 1922, Otto Stern and Walther Gerlach* completed a beautiful experiment to investigate the polarization of matter quantons. They knew that inhomogeneous magnetic fields act as polarizers for rotating charges. Rotating charges are
present in every atom. Therefore they let a beam of silver atoms, extracted from an oven
by evaporation, pass an inhomogeneous magnetic field. They found that the beam splits
into two separate beams, as shown in Figure 42. No atoms leave the magnetic field region in intermediate directions. This is in full contrast to what would be expected from
classical physics.
The splitting into two beams is an intrinsic property of silver atoms; today we know
that it is due to their spin. Silver atoms have spin ℏ/2, and depending on their orientation in space, they are deflected either in the direction of the field inhomogeneity or
against it. The splitting of the beam is a pure quantum effect: there are no intermediate
options. Indeed, the Stern–Gerlach experiment provides one of the clearest demonstrations that classical physics does not work well in the microscopic domain. In 1922, the
* Otto Stern (1888–1969) and Walther Gerlach (1889–1979) worked together at the University of Frankfurt.
For his subsequent measurement of the anomalous magnetic moment of the proton, Stern received the
Nobel Prize in Physics in 1943, after he had to flee National Socialism.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Silver, Stern and Gerlach – polarization of quantons

Motion Mountain – The Adventure of Physics

can be measured depends on the precision of the rotation angle. But if a microscopic
particle rotates, this rotation might be unobservable: a situation in fundamental contrast
with the case of macroscopic objects. Experiments indeed confirm that many microscopic
particles have unobservable rotation angles. For example, in many (but not all) cases, an
atomic nucleus rotated by half a turn cannot be distinguished from the unrotated nucleus.
If a microscopic particle has a smallest unobservable rotation angle, the quantum of
action implies that the angular momentum of that particle cannot be zero. It must always
be rotating. Therefore we need to check, for each particle, what its smallest unobservable
angle of rotation is. Physicists have checked all particles in nature in experiments, and
found smallest unobservable angles (depending on the particle type) of 0, 4π, 2π, 4π/3,
π, 4π/5, 2π/3 etc.
Let us take an example. Certain nuclei have a smallest unobservable rotation angle of
half a turn. This is the case for a prolate nucleus (one that looks like a rugby ball) turning
around its short axis, such as a 23 Na nucleus. In this case, both the largest observable
rotation angle and the indeterminacy are thus a quarter turn. Since the change, or action,
produced by a rotation is the number of turns multiplied by the angular momentum, we
find that the angular momentum of this nucleus is 2 ⋅ ℏ.
As a general result, we deduce from the minimum angle values that the angular momentum of a microscopic particle can be 0, ℏ/2, ℏ, 3ℏ/2, 2ℏ, 5ℏ/2, 3ℏ etc. In other words,
the intrinsic angular momentum of a particle, usually called its spin, is an integer multiple
of ℏ/2. Spin describes how a particle behaves under rotations.
How can a particle rotate? At this point, we do not yet know how to picture the rotation. But we can feel it – just as we showed that light is made of rotating entities: all matter,
including electrons, can be polarized. This is shown clearly by the famous Stern–Gerlach
experiment.

84

3 motion of matter – beyond classical physics

observation
classical
prediction

silver
beam

?
N

∂?
∂?

S

oven

F I G U R E 42 The
Stern–Gerlach
experiment.

result seemed so strange that it was studied in great detail all over the world.
When one of the two beams – say the ‘up’ beam – is passed through a second set-up,
all the atoms end up in the ‘up’ beam. The other possible exit, the ‘down’ beam, remains
unused in this case. In other words, the up and down beams, in contrast to the original
beam, cannot be split further. This is not surprising.
But if the second set-up is rotated by π/2 with respect to the first, again two beams
– ‘right’ and ‘left’ – are formed, and it does not matter whether the incoming beam is
directly from the oven or from the ‘up’ part of the beam. A partially-rotated set-up yields
a partial, uneven split. The proportions of the two final beams depend on the angle of
rotation of the second set-up.
We note directly that if we split the beam from the oven first vertically and then horizontally, we get a different result from splitting the beam in the opposite order. (You can
check this yourself.) Splitting processes do not commute. When the order of two operations makes a difference to the net result, physicists call them non-commutative. Since
all measurements are also physical processes, we deduce that, in general, measurements
and processes in quantum systems are non-commutative.
Beam splitting is direction-dependent. Matter beams behave almost in the same way
as polarized light beams. Indeed, the inhomogeneous magnetic field acts on matter
somewhat like a polarizer acts on light. The up and down beams, taken together, define a
polarization direction. Indeed, the polarization direction can be rotated, with the help of
a homogeneous magnetic field. And a rotated beam in a unrotated magnet behaves like
an unrotated beam in a rotated magnet.
In summary, matter quantons can be polarized. We can picture polarization as the
orientation of an internal rotation axis of the massive quanton. To be consistent, the

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 66 e

silver
beam

Motion Mountain – The Adventure of Physics

aperture

3 motion of matter – beyond classical physics

85

F I G U R E 43 An idealized graph of
the heat capacity of hydrogen over
temperature (© Peter Eyland).

Curiosities and fun challenges ab ou t quantum mat ter

“

It is possible to walk while reading, but not to
read while walking.
Serge Pahaut

”

∗∗

Ref. 51

Can atoms rotate? Can an atom that falls on the floor roll under the table? Can atoms be
put into high-speed rotation? The answer is ‘no’ to all these questions, because angular
momentum is quantized; moreover, atoms are not solid objects, but clouds. The macroscopic case of an object turning more and more slowly until it stops does not exist in the
microscopic world. The quantum of action does not allow it.
∗∗

Ref. 52

Light is refracted when it enters dense matter. Do matter waves behave similarly? Yes,
they do. In 1995, David Pritchard showed this for sodium waves entering a gas of helium
and xenon.
∗∗
Many quantum effects yield curves that show steps. An important example is the molar
heat of hydrogen H2 gas, shown in Figure 43. In creasing the temperature from 20 to
8 000 K, the molar heat is shows two steps, first from 3?/2 to 5?/2, and then to 7?/2.
Can you explain the reason?
∗∗
Most examples of quantum motion given so far are due to electromagnetic effects. Can

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The quantum of action implies that there are no fractals in nature. Everything is made of
particles. And particles are clouds. Quantum theory requires that all shapes in nature be
‘fuzzy’ clouds.

Motion Mountain – The Adventure of Physics

rotation axis must be imagined to precess around the direction of polarization. Thus,
massive quantum particles resemble photons also in their polarizability.

86

Challenge 67 s

3 motion of matter – beyond classical physics

you argue that the quantum of action must also apply to nuclear motion, and in particular, to the nuclear interactions?
∗∗
There are many other formulations of the indeterminacy principle. An interesting one
is due to de Sabbata and Sivaram, who explained in 1992 that the following intriguing
relation between temperature and time also holds:
Δ?Δ? ⩾ ℏ/?.

Ref. 53

(31)

Here, ? is the Boltzmann constant. All experimental tests so far have confirmed the result.
First summary on the motion of quantum particles
Motion Mountain – The Adventure of Physics

In summary, the ‘digital’ beam splitting seen in the Stern–Gerlach experiment and the
wave properties of matter force us to rethink our description of motion. They show that
microscopic matter motion follows from the quantum of action, the smallest observable action value. In special relativity, the existence of a maximum speed forced us to
introduce the concept of space-time, and then to refine our description of motion. In
general relativity, the maximum force obliged us to introduce the concepts of horizon
and curvature, and then again to refine our description of motion. At the present point,
the existence of the quantum of action and the wave behaviour of matter force us to take
two similar steps: we first introduce the concept of a wave function, and then we refine
our description of matter motion.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Chapter 4

T H E QUA N T UM DE S C R I P T ION OF
M AT T E R A N D I T S MOT ION
Ref. 54

States and measurements – the wave function
Page 83

The Stern–Gerlach experiment shows that the measured values of spin orientation are
not intrinsic, but result from the measurement process (in this case, from the interaction
with the applied inhomogeneous field). This is in contrast to the spin magnitude, which
is intrinsic and independent of state and measurement. In short, the quantum of action
forces us to distinguish carefully three concepts:
— the state of the system;
— the operation of measurement;
— the result or outcome of the measurement.
In contrast to the classical, everyday case, the state of a quantum system – the properties a system ‘has’ – is not described by the outcomes of measurements. The simplest
illustration of this difference is the system made of a single particle in the Stern–Gerlach
experiment. The experiment shows that a spin measurement on a general (oven) particle
state sometimes gives ‘up’ (say +1), and sometimes gives ‘down’ (say −1). So a general
atom, in an oven state, has no intrinsic orientation. Only after the measurement, an atom
is either in an ‘up’ state or in a ‘down’ state.
** ‘Those quanta are a hopeless dirty mess!’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

n everyday life and in classical physics, we say that a system has a position, that
t is oriented in a certain direction, that it has an axis of rotation, and that
t is in a state with specific momentum. In classical physics, we can talk in this
way because the state – the situation a system ‘is’ in and the properties a system ‘has’
– coincide with the results of measurement. They coincide because measurements can
always be imagined to have a negligible effect on the system.
However, because of the existence of a smallest action, the interaction necessary to
perform a measurement on a system cannot be made arbitrarily small. Therefore, the
quantum of action makes it impossible for us to continue saying that a system has momentum, has position or has an axis of rotation. The quantum of action forces us to use
the idea of the rotating arrow and to introduce the concept of wave function or state
function. Let us see why and how.

Motion Mountain – The Adventure of Physics

I

“

Die Quanten sind doch eine hoffnungslose
Schweinerei!** Max Born

88

4 the quantum description of matter

It is also found that feeding ‘up’ states into a second measurement apparatus gives
only ‘up’ states: thus certain special states, called eigenstates, do remain unaffected by
measurement.
Finally, the Stern–Gerlach experiment and its variations show that states can be rotated by applied fields: atom states have a direction or orientation in space. The experiments also show that the states rotate as the atoms move through space.
The experimental observations can be described in a straightforward way. Since measurements are operations that take a state as input and produce an output state and a
measurement result, we can say:
⊳ States are described by rotating arrows, or rotating vectors.
⊳ Measurements of observables are operations on the state vectors.
⊳ Measurement results are real numbers; and like in classical physics, they usually depend on the observer.

⊳ Mathematically, states are complex vectors, or rotating arrows, in an abstract
space. This space of all possible states or arrows is a Hilbert space.
⊳ Mathematically, measurements are linear transformations, more precisely,
they are described by self-adjoint, or Hermitean, operators (or matrices).
⊳ Mathematically, changes of viewpoint are described by unitary operators (or
matrices) that act on states, or arrows, and on measurement operators.
Quantum-mechanical experiments also show that a measurement of an observable can
only give a result that is an eigenvalue of the corresponding transformation. The resulting states after the measurement, those exceptional states that are not influenced when
the corresponding variable is measured, are the eigenvectors. In short, every expert on
motion must know what an eigenvalue and an eigenvector is.
For any linear transformation ?, those special vectors ? that are transformed into
multiples of themselves,
?? = ??
(32)
are called eigenvectors (or eigenstates), and the multiplication factor ? is called the associated eigenvalue. Experiments show:
⊳ The state of the system after a measurement is given by the eigenvector corresponding to the measured eigenvalue.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 237

Motion Mountain – The Adventure of Physics

In particular, we have distinguished two quantities that are not distinguished in classical
physics: states and measurement results. Given this distinction, quantum theory follows
quite simply, as we shall see.
Given that the quantum of action is not vanishingly small, any measurement of an
observable quantity is an interaction with a system and thus a transformation of its state.
Therefore, quantum physics describes physical observables as operators, or equivalently,
as transformations. The Stern–Gerlach experiment shows this clearly: the interaction
with the field influences the atoms: some in one way, and some in another way. In fact,
all experiments show:

4 the quantum description of matter

89

In the Stern–Gerlach experiment, the eigenstates are the ‘up’ and the ‘down’ states. In
general, the eigenstates are those states that do not change when the corresponding variable is measured. Eigenvalues of Hermitean operators are always real, so that consistency
is ensured: all measurement results are real numbers.
In summary, the quantum of action obliges us to distinguish between three concepts
that are mixed together in classical physics: the state of a system, a measurement on
the system, and the measurement result. The quantum of action forces us to change the
vocabulary with which we describe nature, and obliges to use more differentiated concepts. Now follows the main step: the description of motion with these concepts. This is
what is usually called ‘quantum theory’.
Visualizing the wave function: rotating arrows and probability
clouds

⊳ The state of a quantum particle is described by a spatial distribution of arrows, a so-called wave function.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 225

To develop a visual image of the wave function, we first imagine a quantum particle that
is localized as much as possible. In this case, the wave function for a free quanton can be
described simply by a single rotating arrow.
Experiments show that when a localized quanton travels through space, the attached
arrow rotates. If the particle is non-relativistic and if spin can be neglected, the rotation
takes place in a plane perpendicular to the direction of motion. The end of the arrow
then traces a helix around the direction of motion. In this case, the state at a given time
is described by the angle of the arrow. This angle is the quantum phase. The quantum
phase is responsible for the wave properties of matter, as we will see. The wavelength and
the frequency of the helix are determined by the momentum and the kinetic energy of
the particle.
If the particle is not localized – but still non-relativistic and still with negligible spin
effects – the state, or the wave function, defines a rotating arrow at each point in space.
The rotation still takes place in a plane perpendicular to the direction of motion. But now
we have a distribution of arrows that all trace helices parallel to the direction of motion.
At each point in space and time, the state has a quantum phase and a length of the arrow.
The arrow lengths decrease towards spatial infinity.
Figure 44 shows an example of evolution of a wave function for non-relativistic
particles with negligible spin effects. The direction of the arrow at each point is shown
by the colour at the specific point. The length of the arrow is shown by the brightness of
the colour. For non-relativistic particles with negligible spin effects, the wave function
?(?, ?) is thus described by a length and a phase: it is a complex number at each point in
space. The phase is essential for interference and many other wave effects. What measurable property does the amplitude, the length of the local arrow, describe? The answer
was given by the famous physicist Max Born:

Motion Mountain – The Adventure of Physics

We just described the state of a quanton with an arrow. In fact, this is an approximation
for localized quantons. More precisely,

90

4 the quantum description of matter

⊳ The amplitude of the wave function is a probability amplitude. The square
of the amplitude, i.e., the quantity |?(?, ?)|2 , gives the probability to find the
particle at the place ? at time ?.

⊳ A wave function is a cloud of rotating arrows.
Describing the state of a matter particle with a cloud of rotating arrows is the essential
step to picture the wave properties of matter.
We can clarify the situation further.
⊳ In every process in which the phase of the wave function is not important,
the cloud image of the wave function is sufficient and correct.
For example, the motion of atoms of molecules in gases or liquids can be imagined as
the motion of cloudy objects. It needs to be stressed that the clouds in question are quite
hard: it takes a lot of energy to deform atomic clouds. The hardness of a typical crystal
is directly related to the hardness of the atomic clouds that are found inside. Atoms are
extremely stiff, or hard clouds.
On the other hand,
⊳ In every process in which the phase of the wave function does play a role,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In other terms, a wave function is a combination of two ideas: on the one hand, a wave
function is a cloud; on the other hand, at each point of the cloud one has to imagine an
arrow. Over time, the arrows rotate and the cloud changes shape.

Motion Mountain – The Adventure of Physics

F I G U R E 44 The motion of a wave
function, the quantum state, through a
double slit, showing both the particle and
the wave properties of matter. The
density of the state, related to the arrow
length, is displayed by brightness, and
the local phase is encoded in the colour.
(QuickTime film © Bernd Thaller)

4 the quantum description of matter

91

the cloud image of the wave function needs to be expanded with rotating
arrows at each point.

Vol. VI, page 172

The state evolu tion – the S chrödinger equation
The description of the state of a non-relativistic quanton with negligible spin effects as a
rotating cloud completely determines how the wave function evolves in time. Indeed, for
such quantum particles the evolution follows from the total energy, the sum of kinetic
and potential energy ? + ?, and the properties of matter waves:

⊳ The local rate of change of the state arrow ? is produced by the local total

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 68 e

Motion Mountain – The Adventure of Physics

Page 237

This is the case for interference processes of quantons, but also for the precise description
of chemical bonds.
Teachers often discuss the best way to explain wave functions. Some teachers prefer
to use the cloud model only, others prefer not to use any visualization at all. Both approaches are possible; but the most useful and helpful approach is to imagine the state
or wave function of a non-relativistic quantum particle as an arrow at every point in
space. The rotation frequency of the set of arrows is the kinetic energy of the particle; the
wavelength of the arrow motion – the period of the helical curve that the tip of the arrows – or of the average arrow – traces during motion – is the momentum of the quantum
particle.
An arrow at each point in space is a (mathematical) field. The field is concentrated
in the region where the particle is located, and the amplitude of the field is related to
the probability to find the particle. Therefore the state field, the wave function or state
function, is an arrow cloud.
Note that even though the wave function can be seen as defining an arrow at every
point in space, the wave function as a whole can also be described as one, single vector, this time in a Hilbert space. For free particles, i.e., particles that are not subject to
external forces, the Hilbert space is infinite dimensional! Nevertheless, it is not hard to
calculate in such spaces. The scalar product of two wave functions is the spatial integral of the product of the complex conjugate of the first function and the (unconjugated)
second function. With this definition, all vector concepts (unit vectors, null vectors, basis
vectors, etc.) can be meaningfully applied to wave functions.
In summary, for non-relativistic particles without spin effects, the state or wave function of a quantum particle is a cloud, or a distributed wave, of rotating arrows. This aspect
of a quantum cloud is unusual. Since a quantum cloud is made of little arrows, every
point of the cloud is described by a local density and a local orientation. This latter property does not occur in any cloud of everyday life.
For many decades it was tacitly assumed that a wave function cannot be visualized more simply than with a cloud of rotating arrows. Only the last years have shown
that there are other visualization for such quantum clouds; one possible visualization is
presented in the last volume of this series.

92

4 the quantum description of matter

energy, or Hamiltonian, ? = ? + ?:
?ℏ

Ref. 55
Ref. 56

∂
? = ?? .
∂?

This famous equation is Schrödinger’s equation of motion.* This evolution equation applies to all quantum systems and is one of the high points of modern physics.
In fact, Erwin Schrödinger had found his equation in two different ways. In his first
paper, he deduced it from a variational principle. In his second paper, he deduced the
evolution equation directly, by asking a simple question: how does the state evolve? He
knew that the state of a quanton behaves both like a wave and like a particle. A wave is
described by a field, which he denoted ?(?, ?). If the state ? behaves like a wave, then
the corresponding wave function must be an amplitude ? multiplied by a phase factor
e???−?? . The state can thus be written as

The amplitude ? is the length of the local arrow; the phase is the orientation of the local
arrow. Equivalently, the amplitude is the local density of the cloud, and the phase is the
local orientation of the cloud.
We know that the quantum wave must also behave like a particle of mass ?. In particular, the non-relativistic relation between energy and momentum ? = ?2 /2? + ?(?)
– where ?(?) is the potential at position ? – must be fulfilled for these waves. The two
de Broglie relations (22) for matter wavelength and matter frequency then imply
∂?
−ℏ2 2
= ?? =
∇ ? + ?(?)? .
∂?
2?

(35)

This is the complete form of Schrödinger’s wave equation. ∇2 is the Laplace operator, essentially the second derivative over space. It states how the arrow wave, the wave function
? associated to a particle, evolves over time. In 1926, this wave equation for the complex
field ? became instantly famous when Schrödinger used it, by inserting the potential felt
by an electron near a proton, to calculate the energy levels of the hydrogen atom. In a hydrogen atom, light is emitted by the single electron inside that atom; therefore a precise
description of the motion of the electron in a hydrogen atom allows us to describe the
light frequencies it can emit. (We will perform the calculation and the comparison with
experiment below.) First of all, the Schrödinger equation explained that only discrete colours are emitted by hydrogen. In addition, the frequencies of the emitted light were found
to be in agreement with the prediction of the equation to five decimal places. Finally, the
* Erwin Schrödinger (b. 1887 Vienna, d. 1961 Vienna) was famous for being a physicien bohémien, always
living in a household with two women. In 1925 he discovered the equation that brought him international
fame, and the Nobel Prize in Physics in 1933. He was also the first to show that the radiation discovered
by Victor Hess in Vienna was indeed coming from the cosmos. He left Germany, and then again Austria,
out of dislike for National Socialism, and was a professor in Dublin for many years. There he published his
famous and influential book What is life?. In it, he came close to predicting the then-unknown nucleic acid
DNA from theoretical insight alone.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?ℏ

Page 181

(34)

Motion Mountain – The Adventure of Physics

?(?, ?) = ?(?, ?) e???−?? .

Page 76

(33)

4 the quantum description of matter

93

F I G U R E 45 Erwin Schrödinger (1887 –1961)

Page 76

Page 90

Page 153

Waves interfere. All experiments, including the examples shown in Figure 38 and Figure 39, confirm that all quantum particles, and in particular all matter quantons, show
interference. Interference is a direct consequence of the Schrödinger equation, as the
film of Figure 44 shows. The film illustrates the solution of the Schrödinger equation for
a quantum particle moving through a double slit. The film visualizes how a double slit
induces diffraction and interference for a matter particle.
It turns out that the Schrödinger equation completely reproduces and explains the
observations of matter interference: also the interference of matter quantons is due to
the evolution of clouds of rotating arrows. And like in all interference phenomena, the
local intensity of the interference pattern turns out to be proportional to the square |?|2
of the local wave amplitude. And the local wave amplitude results from the phase of the
interfering wave trains. The analogy with light interefence is complete; even the formulae
are the same.
We note that even though the wave function is spread out over the whole detection
screen just before it hits the screen, it nevertheless yields only a localized spot on the
screen. This effect, the so-called collapse of the wave function, is explored in detail below.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Self-interference of quantons

Motion Mountain – The Adventure of Physics

Page 188

size of atoms was predicted correctly. These were important results, especially if we keep
in mind that classical physics cannot even explain the existence of atoms, let alone their
light emission! In contrast, quantum physics explains all properties of atoms and their
colours to high precision. In other words, the discovery of the quantum of action led the
description of the motion of matter to a new high point.
In fact, the exact description of matter quantons is only found when both spin effects and the relativistic energy–momentum relation are taken into account. We do this
below. No deviations between the full relativistic calculations and experiments have ever
been found. And even today, predictions and measurements of atomic spectra remain the
most precise and accurate in the whole study of nature: in the cases that experimental
precision allows it, the calculated values agree with experiments to 13 decimal places.

94

4 the quantum description of matter

The speed of quantons

Dispersion of quantons
For free quantum particles, the Schrödinger’s evolution equation implies dispersion, as
illustrated in Figure 46. Imagine a wave function that is localized around a given starting
position. Such a wave function describes a quantum system at rest. When time passes,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 153

Let us delve a little into the details of the description given by the Schrödinger equation
(35). The equation expresses a simple connection: the classical speed of a matter particle
is the group velocity of the wave function ?. Seen from far away, the wave function thus
moves like a classical particle would.
But we know from classical physics that the group velocity is not always well defined:
in cases where the group dissolves into several peaks, the concept of group velocity is not
of much use. These are also the cases in which quantum motion is very different from
classical motion, as we will soon discover. But for well-behaved cases, such as free or
almost free particles, we find that the wave function moves in the same way as a classical
particle does.
The Schrödinger equation makes another point: velocity and position of matter are
not independent variables, and cannot be chosen at will. The initial condition of a system
is given by the initial value of the wave function alone. No derivatives have to be (or
can be) specified. Indeed, experiments confirm that quantum systems are described by
a first-order evolution equation, in stark contrast to classical systems. The reason for this
contrast is the quantum of action and the limit it poses on the possible state variables of
a particle.

Motion Mountain – The Adventure of Physics

F I G U R E 46 The evolution of a wave
function (lowest curve) with zero
momentum, and the motion of its
parts with positive and negative
momenta. Local phase is encoded in
the colour. (QuickTime film © Bernd
Thaller)

4 the quantum description of matter

95

Challenge 70 ny

Tunnelling and limits on memory – damping of quantons
‘Common sense’ says that a slow ball cannot roll over a high hill. More precisely, classical
physics says that if the kinetic energy ? is smaller than the potential energy ? that the
ball would have at the top of the hill, then the ball cannot reach the top of the hill. In
contrast, according to quantum theory, there is a positive probability of passing the hill
for any energy of the ball.
In quantum theory, hills and obstacles are described by potential barriers, and objects
by wave functions. Any initial wave function will spread beyond any potential barrier of
finite height and width. The wave function will also be non-vanishing at the location of
the barrier. In short, any object can overcome any hill or barrier, as shown in Figure 48.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 69 e

this wave function will spread out in space. Indeed, Schrödinger’s evolution equation
is similar, mathematically, to a diffusion equation. In the same way that a drop of ink
in water spreads out, also the state of a localized quantum particle will spread out in
space. True, the most probable position stays unchanged, but the probability to find the
particle at large distances from the starting position increases over time. For quantum
particles, this spreading effect is indeed observed by all experiments. The spread is a
consequence of the wave aspect of matter, and thus of the quantum of action ℏ. It occurs
for quantons at rest and therefore also for quantons in motion. For macroscopic objects,
the spreading effect is not observed, however: cars rarely move away from parking spaces.
Indeed, quantum theory predicts that for macroscopic systems, the effect of spreading is
negligibly small. Can you show why?
In summary, the wave aspect of matter leads to the spreading of wave functions. Wave
functions show dispersion.

Motion Mountain – The Adventure of Physics

F I G U R E 47 The tunnelling of a wave
function through a potential hill (the
rectangular column): most of the wave
function is reflected, and part of the
wave function passes to the other side.
Local phase is encoded in the colour.
(QuickTime film © Bernd Thaller)

96

4 the quantum description of matter

E
m

p
Δx

0

F I G U R E 48 Climbing a hill.

(36)

where ? is the width of the hill, ? its height, and ? and ? the mass and the kinetic energy
of the particle. For a system of large number of particles, the probability is (at most) the
product of the probabilities for the different particles.
Let us take the case of a car in a garage, and assume that the car is made of 1028 atoms
at room temperature. A typical garage wall has a thickness of 0.1 m and a potential height
of ? = 1 keV = 160 aJ for the passage of an atom. We get that the probability of finding
the car outside the garage is
12

? ≈ (10−(10 ) )
Challenge 72 e

Challenge 73 s

(1028 )

40

≈ 10−(10

)

.

(37)

The smallness of this value (just try to write it down, to be convinced) is the reason why
it is never taken into account by the police when a car is reported missing. (Actually, the
probability is even considerably smaller. Can you name at least one effect that has been
forgotten in this simple calculation?)
Obviously, tunnelling can be important only for small systems, made of a few
particles, and for thin barriers, with a thickness of the order of ℏ/√2?(? − ?) . For example, tunnelling of single atoms is observed in solids at high temperature, but is not
important in daily life. For electrons, the effect is more pronounced: the barrier width ?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

2?
16?(? − ?) − √2?(? − ?)
e ℏ
?≈
?2

Motion Mountain – The Adventure of Physics

Challenge 71 ny

This effect is called the tunnelling effect. It is in complete contrast to everyday experience
– and to classical mechanics.
The tunnelling effect results from a new aspect contained in the quantum description of hills: in nature, any obstacle can be overcome with a finite effort. No obstacle is
infinitely difficult to surmount. Indeed, only for a potential of infinite height would the
wave function vanish and fail to spread to the other side. But such potentials exist only
as approximations; in nature potentials are always of finite value.
How large is the tunnelling effect? Calculation shows that the transmission probability
? is given approximately by

4 the quantum description of matter

Farady cage
with high
electric
potential

97

screen with
intereference
pattern that
depends on
potential

beam splitter
F I G U R E 49 A localized electric potential
in an interferometer leads to a shift of the
interference pattern.

charged matter beam

?≈

(38)

At room temperature, the kinetic energy ? is of the order of 6 zJ; increasing the temperature obviously increases the tunnelling. As a result, electrons tunnel quite easily through
barriers that are a few atoms in width. Indeed, every TV tube uses tunnelling at high
temperature to generate the electron beam producing the picture. The necessary heating
is the reason why in the past, television tubes took some time to switch on.
The tunnelling of electrons also limits the physical size of computer memories. Memory chips cannot be made arbitrary small. Silicon integrated circuits with one terabyte of
random-access memory (RAM) will probably never exist. Can you imagine why? In fact,
tunnelling limits the working of any type of memory, including that of our brain. Indeed,
if we were much hotter than 37°C, we could not remember anything!
Since light is made of particles, it can also tunnel through potential barriers. The best
– or highest – potential barriers for light are mirrors; mirrors have barrier heights of the
order of one attojoule. Tunnelling implies that light can be detected behind any mirror.
These so-called evanescent waves have indeed been detected; they are used in various
high-precision experiments and devices.
The quantum phase
We have seen that the amplitude of the wave function, the probability amplitude, shows
the same effects as any wave: dispersion and damping. We now return to the phase of the
wave function and explore it in more detail.
Whereas the amplitude of a wave function is easy to picture – just think of the (square
root of the) density of a real cloud – the phase takes more effort. As mentioned, states
or wave functions are clouds with a local phase: they are clouds of rotating arrows, i.e.,
clouds of objects that rotate and can be rotated. In case of an everyday water cloud, a
local rotation of droplets has no effect of the cloud. In contrast, in quantum theory, the
local rotation of the cloud, thus the local change of its phase, does have a measurable

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 74 s

0.5 nm √aJ
.
√? − ?

Motion Mountain – The Adventure of Physics

for an appreciable tunnelling effect is

98

4 the quantum description of matter

solenoid
with
current

screen with
intereference
pattern that
depends on
magnetic field

beam splitter
neutral matter beam

effect. Let us explore this point.
The phase of free matter waves behaves like the phase of photons: it evolves with time,
and thus increases along the path of a moving particle. The phase can be pictured by a
small rotating arrow. The angular velocity with which the phase rotates is given by the
famous relation ? = ?/ℏ. In short,
⊳ We can picture the wave function of a free quantum particle as a moving
cloud of arrows; the arrows rotate with constant frequency while the cloud
disperses at the same time.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 57

Above all, the phase is that aspect of the wave function that leads to interference effects.
When two partial wave functions are separated and recombined after a relative phase
change, the phase change will determine the interference pattern. This is the origin of
the electron beam interference observations shown in Figure 38. Without the quantum
phase, there would be no extinction and no interference.
The phase of a wave function can be influenced in many ways. The simplest way is
the use of electric fields. If the wave function of a charged particle is split, and one part is
led through a region with an electric field, a phase change will result. The arrangement
is shown in Figure 49. A periodic change of the electric potential should yield a periodic
shift of the interference pattern. This is indeed observed.
Another simple case of phase manipulation is shown in Figure 50: also a magnetic
field changes the phase of a spinning charged particle, and thus influences the interference behaviour.
A famous experiment shows the importance of the phase in an even more surprising way: the Aharonov–Bohm effect. The effect is famous for two reasons: it is counterintuitive and it was predicted before it was observed. Look at the set-up shown in Figure 51. A matter wave of charged particles is split into two by a cylinder – positioned at a
right angle to the matter’s path – and the matter wave recombines behind it. Inside the
cylinder there is a magnetic field; outside, there is none. (A simple way to realize such a
cylinder is a long solenoid.) Quantum physics predicts that an interference pattern will
be observed, and that the position of the stripes will depend on the value of the mag-

Motion Mountain – The Adventure of Physics

Page 56

F I G U R E 50 Magnetic fields change the
phase of a spinning particle.

4 the quantum description of matter

magnetic field (even
if only inside the solenoid)

99

screen

current

vector
potential

charged matter beam

netic field. This happens even though the wave never enters the region with the field!
The surprising effect has been observed in countless experiments.
The reason for the Aharonov–Bohm effect is simple: for a charged particle, the phase

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 52 The motion of a wave
function around a solenoid showing the
Aharonov–Bohm effect. The density of
the state is displayed by brightness, and
the local phase is encoded in the colour.
(QuickTime film © Bernd Thaller)

Motion Mountain – The Adventure of Physics

F I G U R E 51 The Aharonov–Bohm effect: the influence of the magnetic vector potential on interference
(left) and a measurement confirmation (right), using a microscopic sample that transports electrons in
thin metal wires (© Doru Cuturela).

100

4 the quantum description of matter

electrically
charged
wire

beam splitter
polarized neutron beam

Challenge 75 e

of a wave function is determined by the vector potential ?, not by the magnetic field ?.
The vector potential around a solenoid does not vanish – as we know from the section
on electrodynamics – but circulates around the solenoid. This circulation distinguishes
the two sides of the solenoid and leads to a phase shift – one that indeed depends on the
magnetic field value – and thus produces interference, even though the particle never
interacts with the magnetic field itself.
A further example for phase manipulation is the so-called Aharonov–Casher effect,
which even occurs for neutral particles, as long as they have a magnetic moment, such as
neutrons have. The phase of a polarized neutron will be influenced by an electric field, so
that the arrangement shown in Figure 53 will show an interference pattern that depends
on the applied electric potential.
Another case of phase manipulation will be presented later on: also gravitational fields
can be used to rotate wave functions. Even the acceleration due to rotational motion can
do so. In fact, it has been possible to measure the rotation of the Earth by observing the
change of neutron beam interference patterns.
Another important class of experiments that manipulate the phase of wave functions
are possible with macroscopic quantum phenomena. In superconductivity and in superfluidity, the phase of the wave function is regularly manipulated with magnetic and electric fields. This possibility has many important technical applications. For example, the
so-called Josephson effect is used to measure electric potential differences by measuring
the frequency of emitted radio waves, and so-called superconducting quantum interference devices, or SQIDs, are used to measure tiny magnetic fields.
We note that all these experiments confirm that the absolute phase of a wave function
cannot be measured. However, relative phases – phase differences or phase changes – can
be measured. Can you confirm this?
All the phase shift effects just presented have been observed in numerous experiments. The phase is an essential aspect of the wave function: the phase leads to interference and is the main reason for calling it wave function in the first place. Like in any
wave, the phase evolves over time and it can be influenced by various external influences. Above all, the experiments show that a localized quantum particle – thus when
the spread of the wave function can be neglected – is best imagined as a rotating arrow;

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 58

F I G U R E 53 The Aharonov–Casher effect:
the influence of charge on the phase
leads to interference even for interfering
neutrons.

Motion Mountain – The Adventure of Physics

Vol. III, page 84

screen with
intereference
pattern that
depends on
wire charge

4 the quantum description of matter

101

in contrast, whenever the spread cannot be neglected, the wave function is best imagined
as a wave of arrows rotating at each point in space.

Ref. 59

Do coherent electron sources exist? The question is tricky. Results in the literature, such
as the one illustrated in Figure 54, state that is possible to make holograms with electron
beams.* However, when one asks these authors about the meaning of coherence, they
answer that electron coherence is only transversal, not longitudinal. Transversal coherence is determined by the possible size of wavefronts with a given phase. The upper limit
of this size is given by the interactions such a state has with its environment. All this
behaviour is as expected for actual coherence.
However, the concept of ‘transversal coherence’ is a misnomer. The ability to interfere
with oneself, as implies in the term ‘transversal coherence’ is not the correct definition of
coherence. Transversal coherence, be it for photons or for matter particles, only expresses
the smallness of the particle source. Both small lamps (and lasers) can show interference
when the beam is split and recombined with identical path length; this is not a proof of
coherence of the light field. A similar reasoning shows that monochromaticity is not a
proof for coherence either.
A state is called coherent if it possesses a well-defined phase throughout a given domain of space or time. The size of the spatial region or of the time interval defines the
degree of coherence. This definition yields coherence lengths of the order of the source
size for small ‘incoherent’ sources. Even for a small coherence length, the size of an interference pattern or the distance ? between its maxima can be much larger than the

Ref. 60

* In 2002, the first holograms have been produced that made use of neutron beams.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

C an t wo electron beams interfere? Are there coherent electron
beams?

Motion Mountain – The Adventure of Physics

F I G U R E 54 An electron hologram of DNA molecules (© Hans-Werner Fink/Wiley VCH).

102

Page 47

4 the quantum description of matter

coherence length ? or the source size ?. In short, a large size (or a persistent duration in
time) of an interference pattern alone is not a proof of coherence.
Let us recall the situation for light. A light source is coherent if it produces an approximate sine wave over a certain length or time. Due to the indeterminacy relation, in
any coherent beam of light, the photon number is undetermined. The same requirement
applies to coherent electron beams: an undetermined electron number is needed for coherence. That is impossible, as electrons carry a conserved charge. Coherent electron
beams do not exist.
In summary, even though an electron can interfere with itself, and even though it is
possible to produce interference between two light sources, interference between two
electron sources is impossible. Indeed, nobody has every managed to produce interference between two electron sources. There is no conventional concept of coherence for
electron beams.

Vol. I, page 252

Vol. I, page 247

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 56

In nature, motion happens in a way that minimizes change. Indeed, in classical physics, the principle of least action – or principle of cosmic lazyness – states: in nature, the
motion of a particle happens along that particular path – out of all possible paths with
the same end points – for which the action is minimal. This principle of cosmic laziness
or cosmic efficiency was stated mathematically by saying that in nature, the variation
?? of the action is zero. Action or change minimization explains all classical evolution
equations. We now transfer this idea to the quantum domain.
For quantum systems, we need to redefine both the concept of action and the concept
of variation: first of all, we have to find a description of action that is based on operators;
secondly, we need to define the action variation without paths, as the concept of ‘path’
does not exist for quantum systems; thirdly, since there is a smallest action in nature, a
vanishing variation is not a clearly defined concept, and we must overcome this hurdle.
There are two main ways to achieve this goal: to describe the motion of quantum systems as a superposition of all possible paths, or to describe action with the help of wave
functions. Both approaches are equivalent.
In the first approach, the path integral formulation, the motion of a quantum particle
is described as a democratic superposition of motions along all possible paths. (We called
it the ‘arrow model’ above.) For each path, the evolution of the arrow is determined, and
at the end point, the arrows from all paths are added. The action for each path is the
number of turns that the arrow performs along the path. The result from this exercise
is that the path for which the arrow makes the smallest number of turns is usually (but
not always!) the most probable path. A more precise investigation shows that classical,
macroscopic systems always follow only the path of smallest action, whereas quantum
systems follow all paths.
In the second approach to quantum physics, action is defined with the help of wave
functions. In classical physics, we defined the action (or change) as the integral of the
Lagrangian between the initial and final points in time, and the Lagrangian itself as the
difference between kinetic and potential energy. In quantum physics, the simplest definition is the quantum action defined by Julian Schwinger. Let us call the initial and final

Motion Mountain – The Adventure of Physics

The least action principle in quantum physics

4 the quantum description of matter

103

states of the system ?i and ?f . The action ? between these two states is defined as
? = ⟨?i | ∫? d? | ?f ⟩ ,

(39)

where ? is the Lagrangian (operator). The angle brackets represent the ‘multiplication’
of states and operators as defined in quantum theory.* In simple words, also in quantum
theory, action – i.e., the change occurring in a system – is the integral of the Lagrangian.
The Lagrangian operator ? is defined in the same way as in classical physics: the Lagrangian ? = ?−? is the difference between the kinetic energy ? and the potential energy
? operators. The only difference is that, in quantum theory, the momentum and position
variables of classical physics are replaced by the corresponding operators of quantum
physics.**
To transfer the concept of action variation ?? to the quantum domain, Julian
Schwinger introduced the straightforward expression
(40)

?? = ⟨?i | ?∫? d?| ?f ⟩ = −?ℏ ?⟨?i |?f ⟩ .

(41)

This so-called quantum action principle describes all motion in the quantum domain.
Classically, the right-hand side is zero – since ℏ is taken to be zero – and we then recover the minimum-action principle ?? = 0 of classical physics. But in quantum theory, whenever we try to achieve small variations, we encounter the quantum of action
and changes of (relative) phase. This is expressed by the right-hand side of the expression. The right side is the reason that the evolution equations for the wave function –
Schrödinger’s equation for the spinless non-relativistic case, or Dirac’s equation for the
spin 1/2 relativistic case – are valid in nature.
In other words, all quantum motion – i.e., the quantum evolution of a state ? or |?⟩
– happens in such a way that the action variation is the same as −? times the quantum
of action ℏ times the variation of the scalar product between initial and final states. In
* We skip the details of notation and mathematics here; in the simplest description, states are wave functions, operators act on these functions, and the product of two different brackets is the integral of the function product over space.
** More precisely, there is also a condition governing the ordering of operators in a mixed product, so that
the non-commutativity of operators is taken into account. We do not explore this issue here.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The concept of path is not needed in this expression, as the variation of the action is
based on varying wave functions instead of varying particle paths.
The last classical requirement to be transferred to the quantum domain is that, because nature is lazy, the variation of the action must vanish. However, in the quantum
domain, the variation of the action cannot be zero, as the smallest observable action is
the quantum of action. As Julian Schwinger discovered, there is only one possible way to
express the required minimality of action:

Motion Mountain – The Adventure of Physics

?? = ⟨?i | ?∫? d?| ?f ⟩ .

104

4 the quantum description of matter

simple terms, in the actual motion, the intermediate states are fixed by the requirement
that they must lead from the initial state to the final state with the smallest number of
effective turns of the state phase. The factor −? expresses the dependence of the action on
the rotation of the wave function.
In summary, the least action principle is also valid in quantum physics, provided one
takes into account that action values below ℏ cannot be found in experiments. The least
action principle governs the evolution of wave function. The least action principle thus
explains the colour of all things, all other material science, all chemistry and all biology,
as we will see in the following.
The motion of quantons with spin

Page 82

Ref. 61

Page 82

?=?

Page 107

”

What is the origin of the quantum phase? Classical physics helps to answer the question.
Like everyday objects, also quantons can rotate around an axis: we speak of particle spin.
But if quantum particles can spin, they should possess angular momentum. And indeed,
experiments confirm this deduction.
In particular, electrons have spin. The full details of electron spin were deduced from
experiments by two Dutch students, George Uhlenbeck and Samuel Goudsmit, in 1925.
They had the guts to publish what Ralph Kronig had also suspected: that electrons rotate
around an axis with a projected component of the angular momentum given by ℏ/2.
In fact, this value – often called spin 1/2 for short – is valid for all elementary matter
particles. (In contrast, all known elementary radiation particles have spin values of ℏ, or
spin 1 for short.)
If a spinning particle has angular momentum, it must be possible to rearrange the axis
by applying a torque, to observe precession, to transfer the spin in collisions, etc. All these
effects are indeed observed; for example, the Stern–Gerlach experiment already allows
all these observations. The only difference between particle spin and classical angular
momentum is that particle spin is quantized, as we deduced above.
In other words, the spin ? of a quantum particle has all the properties of a rotation
around an axis. As a consequence, spinning charged quantum particles act as small dipole magnets, with the magnet oriented along the axis of rotation. The observed strength
of the dipole magnet, the magnetic moment, is proportional to the spin and to the conversion factor −?/2?? , as expected from classical physics. Therefore, the natural unit for
the magnetic moment of the electron is the quantity ?B = ?ℏ/2?? ; it is called Bohr’s magneton. It turns out that the magnetic moment ? of quantons behaves differently from that
of classical particles. The quantum effects of spin are described by the so-called ?-factor,
which is a pure number:
−?
?
? = −??B ,
2??
ℏ

with ?B =

?ℏ
.
2??

(42)

From the observed optical spectra, Uhlenbeck and Goudsmit deduced a ?-factor of 2 for
the electron. Classically, one expects a value ? = 1. The experimental value ? = 2 was

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 83

Anonymous

Motion Mountain – The Adventure of Physics

“

Everything turns.

4 the quantum description of matter

Ref. 62

105

R elativistic wave equations

Ref. 63

* Wolfgang Ernst Pauli (b. 1900 Vienna, d. 1958 Zürich), at the age of 21, wrote one of the best texts on special
and general relativity. He was the first to calculate the energy levels of hydrogen using quantum theory,
discovered the exclusion principle, incorporated spin into quantum theory, elucidated the relation between
spin and statistics, proved the CPT theorem, and predicted the neutrino. He was admired for his intelligence,
and feared for his biting criticisms, which led to his nickname, ‘conscience of physics’. Despite this trait,
he helped many people in their research, such as Heisenberg with quantum theory, without claiming any
credit for himself. He was seen by many, including Einstein, as the greatest and sharpest mind of twentiethcentury physics. He was also famous for the ‘Pauli effect’, i.e., his ability to trigger disasters in laboratories,
machines and his surroundings by his mere presence. As we will see shortly, one can argue that Pauli actually
received the Nobel Prize in Physics in 1945 – officially ‘for the discovery of the exclusion principle’ – for
finally settling the question of how many angels can dance on the tip of a pin.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In 1899, Max Planck had discovered the quantum of action. In 1905, Albert Einstein published the theory of special relativity, which was based on the idea that the speed of light
? is independent of the speed of the observer. The first question Planck asked himself
was whether the value of the quantum of action would be independent of the speed of
the observer. It was his interest in this question that led him to invite Einstein to Berlin.
With this invitation, he made the patent-office clerk famous in the world of physics.
Experiments show that the quantum of action is indeed independent of the speed of
the observer. All observers find the same minimum value. To include special relativity
into quantum theory, we therefore need to find the correct quantum Hamiltonian ?, i.e.,
the correct energy operator.

Motion Mountain – The Adventure of Physics

explained by Llewellyn Thomas as a relativistic effect a few months after its experimental
discovery.
By 2004, experimental techniques had become so sensitive that the magnetic effect
of a single electron spin attached to an impurity (in an otherwise non-magnetic material) could be detected. Researchers now hope to improve these so-called ‘magneticresonance-force microscopes’ until they reach atomic resolution.
In 1927, Wolfgang Pauli* discovered how to include spin 1/2 in a quantum-mechanical
description: instead of a state function described by a single complex number, a state
function with two complex components is needed. The reason for this expansion is
simple. In general, the little rotating arrow that describes a quantum state does not rotate around a fixed axis, as is assumed by the Schrödinger equation; the axis of rotation
has also to be specified at each position in space. This implies that two additional parameters are required at each space point, bringing the total number of parameters to four
real numbers, or, equivalently, two complex numbers. Nowadays, Pauli’s equation for
quantum mechanics with spin is mainly of conceptual interest, because – like that of
Schrödinger – it does not comply with special relativity.
In summary, the non-relativistic description of a quanton with spin implies the use
of wave functions that specify two complex numbers at each point in space and time.
The additional complex number describe the local rotation plane of the spin. The idea
of including the local rotation plane was also used by Dirac when he introduced the relativistic description of the electron, and the idea is also used in all other wave equations
for particles with spin.

106

4 the quantum description of matter

For a free relativistic particle, the classical Hamiltonian function – that is, the energy
of the particle – is given by
? = ±√?4 ?2 + ?2 ?2

Ref. 64

2

2 2

? = ?√? ? + ? ?

0
0 0
1
0 0
) .
0 −1 0
0
0 −1

(44)

The signs appearing in the matrix operator ? distinguish, as we will see, between particles
and antiparticles. The numbers +1 and −1 appear twice, to take care of the two possible
spin directions for each case.
With this relativistic Hamiltonian operator for spin 1/2 particles – and with all others
– the wave function is described by four complex numbers, two for particles and two
for antiparticles. Why? We saw above that a quantum particle with spin requires two
complex components for its state; this followed from the requirement to specify, at each
point in space, the length of the arrow, its phase, and its plane of rotation. Earlier on we
also found that relativity automatically introduces antimatter. (We will explore the issue
in more detail below.) Both matter and antimatter are thus part of any relativistic description of quantum effects. The wave function for a particle has vanishing antiparticle
components, and vice versa. In total, the wave function for relativistic spin 1/2 particle
has thus four complex components.
The Hamilton operator yields the velocity operator ? through the same relation that
is valid in classical physics:
?=

?
d
.
?=?
d?
√?4 ?2 + ?2 ?2

(45)

This velocity operator shows a continuum of eigenvalues, from minus to plus the speed
of light. The velocity ? is a constant of motion, as are the momentum ? and the energy
? = √?4 ?2 + ?2 ?2 .

(46)

Also the orbital angular momentum ? is defined as in classical physics, through
? =?×?.
Ref. 65

(47)

The orbital angular momentum ? and the spin ? are separate constants of motion. A
particle (or antiparticle) with positive (or negative) angular momentum component has

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 192

1
0
with ? = (
0
0

Motion Mountain – The Adventure of Physics

Vol. II, page 72

(43)

Thus we can ask: what is the corresponding Hamilton operator for the quantum world?
The simplest answer was given, in 1949 by T.D. Newton and E.P. Wigner, and in 1950, by
L.L. Foldy and S.A. Wouthuysen. The operator is almost the same one:

4

Page 105

with ? = ??? .

4 the quantum description of matter

Page 189

107

a wave function with only one non-vanishing component; the other three components
vanish.
But alas, the representation of relativistic motion named after Foldy and Wouthuysen
is not the simplest when it comes to take electromagnetic interactions into account. The
simple identity between the classical and quantum-mechanical descriptions is lost when
electromagnetism is included. We will solve this problem below, when we explore Dirac’s
evolution equation for relativistic wave functions.
B ound motion, or composite vs. elementary quantons

Ref. 66
Page 189

Vol. V, page 162

Vol. V, page 162

The criterion for being elementary can thus be reduced to a condition on the value of the
dimensionless number ?, the so-called ?-factor. (The expression ?ℏ/2? is often called
the magneton of the particle.) If the ?-factor differs from the value predicted by quantum
electrodynamics for point particles – about 2.0 – the object is composite. For example,
a 4 He+ helium ion has spin 1/2 and a ? value of 14.7 ⋅ 103 . Indeed, the radius of the
helium ion is 3 ⋅ 10−11 m, obviously a finite value, and the ion is a composite entity. For the
proton, one measures a ?-factor of about 5.6. Indeed, experiments yield a finite proton
radius of about 0.9 fm and show that it contains several constituents.
The neutron, which has a magnetic moment despite being electrically neutral, must
therefore be composite. Indeed, its radius is approximately the same as that of the proton.
Similarly, molecules, mountains, stars and people must be composite. According to this
first criterion, the only elementary particles are leptons (i.e., electrons, muons, tauons
and neutrinos), quarks, and intermediate bosons (i.e., photons, W-bosons, Z-bosons and
gluons). More details on these particles will be revealed in the chapters on the nucleus.
Another simple criterion for compositeness has just been mentioned: any object with
a measurable size is composite. This criterion yields the same list of elementary particles as
the first. Indeed, the two criteria are related. The simplest model for composite structures

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 189

The gyromagnetic ratio ? is measured in units of s−1 T−1 , i.e., C/kg, and determines the
energy levels of magnetic spinning particles in magnetic fields; it will reappear later in
the context of magnetic resonance imaging. All candidates for elementary particles have
spin 1/2. The gyromagnetic ratio for spin-1/2 particles of magnetic moment ? and mass
? can be written as
?
?
?=
=?
.
(49)
ℏ/2
2?

Motion Mountain – The Adventure of Physics

Challenge 76 e

When is an object composite, and not elementary? Whenever it contains internal, or
bound motion. When is this the case? Quantum theory gives several pragmatic answers.
The first criterion for compositeness is somewhat strange: an object is composite when its gyromagnetic ratio is different from the one predicted by quantum
electrodynamics. The gyromagnetic ratio ? – not to be confused with the relativistic
dilation factor – is defined as the ratio between the magnetic moment ? and the
angular momentum ?:
? = ?? .
(48)

108

Ref. 67

4 the quantum description of matter

predicts that the ?-factor obeys
?−2=

Challenge 77 e
Vol. V, page 342

?
?C

(50)

where ? is the radius and ? C = ℎ/?? is the Compton wavelength of the system. This
expression is surprisingly precise for helium-4 ions, helium-3, tritium ions and protons,
as you may wish to check. The tables in Appendix B in the next volume make the same
point. In short, the second criterion for compositeness is equivalent to the first.
A third criterion for compositeness is more general: any object larger than its Compton
length is composite. The argument is simple. An object is composite if one can detect
internal motion, i.e., motion of some components. Now the action of any part with mass
?part moving inside a composed system of size ? obeys
?part < 2π ? ?part ? < π ? ? ?

(51)

?>

ℏ
.
2π ? ?

(52)

The right-hand side differs only by a factor 4π2 from the so-called Compton (wave)length

Challenge 79 e

Vol. V, page 261
Challenge 80 s
Vol. VI, page 313

Challenge 78 ny

ℎ
??

(53)

of an object. Thus any object larger than its own Compton wavelength is composite; and
any object smaller than the right-hand side of expression (52) is elementary. Again, only
leptons, quarks and intermediate bosons passed the test. (For the Higgs boson discovered
in 2012, the test has yet to be performed, but it is expected to comply as well.) All other
objects are composite. In short, this third criterion produces the same list as the previous
ones. Can you explain why?
A fourth criterion for compositeness is regularly cited by Steven Weinberg: a particle
is elementary if it appears in the Lagrangian of the standard model of particle physics,
i.e., in the description of the fundamental building blocks of nature. Can you show that
this criterion follows from the previous ones?
Interestingly, we are not yet finished with this topic. Even stranger statements about
compositeness will appear when gravity is taken into account. Just be patient: it is
worth it.

* Can you find the missing factor of 2? And is the assumption that the components must always be lighter
than the composite a valid one?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?=

Motion Mountain – The Adventure of Physics

where ? is the mass of the composite object. On the other hand, following the principle
of quantum theory, this action, to be observable, must be larger than ℏ/2. Inserting this
condition, we find that for any composite object*

4 the quantum description of matter

109

Curiosities and fun challenges ab ou t quantum motion of mat ter

“

Die meisten Physiker sind sehr naiv, sie glauben
immer noch an wirkliche Wellen oder
Teilchen.*
Anton Zeilinger

”

Take the sharpest knife edge or needle tip you can think of: the quantum of action implies
that their boundaries are not sharp, but fuzzy, like the boundaries of clouds. Take the
hardest or most solid object you can think of, such as diamond or a block of tungsten:
the quantum of action implies that its surface is somewhat soft. All experiments confirm
these statements. Nothing in nature is really sharp or really solid. Quantum physics thus
disagrees with several ideas of the ancient Greek atomists.
∗∗

Challenge 81 s

∗∗

∗∗

Challenge 83 d

Two observables can commute for two different reasons: either they are very similar –
such as the coordinates ? and ?2 – or they are very different – such as the coordinate ?
and the momentum ?? . Can you give an explanation for this?
∗∗

Challenge 84 ny

Space and time translations commute. Why then do the momentum operator and the
Hamiltonian not commute in general?
∗∗
* ‘Most physicists are very naive; they still believe in real waves or real particles.’ Anton Zeilinger, physicist
at the University of Vienna, is well-known for his experiments on quantum mechanics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 82 e

Is the wave function ’real’? More precisely, is the wave function really a cloud? Some
physicists still doubt this. This dying group of physicists, often born around the middle
of the twentieth century, have heard so often – incorrectly and usually from questionable
authorities – that a wave function has no reality that they stopped asking and answering
the simplest questions. To dispel their doubts, ask them whether they have a non-zero
height or whether they think that atoms are round. If they agree, they have admitted that
wave functions have some sort of reality. All everyday objects are made of elementary
particles that are so unmeasurably small that we can call them point-like. Therefore, the
size, surface area and volume of all everyday objects are exclusively due to wave functions. Every length, area and volume is a proof that wave functions have some sort of
reality.

Motion Mountain – The Adventure of Physics

Ref. 68

Do hydrogen atoms exist? Most types of atom have been imaged with microscopes, photographed under illumination, levitated one by one, and even moved with needles, one by
one, as the picture on page 343 in volume I shows. Researchers have even moved single
atoms by using laser beams to push them. However, not a single one of these experiments has measured or imaged hydrogen atoms. Is that a reason to doubt the existence
of hydrogen atoms? Taking this not-so-serious discussion seriously can be a lot of fun.

110

4 the quantum description of matter

F I G U R E 55 A special potential well
that does not disturb a wave function.
Colour indicates phase. (QuickTime
film © Bernd Thaller)

∗∗
Ref. 69

where ? is the reduced mass and ? the kinetic energy of the components, and ? is the size
of the system. Can you deduce this result, and check it for the ground state of hydrogen?
Challenge 85 s

∗∗
In high school, it often makes sense to visualize electron wave functions as a special type
of fluid-like matter, called electronium, that has a negative charge density. In this visualization, an atom is a positive nucleus surrounded by an electronium cloud. Deforming
the electronium cloud around a nucleus requires energy; this happens when a photon of
the correct frequency is absorbed, for example. When atoms of the right kind approach
each other, the electronium clouds often form stable bridges – chemical bonds.
∗∗

Vol. I, page 318

Ref. 70

Quantum theory allows for many unusual bound states. Usually we think of bound states
as states of low energy. But there are situations in which bound states arise due to forcing in oscillating potentials. We encountered such a situation in classical physics: the
vertically driven, upside-down pendulum that remain vertical despite being unstable.
Similar situations also occur in quantum physics. Examples are Paul traps, the helium
atom, negative ions, Trojan electrons and particle accelerators.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Any bound system in a non-relativistic state with no angular momentum obeys the
relation
9ℏ2
⟨?2 ⟩ ⟨?⟩ ⩾
,
(54)
8?

Motion Mountain – The Adventure of Physics

There exist special potentials that have no influence on a wave function. Figure 55 shows
an example. This potential has reflection coefficient zero for all energies; the scattered
wave has no reflected part. The mathematical reason is fascinating. The potential well
has the shape of a soliton of the Korteweg–de Vries equation; this equation is related to
the Schrödinger equation.

4 the quantum description of matter

111

∗∗
Challenge 86 s

One often reads that the universe might have been born from a quantum fluctuation.
Can you explain why this statement make no sense?
A summary on motion of mat ter quantons
In summary, the motion of massive quantons, i.e., of quantum matter particles, can be
described in two ways:

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The single rotating arrow results from a cloud average. The single arrow combines
particle and wave properties. A full rotation of the arrow corresponds to the quantum
of action ℏ. This central feature implies that a non-relativistic particle whose spin can be
neglected follows the Schrödinger equation, and that a relativistic electron follows the
Dirac equation. The Dirac equation agrees with all known experiments. In particular,
the Dirac equation describes all of materials science, chemistry and biology, as we will
find out.
To continue with the greatest efficiency on our path across quantum physics, we explore three important topics: the indistinguishability of particles of the same kind, the
spin of quantum particles, and the meaning of probabilities.

Motion Mountain – The Adventure of Physics

— At high magnification, quantum matter particles are described by wave functions that
move like advancing, rotating and precessing clouds of arrows. The local cloud orientation, or local phase, follows a wobbling motion. The square of the wave function, i.e.,
the density of the cloud, is the probability for finding the particle at a given spot.
— Seen from far away, at low magnification, a moving massive quantum particle behaves
as a single advancing, rotating and precessing arrow. The details of the rotation and
precession of the arrow depend on the energy and momentum of the particle and the
potential it is subjected to. The arrow is a probability amplitude: the squared length of
the arrow is the probability to observe the particle. If a particle can get from a starting
point to a final point in several ways, the probability amplitudes for each way add up.

Chapter 5

PE R M U TAT ION OF PA RT IC L E S – A R E
PA RT IC L E S L I K E G LOV E S ?

W

A number of important properties of objects are highlighted by studying a combinatorial
puzzle: the glove problem. It asks:
How many surgical gloves (for the right hand) are necessary if ? doctors
need to operate ? patients in a hygienic way, so that nobody gets in contact
with the body fluids of anybody else?

Ref. 71

Challenge 88 s
Challenge 89 e

Ref. 72

The same problem also appears in other settings. For example, it also applies to computers, interfaces and computer viruses or to condoms, men and women – and is then
called the condom problem. To be clear, the optimal number of gloves is not the product
??. In fact, the problem has three subcases.
— The simple case ? = ? = 2 already provides the most important ideas needed. Are
you able to find the optimal solution and procedure?
— In the case ? = 1 and ? odd, the solution is (? + 1)/2 gloves. The corresponding
expression (? + 1)/2 holds for the case ? = 1 and ? odd. This is the optimal solution,
as you can easily check yourself.
— A solution with a simple procedure for all other cases is given by ⌈2?/3+?/2⌉ gloves,
where ⌈?⌉ means the smallest integer greater than or equal to ?. For example, for two

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Distinguishing macroscopic objects

Motion Mountain – The Adventure of Physics

Challenge 87 s

hy are we able to distinguish twins from each other? Why can we distinguish
hat looks alike, such as a copy from an original? Most of us are convinced that
henever we compare an original with a copy, we can find a difference. This conviction turns out to be correct also in the quantum domain, but the conclusion is not
straightforward.
Think about any method that allows you to distinguish objects: you will find that it
runs into trouble for point-like particles. Therefore, in the quantum domain something
must change about our ability to distinguish particles and objects.
We could argue that differences between an original object and a copy can always be
made to disappear: it should be sufficient to use the same number and type of atoms. In
fact, the quantum of action shows that this is not sufficient, even though all atoms of the
same type are indeed indistinguishable copies of each other! In the following we explore
the most important consequences on motion of the indistinguishability of atoms and of
the distinguishability of macroscopic objects.

5 permutation of particles

Challenge 90 e

Vol. VI, page 112

Vol. V, page 245

113

doctors and three patients this gives only three gloves. (However, this formula does
not always give the optimal solution; better values exist in certain subcases.)

Vol. I, page 401

What is the entropy of a gas? The calculation of the entropy ? of a simple gas, made of ?
simple particles* of mass ? moving in a volume ?, gives
?
3 ln ?
?
= ln [ 3 ] + +
.
??
Λ
2
?

(55)

Here, ? is the Boltzmann constant, ln the natural logarithm, ? the temperature, and Λ =
√2πℏ2 /??? is the thermal wavelength (approximately the de Broglie wavelength of the
* Particles are simple if they are fully described by their momentum and position; atoms are simple particles.
Molecules are not simple, as they are describe also by their orientation.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Distinguishing atoms

Motion Mountain – The Adventure of Physics

Enjoy working on the puzzle. You will find that three basic properties of gloves determine
the solution. First, gloves have two sides, an interior and an exterior one, that can be
distinguished from each other. Secondly, gloves turned inside out exchange left and right
and can thus be distingusihed from gloves that are not reversed. Thirdly, gloves can be
distinguished from each other.
Now we come back to our original aim: Do the three basic properties of gloves also
apply to quantum particles? We will explore the issue of double-sidedness of quantum
particles in the last part of our mountain ascent. The question whether particles can be
turned inside out will be of importance for their description and their motion. We will
also explore the difference between right- and left-handed particles, though in the next
part of our adventure. In the present chapter we concentrate on the third issue, namely
whether objects and particles can always be distinguished from copies. We will find that
elementary particles do not behave like gloves – but in a much more surprising manner.
In everyday life, distinction of macroscopic objects can be achieved in two ways. On
the one hand, we are able to distinguish objects – or people – from each other because
they differ in their intrinsic properties, such as their mass, colour, size or shape. On the
other hand, we are able to distinguish objects even if they have the same intrinsic properties. Any game of billiard shows us that by following the path of each ball, we can
distinguish it from the other balls. In short, we can distinguish objects with identical
properties also using their state.
The state of a billiard ball is given by its position, its linear and its angular momentum.
We are able to distinguish two identical billiard balls because the measurement error for
the position of each ball is much smaller than the size of the ball itself. The different states
of two billiard balls allow us to track each ball. However, in the microscopic domain, this
is not possible! Let us take two atoms of the same type. Two such atoms have exactly the
same intrinsic properties. To distinguish them in collisions, we would need to keep track
of their motion. But due to the quantum of action and the ensuing indeterminacy relation, we have no chance to achieve this. In fact, a simple experiment from the nineteenth
century showed that even nature itself is not able to do it! This profound result was discovered studying systems which incorporate a large number of colliding atoms of the
same type: gases.

114

5 permutation of particles

F I G U R E 56 Willard Gibbs (1839 –1903)

Challenge 92 e
Ref. 73

* Indeed, the entropy values observed by experiment, for a monoatomic gas, are given by the so-called
Sackur–Tetrode formula
?
5
?
]+
= ln [
(56)
??
?Λ3
2
which follows when ? = 1/?! is inserted above. It was deduced independently by the German physicist Otto
Sackur (1880–1914) and the Dutch physicist Hugo Tetrode (1895–1931). Note that the essential parameter
is the ratio between ?/?, the classical volume per particle, and Λ3 , the de Broglie volume of a quantum
particle.
** Josiah Willard Gibbs (1839–1903), US-American physicist who was, with Maxwell and Planck, one of the
three founders of statistical mechanics and thermodynamics; he introduced the concept of ensemble and
the term thermodynamic phase.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 93 d

Motion Mountain – The Adventure of Physics

Challenge 91 e

particles making up the gas). In this result, the pure number ? is equal to 1 if the particles
are distinguishable like billiard balls, and equal to 1/?! if they are not distinguishable at
all. Measuring the entropy of a simple gas thus allows us to determine ? and therefore to
test experimentally whether particles are distinguishable.
It turns out that only the second case, ? = 1/?!, describes nature. We can easily check
this without even performing the measurement: only in the second case does the entropy
of two volumes of identical gas add up.* The result, often called Gibbs’ paradox,** thus
proves that the microscopic components of matter are indistinguishable: in a system of
quantum particles – be they electrons, protons, atoms or small molecules – there is no
way to say which particle is which.
Indistinguishability of particles is thus an experimental property of nature. It holds
without exception. For example, when radioactivity was discovered, people thought that
it contradicted the indistinguishability of atoms, because decay seems to single out certain atoms compared to others. But quantum theory then showed that this is not the case
and that even atoms and molecules are indistinguishable.
Since ℏ appears in the expression for the entropy, indistinguishability is a quantum
effect. Indeed, indistinguishability plays no role if quantum effects are negligible, as is the
case for billiard balls. Nevertheless, indistinguishability is important in everyday life. We
will find out that the properties of everyday matter – plasma, gases, liquids and solids –
would be completely different without indistinguishability. For example, we will discover
that without it, knifes and swords would not cut. In addition, the soil would not carry
us; we would fall right through it. To illuminate the issue in more detail, we explore the
following question.

5 permutation of particles

115

m

m
F I G U R E 57 Identical objects with
crossing paths.

Take two quantum particles with the same mass, the same composition and the same
shape, such as two atoms of the same kind. Imagine that their paths cross, and that they
approach each other to small distances at the crossing, as shown in Figure 57. In a gas,
both a collision of atoms or a near miss are examples. Now, all experiments ever performed show that at small distances it is impossible to say whether the two quantons
have switched roles or not.

The impossibility to distinguish nearby particles is a direct consequence of the quantum
of action ℏ. For a path that brings two approaching particles very close to each other, a
role switch requires only a small amount of change, i.e., only a small (physical) action.
However, we know that there is a smallest observable action in nature. Keeping track of
each quantum particle at small distances would require action values smaller than the
quantum of action. The existence of the quantum of action thus makes it impossible to
keep track of quantum particles when they come too near to each other. Any description
of systems with several quantons must thus take into account that after a close encounter,
it is impossible to say which quanton is which.
If we remember that quantum theory describes quantons as clouds, the indistinguishability appears even more natural. Whenever two clouds meet and depart again,
it is impossible to say which cloud is which. On the other hand, if two particles are kept
distant enough, one does have an effective distinguishability; indistinguishability thus
appears only when the particles come close.
In short, indistinguishability is a natural, unavoidable consequence of the existence of
a smallest action value in nature. This result leads us straight away to the next question:
C an quantum particles be counted?
In everyday life, we can count objects because we can distinguish them. Since quantum
particles cannot always be distinguished, we need some care in determining how to count

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ It is impossible in a gas to follow quantum particles moving around and to
determine which one is which. Tracking colliding quantons is impossible.

Motion Mountain – The Adventure of Physics

Why d oes indistinguishability appear in nature?

116

Vol. II, page 72

What is permu tation symmetry?
Since quantum particles are countable but indistinguishable, there exists a symmetry of
nature for systems composed of several identical quantons. Permutation symmetry, also
called exchange symmetry, is the property of nature that observations are unchanged under exchange of identical particles. Permutation symmetry forms one of the four pil-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

them. The first step in counting particles is the definition of what is meant by a situation
without any particle at all. This seems an easy thing to do, but later on we will encounter
situations where already this step runs into difficulties. In any case, the first step of counting is thus the specification of the vacuum. Any counting method requires that the situation without particles is clearly separated from situations with particles.
The second step necessary for counting is the specification of an observable useful
for determining quantum particle number. The easiest way is to choose one of those
conserved quantum numbers that add up under composition, such as electric charge.
Counting itself is then performed by measuring the total charge and dividing by the unit
charge.
In everyday life, the weight or mass is commonly used as observable. However, it
cannot be used generally in the quantum domain, except for simple cases. For a large
number of particles, the interaction energy will introduce errors. For very large particle
numbers, the gravitational binding energy will do so as well. But above all, for transient
phenomena, unstable particles or short measurement times, mass measurements reach
their limits. In short, even though counting stable atoms through mass measurements
works in everyday life, the method is not applicable in general; especially at high particle
energies, it cannot be applied.
Counting with the help of conserved quantum numbers has several advantages. First
of all, it works also for transient phenomena, unstable particles or short measurement
times. Secondly, it is not important whether the particles are distinguishable or not;
counting always works. Thirdly, virtual particles are not counted. This is a welcome
state of affairs, as we will see, because for virtual particles, i.e., particles for which
?2 ≠ ?2 ?2 + ?2 ?4 , there is no way to define a particle number anyway. Using a conserved
quantity is indeed the best particle counting method possible.
The side effect of counting with the help of quantum numbers is that antiparticles
count negatively! Also this consequence is a result of the quantum of action. We saw
above that the quantum of action implies that even in vacuum, particle–antiparticle pairs
are observed at sufficiently high energies. As a result, an antiparticle must count as minus
one particle. In other words, any way of counting quantum particles can produce an error
due to this effect. In everyday life this limitation plays no role, as there is no antimatter
around us. The issue does play a role at higher energies, however. It turns out that there is
no general way to count the exact number of particles and antiparticles separately; only
the sum can be defined. In short, quantum theory shows that particle counting is never
perfect.
In summary, nature does provide a way to count quantum particles even if they cannot
be distinguished, though only for everyday, low energy conditions; due to the quantum
of action, antiparticles count negatively. Antiparticles thus provide a limit to the counting
of particles at high energies, when the mass–energy equivalence becomes important.

Motion Mountain – The Adventure of Physics

Vol. II, page 72
Vol. V, page 127

5 permutation of particles

5 permutation of particles

117

lars of quantum theory, together with space-time symmetry, gauge symmetry and the
not yet encountered renormalization symmetry. Permutation symmetry is a property of
composed systems, i.e., of systems made of many (identical) subsystems. Only for such
systems does indistinguishability play a role.
In other words, ‘indistinguishable’ is not the same as ‘identical’. Two quantum
particles of the same type are not the same; they are more like exact copies of each other.
On the other hand, everyday life experience shows us that two copies can always be distinguished under close inspection, so that the term ‘copy’ is not fully appropriate either.
⊳ Quantons, quantum particles, are countable and completely indistinguishable.* Quantum particles are perfect copies of each other.

Indistinguishabilit y and wave function symmetry

Challenge 95 s

for some unknown angle ?. Applying this expression twice, by exchanging the same
couple of indices again, allows us to conclude that e2?? = 1. This implies that
Ψ1...?...?...? = ± Ψ1...?...?...? ,

(58)

in other words, a wave function is either symmetric or antisymmetric under exchange of
indices. (We can also say that the eigenvalue for the exchange operator is either +1 or
−1.)
⊳ Quantum theory thus predicts that quantum particles can be indistinguish* The word ‘indistinguishable’ is so long that many physicists sloppily speak of ‘identical’ particles nevertheless. Take care.
** We therefore have the same situation that we encountered already several times: an overspecification of
the mathematical description, here the explicit ordering of the indices, implies a symmetry of this description,
which in our case is a symmetry under exchange of indices, i.e., under exchange of particles.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The indistinguishability of quantum particles leads to important conclusions about the
description of their state of motion. This happens because it is impossible to formulate
a description of motion that includes indistinguishability right from the start. (Are you
able to confirm this?) We need to describe a ?-particle state with a state Ψ1...?...?...? which
assumes that distinction is possible, as expressed by the ordered indices in the notation,
and we introduce the indistinguishability afterwards.
Indistinguishability, or permutation symmetry, means that the exchange of any two
quantum particles results in the same physical observations.** Now, two quantum states
have the same physical properties if they differ at most by a phase factor; indistinguishability thus requires
Ψ1...?...?...? = e?? Ψ1...?...?...?
(57)

Motion Mountain – The Adventure of Physics

Challenge 94 e

Being perfect copies, not even nature can distinguish particles; as a result, permutation
symmetry appears.
In the next chapter, we will discover that permutation is partial rotation. Permutation
symmetry thus is a symmetry under partial rotations. Can you find out why?

118

5 permutation of particles

able in one of two distinct ways.*
⊳ Particles corresponding to symmetric wave functions – those which transform under particle exchange with a ‘+’ in equation (58) – are called** bosons.
⊳ Particles corresponding to antisymmetric wave functions – those which
transform under particle exchange with a ‘−’ in equation (58) – are called***
fermions.

The behaviour of photons

Ref. 74

* This conclusion applies to three-dimensional space. In two dimensions there are more possibilities. Such
possibilities have been and partly still are topic of research.
** ‘Bosons’ are named after the physicist Satyenra Nath Bose (b. 1894 Calcutta, d. 1974 Calcutta) who first
described the statistical properties of photons. The work was later expanded by Albert Einstein, so that one
speaks of Bose–Einstein statistics.
*** The term ‘fermion’ is derived from the name of the physicist and Nobel Prize winner Enrico Fermi
(b. 1901 Rome, d. 1954 Chicago) famous for his all-encompassing genius in theoretical and experimental
physics. He mainly worked on nuclear and elementary particle physics, on spin and on statistics. For his
experimental work he was called ‘quantum engineer’. He is also famous for his lectures, which are still
published in his own hand-writing, and his brilliant approach to physical problems. Nevertheless, his highly
deserved Nobel Prize was one of the few cases in which the prize was given for a discovery which turned
out to be incorrect. He left Italy because of the bad treatment his Jewish wife was suffering and emigrated to
the USA. Fermi worked on the Manhattan project that built the first atomic bombs. After the Second World
War, he organized one of the best physics department in the world, at the University of Chicago, where he
was admired by everybody who worked with him.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 75

A simple experiment, shown in Figure 58, allows observing an important aspect of
photon behaviour. Take a source that emits two indistinguishable photons, i.e., two
photons of identical frequency and polarization, at the same time. The photon pair is
therefore in an entangled state. In the laboratory, such a source can be realized with a
down-converter, a material that converts a photon of frequency 2? into two photons of
frequency ?. The two entangled photons, after having travelled exactly the same distance,
are made to enter the two sides of an ideal beam splitter (for example, a half-silvered mirror). Two detectors are located at the two exits of the beam splitter. Experiments show
that both photons are always detected together on the same side, and never separately on

Motion Mountain – The Adventure of Physics

Experiments show that the exchange behaviour depends on the type of particle. Photons
are found to be bosons. On the other hand, electrons, protons and neutrons are found
to be fermions. Also about half of the atoms are found to behave as bosons (at moderate
energies), the other half are fermions. To determine they type of atom, we need to take
into account the spin of the electron and that of the nucleus.
In fact, a composite of an even number of fermions (at moderate energies) – or of
any number of bosons (at any energy) – turns out to be a boson; a composite of an odd
number of fermions is (always) a fermion. For example, 4 He is a boson, 3 He a fermion.
Also the natural isotopes 23 Na, 41 K, 85 Rb, 87 Rb and 133 Cs are bosons, because they have
odd numbers of electrons and of nucleons; in contrast, 40 K and 134 Cs are fermions (and,
in this case, also radioactive).

5 permutation of particles

119

detectors

mirrors
beam
splitter

source
one photon of
frequency 2f

two photons of
same frequency f
and polarization

possible
light
paths

F I G U R E 58

Two-photon emission
and interference: two
indistinguishable
photons are always
found arriving
together, at the same
detector.

classical
prediction

classical
prediction

F I G U R E 59 Bunching and antibunching of 3 He and 4 He helium!bunching atoms: the measurement

result, the detector and the experiment (from atomoptic.iota.u-psud.fr/research/helium/helium.html,
photo © Denis Boiron, Jerome Chatin).

Page 59

Ref. 76

opposite sides. This happens because the two options where one of the photons is transmitted and the other reflected interfere destructively. (The discussion mentioned above
applies also here: despite two photons being involved, also in this case, when investigating the details, only one photon interferes with itself.)
The experiment shows that photons are bosons. Indeed, in the same experiment, fermions behave in exactly the opposite way; two fermions are always detected separately
on opposite sides, never together on the same side.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

3He shows anti-bunching

Motion Mountain – The Adventure of Physics

4He shows bunching

120

5 permutation of particles

F I G U R E 60 Picturing particles as localized excitations (left) or clouds (right).

Bunching and antibunching
Page 53

If experiments force us to conclude that nobody, not even nature, can distinguish
between two particles of the same type, we deduce that they do not form two separate entities, but some sort of unity. Our naive, classical sense of particle as a separate
entity from the rest of the world is thus an incorrect description of the phenomenon of
‘particle’. Indeed, no experiment can track particles with identical intrinsic properties
in such a way that they can be distinguished with certainty. This impossibility has been
checked experimentally with all elementary particles, with nuclei, with atoms and with
numerous molecules.
How does this fit with everyday life, i.e., with classical physics? Photons do not worry
us much here. Let us focus the discussion on matter particles. We know to be able to
distinguish electrons by pointing to the wire in which they flow, and we can distinguish
our fridge from that of our neighbour. While the quantum of action makes distinction
impossible, everyday life allows it.
The simplest explanation for both observations is to imagine a microscopic particle,
especially an elementary one, as a bulge, i.e., as a localized excitation of the vacuum, or
as a tiny cloud. Figure 60 shows two such bulges and two clouds representing particles.
It is evident that if particles are too near to each other, it makes no sense to distinguish
them; we cannot say any more which is which.
The bulge image shows that either for large distances or for high potential walls separating them, distinction of identical particles does become possible. In such situations,
measurements allowing us to track particles independently do exist – as we know from

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The energy dependence of permu tation symmetry

Motion Mountain – The Adventure of Physics

Ref. 77

Another way to test the exchange character of a particle is the Hanbury Brown–Twiss experiment described earlier on. First of all, this beautiful experiment shows that quantum
particles behave differently than classical particles. In addition, compared to classical
particles, fermions show antibunching – because of Pauli’s exclusion principle – and
bosons show bunching. Hanbury Brown and Twiss performed the experiment with
photons, which are bosons.
In 2005, a French–Dutch research collaboration performed the experiment with
atoms. By using an extremely cold helium gas at 500 nK and a clever detector principle,
they were able to measure the correlation curves typical for the effect. The results, shown
in Figure 59, confirm that 3 He is a fermion and 4 He is a boson, as predicted from the
composition rule of quantum particles.

5 permutation of particles

121

everyday life. In other words, we can specify a limit energy at which permutation symmetry of objects or particles separated by a distance ? becomes important. It is given
by
?ℏ
?=
.
(59)
?
Challenge 96 e

Challenge 97 e

Indistinguishabilit y in quantum field theory

— Field operators for particles with half-integer spin are fermions and imply (local) anticommutation.
— Fields with integer spin are bosons and imply (local) commutation.
— For all field operators at space-like separations, the commutator, respectively anticommutator, vanishes.
— Antiparticles of fermions are fermions, and antiparticles of bosons are bosons.
* Whenever the relation
[?, ?† ] = ??† − ?† ? = 1

(60)

†

holds between the creation operator ? and the annihilation operator ?, the operators describe a boson. The
dagger can thus be seen as describing the operation of adjoining; a double dagger is equivalent to no dagger.
If the operators for particle creation and annihilation anticommute
{?, ?† } = ??† + ?† ? = 1
they describe a fermion. The so defined bracket is called the anticommutator bracket.

(61)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Quantum field theory, as we will see in the next volume, simply puts the cloudy bulge
idea of Figure 60 into mathematical language. A situation without any bulge is called
vacuum state. Quantum field theory describes all particles of a given type as excitations
of a single fundamental field. Particles are indistinguishable because each particle is an
excitation of the same basic substrate and each excitation has the same properties. A
situation with one particle is then described by a vacuum state acted upon by a creation
operator. Adding a second particle is described by adding a second creation operator,
and subtracting a particle by adding a annihilation operator; the latter turns out to be the
adjoint of the former.
Quantum field theory studies how creation and annihilation operators must behave
to describe observations.* It arrives at the following conclusions:

Motion Mountain – The Adventure of Physics

Are you able to confirm the expression? For example, at everyday temperatures we can
distinguish atoms inside a solid from each other, since the energy so calculated is much
higher than the thermal energy of atoms. To have fun, you might want to determine at
what energy two truly identical human twins become indistinguishable. Estimating at
what energies the statistical character of trees or fridges will become apparent is then
straightforward.
To sum up, in daily life we are able to distinguish objects and thus people for two
reasons: because they are made of many parts, and because we live in a low energy environment. The bulge image of particles purveys the idea that distinguishability exists for
objects in everyday life but not for particles in the microscopic domain.
The energy issue immediately adds a new aspect to the discussion. How can we describe fermions and bosons in the presence of virtual particles and of antiparticles?

122

5 permutation of particles

— Virtual particles behave under exchange like their real counterparts.
These connections are at the basis of quantum field theory. They describe how quantons
behave under permutation.
But why are quantum particles identical? Why are all electrons identical? Lead by
experiment, quantum field theory describes electrons as identical excitations of the vacuum, and as such as identical by construction. Of course, this answer is not really satisfying. We will find a better one only in the final part of our mountain ascent.
How accurately is permu tation symmetry verified?
Ref. 78

C opies, clones and gloves

Ref. 79

Can classical systems be indistinguishable? They can: large molecules are examples –
provided they are made of exactly the same isotopes. Can large classical systems, made
of a mole or more particles be indistinguishable? This simple question effectively asks
whether a perfect copy, or (physical) clone, of a physical system is possible.
It could be argued that any factory for mass-produced goods, such as one producing
shirt buttons or paper clips, shows that copies are possible. But the appearance is deceiving. On a microscope there is usually some difference. Is this always the case? In 1982, the
Dutch physicist Dennis Dieks and independently, the US-American physicists Wootters
and Zurek, published simple proofs that quantum systems cannot be copied. This is the
famous no-cloning theorem.
A copying machine is a machine that takes an original, reads out its properties and
produces a copy, leaving the original unchanged. This definition seems straightforward.
However, we know that if we extract information from an original, we have to interact
with it. As a result, the system will change at least by the quantum of action. We thus
expect that due to quantum theory, copies and originals can never be identical.*
* This seems to provide a solution against banknote forgeries. In fact, Stephen Wiesner proposed to use

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

of the total state. In short, electrons are always in an antisymmetric state: they are fermions.
The reasoning behind this elegant experiment is the following. If electrons would not
always be fermions, every now and then an electron could fall into the lowest energy
level of a copper atom, leading to X-ray emission. The lack of such X-rays implies that
electrons are fermions to a very high accuracy. X-rays could be emitted only if they were
bosons, at least part of the time. Indeed, two electrons, being fermions, cannot be in the
same quantum state: this restriction is called the Pauli exclusion principle. It applies to all
fermions and is the topic of the next chapter.

Motion Mountain – The Adventure of Physics

Are electrons perfect fermions? In 1990, a simple but effective experiment testing their
fermion behaviour was carried out by Ramberg and Snow. They sent an electric current
of 30 A through a copper wire for one month and looked for X-ray emission. They did
not find any. They concluded that electrons are always in an antisymmetric state, with a
symmetric component of less than
2 ⋅ 10−26
(62)

5 permutation of particles

123

Quantum theory indeed shows that copying machines are impossible. A copying machine is described by an operator that maps the state of an original system to the state of
the copy. In other words, a copying machine is linear. This linearity leads to a problem.
Simply stated, if a copying machine were able to copy originals either in state |?⟩ or in
state |?⟩, it could not work if the state of the original were a superposition |?⟩ + |?⟩. Let
us see why.
A copy machine is a device described by an operator ? that changes the starting state
|?⟩c of the copy in the following way:
— If the original is in state |?⟩, a copier acts on the copy |?⟩c as
?|?⟩|?⟩c = |?⟩|?⟩c .

(63)

— If the original is in state |?⟩, a copier acts on the copy |?⟩c as
(64)

As a result of these two requirements, an original in the state |? + ?⟩ is treated by the
copier as
?|? + ?⟩|?⟩c = |?⟩|?⟩c + |?⟩|?⟩c .
(65)
This is in contrast to what we want, which would be
?wanted|? + ?⟩|?⟩c = (|?⟩ + |?⟩)(|?⟩c + |?⟩c ) .

Ref. 80
Challenge 98 s

In other words, a copy machine cannot copy a state completely.* This is the so-called
no-cloning theorem.
The impossibility of copying is implicit in quantum theory. If we were able to clone
systems, we could measure a variable of a system and a second variable on its copy. We
would be thus able to beat the indeterminacy relation in both copies. This is impossible.
In short, copies are always imperfect.
The lack of quantum mechanical copying machines is disappointing. Such science
fiction machines could be fed with two different inputs, such as a lion and a goat, and
produce a superposition: a chimaera. Quantum theory shows that all these imaginary
beings or situations cannot be realized.
Other researchers then explored how near to perfection a copy can be, especially in the
case of classical systems. To make a long story short, these investigations show that also
the copying or cloning of macroscopic systems is impossible. In simple words, copying
machines do not exist. Copies can always be distinguished from originals if observations
quantum theory already in 1970; he imagined to use polarizations of stored single photons as bits of serial
numbers. Can you explain why this cannot work?
* The no-cloning theorem puts severe limitations on quantum computers, as computations often need copies of intermediate results. The theorem also shows that faster-than-light communication is impossible in
EPR experiments. In compensation, quantum cryptography becomes possible – at least in the laboratory.
Indeed, the no-cloning theorem shows that nobody can copy a quantum message without being noticed.
The specific ways to use this result in cryptography are the 1984 Bennett–Brassard protocol and the 1991
Ekert protocol.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 81

(66)

Motion Mountain – The Adventure of Physics

?|?⟩|?⟩c = |?⟩|?⟩c .

124

Challenge 99 s

5 permutation of particles

are made with sufficient care. In particular, this is the case for biological clones; biological
clones are identical twins born following separate pregnancies. They differ in their finger
prints, iris scans, physical and emotional memories, brain structures, and in many other
aspects. (Can you specify a few more?) In short, biological clones, like identical twins,
are not copies of each other.
In summary, everyday life objects such as photocopies, billiard balls or twins are always distinguishable. There are two reasons: first, quantum effects play no role in everyday life, so that there is no danger of unobservable exchange; secondly, perfect clones of
classical systems do not exist anyway, so that there always are tiny differences between
any two objects, even if they look identical at first sight. Gloves, being classical systems,
can thus always be distinguished.
Summary
Motion Mountain – The Adventure of Physics

As a consequence of the quantum of action ℏ, quantum particles are indistinguishable.
This happens in one of two ways: they are either bosons or fermions. Not even nature is
able to distinguish between identical quantum particles.
Despite the indistinguishability of quantons, the state of a physical system cannot be
copied to a second system with the same particle content. Therefore, perfect clones do
not exist in nature.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Chapter 6

ROTAT ION S A N D STAT I ST IC S
– V I SUA L I Z I NG SPI N

S

Q uantum particles and symmetry
Ref. 82

Vol. I, page 265

** Eugene Wigner (b. 1902 Budapest, d. 1995 Princeton), theoretical physicist, received the Nobel Prize in
Physics in 1963. He wrote over 500 papers, many about various aspects of symmetry in nature. He was also
famous for being the most polite physicist in the world.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 265

The general background for the appearance of spin was clarified by Eugene Wigner in
1939.** He started by recapitulating that any quantum particle, if elementary, must behave like an irreducible representation of the set of all viewpoint changes. This set of viewpoint changes forms the symmetry group of flat space-time, the so-called inhomogeneous
Lorentz group. Why?
We have seen in the chapter on symmetry, in the first volume of this adventure, that
the symmetry of any composite system leads to certain requirements for the components
of the system. If the components do not follow these requirements, they cannot build a
symmetric composite.
We know from everyday life and precision experiments that all physical systems are
symmetric under translation in time and space, under rotation in space, under boosts,
and – in many cases – under mirror reflection, matter–antimatter exchange and motion
reversal. We know these symmetries from everyday life; for example, the usefulness of
what we call ‘experience’ in everyday life is simply a consequence of time translation
symmetry. The set of all these common symmetries, more precisely, of all these symmetry
transformations, is called the inhomogeneous Lorentz group.
These symmetries, i.e., these changes of viewpoints, lead to certain requirements for
the components of physical systems, i.e., for the elementary quantum particles. In mathematical language, the requirement is expressed by saying that elementary particles must
be irreducible representations of the symmetry group.

Motion Mountain – The Adventure of Physics

Page 104

pin is the observation that matter beams can be polarized: rays can be rotated.
pin thus describes how particles behave under rotations. Particles are thus not
imply point-like: quantum particles can rotate around an axis. This proper rotation
is called spin; like macroscopic rotation, spin is described by an angular momentum.
In the following, we recall that the spin of quantons is quantized in units of ℏ/2. Then
we show a deep result: the value of spin determines whether a quantum particle, and
any general quantum system, is a boson or a fermion. And we will show that spin is the
rotation of quantons.

126

6 rotations and statistics – visualizing spin

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

* To be of physical relevance for quantum theory, representations have to be unitary. The full list of irreducible and unitary representations of viewpoint changes thus provides the range of possibilities for any
particle that wants to be elementary.
** The group of physical rotations is also called SO(3), since mathematically it is described by the group of
Special Orthogonal 3 by 3 matrices.

Motion Mountain – The Adventure of Physics

Every textbook on quantum theory carries out this reasoning in systematic detail.
Starting with the Lorentz group, one obtains a list of all possible irreducible representations. In other words, on eobtains a list of all possible ways that elementary particles can
behave. * Cataloguing the possibilities, one finds first of all that every elementary particle
is described by four-momentum – no news so far – by an internal angular momentum,
the spin, and by a set of parities.
— Four-momentum results from the translation symmetry of nature. The momentum
value describes how a particle behaves under translation, i.e., under position and time
shift of viewpoints. The magnitude of four-momentum is an invariant property, given
by the mass, whereas its orientation in space-time is free.
— Spin results from the rotation symmetry of nature. The spin value describes how an
object behaves under rotations in three dimensions, i.e., under orientation change
of viewpoints.** The magnitude of spin is an invariant property, and its orientation
has various possibilities with respect to the direction of motion. In particular, the
spin of massive quantum particles behaves differently from that of massless quantum
particles.
For massive quantum particles, the inhomogeneous Lorentz group implies that
the invariant magnitude of spin is √?(? + 1) ℏ, often written, by oversimplification,
as ?. It is thus customary to say and write ‘spin J’ instead of the cumbersome ‘spin
√?(? + 1) ℏ’. Since the value of the quantum number ? specifies the magnitude of the
angular momentum, it gives the representation under rotations of a given particle
type. The exploration shows that the spin quantum number ? can be any multiple
of 1/2, i.e., it can take the values 0, 1/2, 1, 3/2, 2, 5/2, etc. As summarized in Table 4,
experiments show that electrons, protons and neutrons have spin 1/2, the W and Z
particles spin 1 and helium atoms spin 0. In addition, the representation of spin ?
is 2? + 1 dimensional, meaning that the spatial orientation of the spin has 2? + 1
possible values. For electrons, with ? = 1/2, there are thus two possibilities; they are
usually called ‘up’ and ‘down’. Spin thus only takes discrete values. This is in contrast
with linear momentum, whose representations are infinite dimensional and whose
possible values form a continuous range.
Also massless quantum particles are characterized by the value of their spin. It can
take the same values as in the massive case. For example, photons and gluons have
spin 1. For massless particles, the representations are one-dimensional, so that massless particles are completely described by their helicity, defined as the projection of
the spin onto the direction of motion. Massless particles can have positive or negative helicity, often also called right-handed and left-handed polarization. There is no
other freedom for the orientation of spin in the massless case.
— To complete the list of particle properties, the remaining, discrete symmetries of the
inhomogeneous Lorentz group must be included. Since motion inversion, spatial parity and charge inversion are parities, each elementary particle has to be described by
three additional numbers, called T, P and C, each of which can only take the values

6 rotations and statistics – visualizing spin

127

TA B L E 4 Particle spin as representation of the rotation group.

Spin System
Massive examples
[ℏ]
unchanged after elementary composite
rotation by

Massless examples
elementary

0

any angle

Higgs
boson

none ?

1/2

2 turns

?, ?, ?, ?, nuclei, atoms,
?? , ?? , ??
molecules,
radicals

1

1 turn

W, Z

mesons, nuclei, photon ?, gluon ?
atoms, molecules,
toasters

3/2

2/3 turn

none ?

baryons, nuclei,
atoms

none ?

2

1/2 turn

none

nuclei

‘graviton’

5/2

2/5 turn

none

nuclei

none

3

1/3 turn

none

nuclei ?

none

etc.

?

etc.

?

etc.

?

etc.

none, as neutrinos have a tiny mass

?

none possible

+1 or −1. Being parities, these numbers must be multiplied to yield the value for a
composed system.
In short, the symmetries nature lead to the classification of all elementary quantum
particles by their mass, their momentum, their spin and their P, C and T parities.
Types of quantum particles

Vol. V, page 162

The spin values observed for all quantum particles in nature are given in Table 4. The
parities and all known intrinsic properties of the elementary particles are given in Table 5.
Spin and parities together are called quantum numbers. All other intrinsic properties of
quantons are related to interactions, such as mass, electric charge or isospin, and we will
explore them in the next volume.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?. Supersymmetry, a symmetry conjectured in the twentieth century, predicts elementary
particles in these and other boxes.
?. The graviton has not yet been observed.
?. Nuclei exist with spins values up to at least 101/2 and 51 (in units of ℏ). Ref. 83

Motion Mountain – The Adventure of Physics

?

mesons, nuclei,
atoms

128

6 rotations and statistics – visualizing spin

TA B L E 5 Elementary particle properties.

Particle

Mass ? ?

Lifetime ?
or energy
width, ?
main decay
modes

Isospin ?,
spin ?, ?
parity ?,
charge
parity ?

Charge,
isospin,
strangeness, ?
charm,
beauty, ?
topness:
??????

Lepton
&
baryon ?
numbers
??

000000

0, 0

±100000

0, 0

000000

0, 0

000000

0, 0

Elementary radiation (bosons)
0 (<10−53 kg)

?±

80.398(25) GeV/?2

?

91.1876(21) GeV/?2

gluon

0

?(??? ) =
0, 1(1−−)
2.124(41) GeV ? = 1
67.60(27) % hadrons,
32.12(36) % ?+ ?
2.65(2) ⋅ 10−25 s ? = 1
or 2.4952(23) GeV/?2
69.91(6) % hadrons,
10.0974(69) % ?+ ?−
stable
?(?? ) = 0(1− )
stable

Elementary matter (fermions): leptons

muon ?

tau ?

9.109 382 15(45) ⋅
> 13 ⋅ 1030 s
? = 12
−100 000
10−31 kg = 81.871 0438(41) pJ/?2
= 0.510 998 910(13) MeV/?2 = 0.000 548 579 909 43(23) u
gyromagnetic ratio ?? /?B = −1.001 159 652 1811(7)
0.188 353 130(11) yg 2.197 03(4) μs ? = 12
−100000
99 % ?− ??̄ ??
= 105.658 3668(38) MeV/?2 = 0.113 428 9256(29) u
gyromagnetic ratio ?? /(?ℏ/2?? ) = −1.001 165 9208(6)
? = 12
−100000
1.776 84(17) GeV/?2 290.6(1.0) fs

el. neutrino < 2 eV/?2
?e
muon
< 2 eV/?2
neutrino ??
tau neutrino < 2 eV/?2
??

1, 0

1, 0

1, 0

?=

1
2

1, 0

?=

1
2

1, 0

?=

1
2

1, 0

Elementary matter (fermions): quarks ?
up ?
down ?
strange ?
charm ?

1.5 to 3.3 MeV/?2
3.5 to 6 MeV/?2
70 to 130 MeV/?2
1.27(11) GeV/?2

see proton
see proton

+

?(?? ) = 12 ( 12 )
+
?(?? ) = 12 ( 12 )
+
?(?? ) = 0( 12 )
+
?(?? ) = 0( 12 )

+ 23 + 12 0000
− 13 − 12 0000
− 13 0−1000
+ 23 00+100

0, 13
0, 13
0, 13
0, 13

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

electron ?

Motion Mountain – The Adventure of Physics

photon ?

6 rotations and statistics – visualizing spin

129

TA B L E 5 (Continued) Elementary particle properties.

Particle

Mass ? ?

Lifetime ?
or energy
width, ?
main decay
modes

bottom ?
top ?

4.20(17) GeV/?2
171.2(2.1) GeV/?2

? = 1.33(11) ps ?(?? ) = 0( 12 )
+
?(?? ) = 0( 12 )

Isospin ?,
spin ?, ?
parity ?,
charge
parity ?

+

Charge,
isospin,
strangeness, ?
charm,
beauty, ?
topness:
??????

Lepton
&
baryon ?
numbers
??

− 13 000−10
+ 23 0000+1

0, 13
0, 13

Observed elementary boson
Higgs boson 126 GeV/?2

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Notes:
?. See also the table of SI prefixes on page 206. About the eV/?2 mass unit, see page 210.
?. The energy width Γ of a particle is related to its lifetime ? by the indeterminacy relation Γ? = ℏ.
There is a difference between the half-life ?1/2 and the lifetime ? of a particle: they are related by
?1/2 = ? ln 2, where ln 2 ≈ 0.693 147 18; the half-life is thus shorter than the lifetime. The unified
atomic mass unit u is defined as 1/12 of the mass of a carbon 12 atom at rest and in its ground
1
?(12 C) = 1.660 5402(10) yg.
state. One has 1 u = 12
?. To keep the table short, its header does not explicitly mention colour, the – confusingly named –
charge of the strong interactions. It has to be added to the list of basic object properties. Quantum
numbers containing the word ‘parity’ are multiplicative; all others are additive. Parity ? and
charge parity ? are written as + or −. Time parity ? (not to be confused with topness ?), better
called motion inversion parity, is equal to CP in all known particles. The isospin ? (or ?Z ) appears
twice in the table; it is defined only for up and down quarks and their composites, such as the
proton and the neutron. In the literature one also sees references to the so-called ?-parity, defined
as ? = (−1)?? .
The table header also does not mention the weak charge of the particles. The details on weak
charge ?, or, more precisely, on the weak isospin, a quantum number assigned to all left-handed
fermions (and right-handed anti-fermions), but to no right-handed fermion (and no left-handed
antifermion), are given in the section on the weak interactions.
?. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called topness. The signs of the quantum numbers ?, ?, ?, ?, ? can be defined in different ways. In the
standard assignment shown here, the sign of each of the non-vanishing quantum numbers is
given by the sign of the charge of the corresponding quark.
?. If supersymmetry existed, ?-parity would have to be added to this column. ?-parity is a multiplicative quantum number related to the lepton number ?, the baryon number ? and the spin
? through the definition ? = (−1)3?+?+2? . All particles from the standard model are ?-even,
whereas their conjectured supersymmetric partner particles would be ?-odd. However, supersymmetry is now known to be in contrast with experiment.
?. For the precise definition and meaning of quark masses, see page 233 in volume V.

Motion Mountain – The Adventure of Physics

Vol. V, page 245

?=0

130

6 rotations and statistics – visualizing spin

F I G U R E 61 Illustrating an argument showing why rotations by 4π are equivalent to no rotation at all

Spin 1/2 and tethered objects

Challenge 100 e

Challenge 101 e

Challenge 102 e

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 49

A central result of quantum theory is that spin 1/2 is a possibility in nature, even though
this value does not appear in everyday life. For a system to have spin 1/2 means that for
such a system only a rotation by two turns is equivalent to none at all, while one by one
turn is not. No simple systems with this property exist in everyday life, but such systems
do exist in microscopic systems: electrons, neutrinos, silver atoms and molecular radicals
all have spin 1/2. Table 4 gives a more extensive list.
The mathematician Hermann Weyl used a simple image to explain that the rotation by
two turns is equivalent to zero turns, whereas one turns differs. Take two cones, touching
each other at their tips as well as along a line, as shown in Figure 61. Hold one cone and
roll the other around it. When the rolling cone, after a full turn around the other cone,
i.e., around the vertical axis, has come back to the original position, it has rotated by some
angle. If the cones are wide, as shown on the left, the final rotation angle is small. The limit
of extremely wide cones gives no rotation at all. If the cones are very thin, like needles,
the moving cone has rotated by (almost) 720 degrees; this situation is like a coin rolling
around a second coin of the same size, both lying on a table. The rolling coins rotates
by two turns, thus by 720 degrees. Also in this case, the final rotation angle is small. The
result for 0 degrees and for 720 degrees is the same. If we imagine the cone angle to
vary continuously, this visualization shows that a 0 degree rotation can be continuously
changed into a 720 degree rotation. In contrast, a 360 degree rotation cannot be ‘undone’
in this way.
There are systems in everyday life that behave like spin 1/2, but they are not simple:
all such systems are tethered. The most well-known system is the belt. Figure 62 and Figure 63 show that a rotation by 4π of a belt buckle is equivalent to no rotation at all: this is
easily achieved by moving the belt around. You may want to repeat the process by yourself, using a real belt or a strip of paper, in order to get a feeling for it. The untangling
process is often called the belt trick, but also scissor trick, plate trick, string trick, Philippine wine dance or Balinese candle dance. It is sometimes incorrectly attributed to Dirac,

Motion Mountain – The Adventure of Physics

(see text).

6 rotations and statistics – visualizing spin

131

Challenge 103 e

because he used it extensively in his lectures.
The human body has such a belt built in: the arm. Just take your hand, put an object
on it for clarity, such as a cup, and turn the hand and object by 2π by twisting the arm.
After a second rotation the whole system will be untangled again, as shown in Figure 64.
The trick is even more impressive when many arms are used. You can put your two hands
(if you chose the correct starting position) under the cup or you can take a friend or two

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 63 The belt trick with a simple belt: a double rotation of the belt buckle is equivalent to no
rotation. (QuickTime film © Greg Egan)

Motion Mountain – The Adventure of Physics

F I G U R E 62 Assume that the belt cannot be observed, but the square object can, and that it represents
a particle. The animation then shows that such a particle (the square object) can return to the starting
position after rotation by 4π (and not after 2π). Such a ‘belted’ particle thus fulfils the defining property
of a spin 1/2 particle: rotating it by 4π is equivalent to no rotation at all. The belt thus represents the
spinor wave function; for example, a 2π rotation leads to a twist; this means a change of the sign of the
wave function. A 4π rotation has no influence on the wave function. You can repeat the trick at home,
with a paper strip. The equivalence is shown here with two attached belts, but the trick works with any
positive number of belts! (QuickTime film © Antonio Martos)

132

?=0

6 rotations and statistics – visualizing spin

? = 2π

? = 4π

F I G U R E 64 The human arm as spin 1/2 model.

or simply rearranging
the bands gives the
other situation

who each keep a hand attached to the cup together with you. The belt trick can still be
performed, and the whole system untangles after two full turns.
This leads us to the most general way to show the connection between tethering and
spin 1/2. Just glue any number of threads, belts or tubes, say half a metre long, to some
object, as shown in Figure 65. (With many such tails, is not appropriate any more to call
it a belt buckle.) Each band is supposed to go to spatial infinity and be attached there.
Instead of being attached at spatial infinity, we can also imagine the belts attached to a
distant, fixed object, like the arms are attached to a human body. If the object, which
represents the particle, is rotated by 2π, twists appear in its tails. If the object is rotated
by an additional turn, to a total of 4π, all twists and tangles can be made to disappear,
without moving or turning the object. You really have to experience this in order to believe it. And the process really works with any number of bands glued to the object. The
website www.evl.uic.edu/hypercomplex/html/dirac.html provides a animation showing
this process with four attached belts.
In short, all these animations show that belt buckles, and in fact all (sufficiently)
tethered systems, return to their original state only after rotations by 4π, and not after
rotations by 2π only. Tethered objects behave like spin 1/2 particles. In fact, tethered ob-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 104 e

F I G U R E 65 The generalized
belt trick, modelling the
rotation behaviour of a spin
1/2 particle: independently of
the number of bands or tubes
or strings attached, the two
situations can be transformed
into each other, either by
rotating the central object by
4π or by keeping the central
object fixed and moving the
bands around it.

Motion Mountain – The Adventure of Physics

rotating the buckle
either by 4π

6 rotations and statistics – visualizing spin

133

F I G U R E 66 Two belt buckles
connected by a belt, one way of
visualizing two spin 1/2 particles.

Challenge 105 e

The extension of the belt trick

Ref. 86
Ref. 87
Page 120

But why do experiments show that all fermions have half-integer spin and that all bosons
have integer spin? In particular, why do electrons obey the Pauli exclusion principle? At
first sight, it is not clear what the spin value has to do with the statistical properties of a
particle. In fact, there are several ways to show that rotations and statistics are connected.
The first proof, due to Wolfgang Pauli, used the details of quantum field theory and was so
complicated that its essential ingredients were hidden. It took several decades to convince
everybody that a further observation about belts was the central part of the proof.
Starting from the bulge model of quantum particles shown in Figure 60, we can imagine a tube connecting two particles, similar to a belt connecting two belt buckles, as
shown in Figure 66. The buckles represent the particles. The tube keeps track of their
relative orientation. If one particle/buckle is rotated by 2π along any axis, a twist is inserted into the belt. As just shown, if the same buckle is rotated by another 2π, bringing the
total to 4π, the ensuing double twist can easily be undone without moving or rotating
the buckles.
Now we look again at Figure 66. If we take the two buckles and simply swap their
positions, a twist is introduced into the belt. If we swap them again, the twist will disappear. In short, two connected belt buckles return to their original state only after a double
exchange, and not after a single exchange.
In other words, if we take each buckle to represent a particle and a twist to mean
a factor −1, the belt exactly describes the phase behaviour of spin 1/2 wave functions,
both under rotation and under exchange. In particular, we see that rotation and exchange
behaviour are related.
Similarly, also the belt trick itself can be extended to exchange. Take two buckles that

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 135

Motion Mountain – The Adventure of Physics

jects, such as belt buckles, are the only systems that reproduce spin 1/2 properties. In the
last part of our adventure we will discover the deep underlying reason for the equivalence
between spin 1/2 particles and tethered systems.
Exploring the symmetries of wave functions, quantum theory shows that rotations
require the existence of spin for all quantum particles. An investigation of the wave function shows that wave functions of elementary matter particles behave under rotation like
tethered objects. For example, a wave function whose tethered equivalent is tangled acquires a negative sign.
In summary, quantum theory implies the existence of the slightly counter-intuitive
spin 1/2 value. In particular, it appears for elementary matter particles.

134

6 rotations and statistics – visualizing spin

F I G U R E 67 Extended
belt models for two
spin 1/2 particles.
Motion Mountain – The Adventure of Physics

Challenge 106 e

are connected with many bands or threads, like in Figure 67 or in Figure 68. The band can
connect the particles, or go to spatial infinity, or both. An exchange of the two buckles
produces quite a messy tangle. But almost incredibly, in all cases, a second exchange leads
back to the original situation, if the belts are properly rearranged. You might want to test

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 68 Assume that the belts cannot be observed, but the square objects can, and that they
represent particles. We know from above that belted buckles behave as spin 1/2 particles. The
animation shows that two such particles return to the original situation if they are switched in position
twice (but not once). Such particles thus fulfil the defining property of fermions. (For the opposite case,
that of bosons, a simple exchange would lead to the identical situation.) You can repeat the trick at
home using paper strips. The equivalence is shown here with two belts per particle, but the trick works
with any positive number of belts attached to each buckle. This animation is the essential part of the
proof that spin 1/2 particles are fermions. This is called the spin–statistics theorem. (QuickTime film
© Antonio Martos)

6 rotations and statistics – visualizing spin

Ref. 88

135

yourself that the behaviour is also valid if additional particles are involved, as long as you
always exchange the same two particles twice.
We conclude that tethered objects behave like fermions under exchange. These observations together form the spin–statistics theorem for spin 1/2 particles: spin and exchange behaviour are related. Indeed, these almost ‘experimental’ arguments can be put
into exact mathematical language by studying the behaviour of the configuration space
of particles. These investigations result in the following statements:
⊳ Objects of spin 1/2 are fermions.*
⊳ Exchange and rotation of spin 1/2 particles are similar processes.

Angels, Pauli ’ s exclusion principle and the hardness of mat ter
Why are we able to knock on a door? Why can stones not fly through tree trunks? How
does the mountain we are walking on carry us? Why can’t we walk across walls? In classical physics, we avoided this issue, by taking solidity as a defining property of matter.
But we cannot do so any more: we have seen that matter consists mainly of low density
electron clouds. The quantum of action thus forces us to explain the quantum of matter.
The explanation of the impenetrability of matter is so important that it led to a Nobel
prize in physics. The interpenetration of bodies is made impossible by Pauli’s exclusion
principle among the electrons inside atoms. Pauli’s exclusion principle states:
⊳ Two fermions cannot occupy the same quantum state.

Challenge 107 e
Page 189

* A mathematical observable behaving like a spin 1/2 particle is neither a vector nor a tensor, as you may
want to check. An additional concept is necessary; such an observable is called a spinor. We will introduce
it in detail later on.
** Of course, knots and tangles do exist in higher dimensions. Instead of considering knotted onedimensional lines, one can consider knotted planes or knotted higher-dimensional hyperplanes. For example, deformable planes can be knotted in four dimensions and deformable 3-spaces in five dimensions.
However, the effective dimensions that produce the knot are always three.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 109 d

Motion Mountain – The Adventure of Physics

Challenge 108 s

In short, objects that behave like spin 1/2 particles under rotations also behave like fermions under exchange. And vice versa. The exchange behaviour of particles determines
their statistical properties; the rotation behaviour determines their spin. By extending the
belt trick to several buckles, each with several belts, we thus visualized the spin–statistics
theorem for fermions.
Note that all these arguments require three dimensions of space, because there are no
tangles (or knots) in fewer or more dimensions.** And indeed, spin exists only in three
spatial dimensions.
The belt trick leads to interesting puzzles. We saw that a spin 1/2 object can be modelled by imagining that a belt leading to spatial infinity is attached to it. If we want to
model the spin behaviour with attached one-dimensional strings instead of bands, what
is the minimum number of strings we need? More difficult is the following puzzle: Can
the belt trick be performed if the buckle is glued into a mattress, thus with the mattress
acting like ‘infinitely many’ belts?

136

Ref. 89

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 90

All experiments known confirm the statement.
Why do electrons and other fermions obey Pauli’s exclusion principle? The answer
can be given with a beautifully simple argument. We know that exchanging two fermions
produces a minus sign in the total wave function. Imagine these two fermions being, as
a classical physicist would say, located at the same spot, or as a quantum physicist would
say, in the same state. If that could be possible, an exchange would change nothing in the
system. But an exchange of fermions must produce a minus sign for the total state. Both
possibilities – no change at all as well as a minus sign – cannot be realized at the same
time. There is only one way out: two fermions must avoid to ever be in the same state.
This is Pauli’s exclusion principle.
The exclusion principle is the reason that two pieces of matter in everyday life cannot
penetrate each other, but have to repel each other. For example, take a bell. A bell would
not work if the colliding pieces that produce the sound would interpenetrate. But in any
example of two interpenetrating pieces, the electrons from different atoms would have to
be at the same spot: they would have to be in the same states. This is impossible. Pauli’s
exclusion principle forbids interpenetration of matter. Bells only work because of the
exclusion principle.
Why don’t we fall through the floor, even though gravity pulls us down, but remain
standing on its surface? Again, the reason is Pauli’s exclusion principle. Why does the
floor itself not fall? It does not fall, because the matter of the Earth cannot interpenetrate
and the atoms cannot made to approach each other than a certain minimal distance. In
other words, Pauli’s exclusion principle implies that atomic matter cannot be compressed
indefinitely. At a certain stage an effective Pauli pressure appears, so that a compression
limit ensues. For this reason for example, planets made of atomic matter – or neutron
stars made of neutrons, which also have spin 1/2 and thus also obey the exclusion principle – do not collapse under their own gravity.
The exclusion principle is the reason that atoms are extended electron clouds and that
different atoms have different sizes. In fact, the exclusion principle forces the electrons
in atoms to form shells. When electrons are added around a nucleus and when one shell
is filled, a new shell is started. This is the origin of the periodic systems of the elements.
The size of any atom is the size of its last shell. Without the exclusion principle, atoms
would be as small as a hydrogen atom. In fact, most atoms are considerably larger. The
same argument applies to nuclei: their size is given by the last nucleon shell. Without the
exclusion principle, nuclei would be as small as a single proton. In fact, they are usually
about 100 000 times larger.
The exclusion principle also settles an old question: How many angels can dance on
the top of a pin? (Note that angels, if at all, must be made of fermions, as you might
want to deduce from the information known about them, and that the top of a pin is a
single point in space.) Both theory and experiment confirm the answer already given by
Thomas Aquinas in the Middle Ages: Only one angel! The fermion exclusion principle
could also be called ‘angel exclusion principle’. To stay in the topic, the principle also
shows that ghosts cannot be objects, as ghosts are supposed to be able to traverse walls.
Let us sum up. Simplifying somewhat, the exclusion principle keeps things around us
in shape. Without the exclusion principle, there would be no three-dimensional objects.
Only the exclusion principle fixes the diameter of atomic clouds, keeps these clouds from
merging, and holds them apart. This repulsion is the origin for the size of soap, planets

Motion Mountain – The Adventure of Physics

Challenge 110 s

6 rotations and statistics – visualizing spin

6 rotations and statistics – visualizing spin

Challenge 111 e
Ref. 82

137

and neutron stars. All shapes of solids and fluids are a direct consequence of the exclusion
principle. In other words, when we knock on a table or on a door, we prove experimentally that these objects and our hands are made of fermions.
So far, we have only considered fermions of spin 1/2. We will not talk much about
particles with odd spin of higher value, such as 3/2 or 5/2. Such particles can all be seen
as being composed of spin 1/2 entities. Can you confirm this?
We did not talk about lower spins than 1/2 either. A famous theorem states that a spin
value between 0 and 1/2 is impossible in three dimensions. Smaller spins are impossible
because the largest rotation angle that can be distinguished and measured in three dimensions is 4π. There is no way to measure a larger angle; the quantum of action makes
this impossible. Thus there cannot be any spin value between 0 and 1/2 in nature.
Is spin a rotation ab ou t an axis?

Now, we just found that spin is exchange behaviour. Since rotation is exchange and spin
is exchange, it follows that
⊳ Spin is rotation.
Challenge 112 ny

* This magnetic moment can easily be measured in an experiment; however, not one of the Stern–Gerlach
type. Why not?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ Rotation and exchange are the same process.

Motion Mountain – The Adventure of Physics

The spin of a particle behaves experimentally like an intrinsic angular momentum, adds
up like angular momentum, is conserved as part of angular momentum, is described like
angular momentum and has a name synonymous with angular momentum. Despite all
this, for many decades a strange and false myth was spread in many physics courses and
textbooks around the world: “Spin 1/2, despite its name, is not a rotation about an axis.”
It is time to finish with this example of incorrect thinking.
Electrons do have spin 1/2 and are charged. Electrons and all other charged particles
with spin 1/2 do have a magnetic moment.* A magnetic moment is expected for any
rotating charge. In other words, spin 1/2 does behave like rotation. However, assuming
that a particle consists of a continuous charge distribution in rotational motion gives the
wrong value for the magnetic moment. In the early days of the twentieth century, when
physicists were still thinking in classical terms, they concluded that charged spin 1/2
particles thus cannot be rotating. This myth has survived through many textbooks. The
correct deduction, however, is that the assumption of continuous charge distribution is
wrong. Indeed, charge is quantized; nobody expects that elementary charge is continuously spread over space, as that would contradict its quantization.
The other reason for the false myth is rotation itself. The myth is based on classical
thinking and maintains that any rotating object must have integer spin. Since half integer
spin is not possible in classical physics, it is argued that such spin is not due to rotation.
But let us recall what rotation is. Both the belt trick for spin 1/2 as well as the integer
spin case remind us: a rotation of one body around another is a fraction or a multiple
of an exchange. What we call a rotating body in everyday life is a body continuously
exchanging the positions of its parts – and vice versa.

138

6 rotations and statistics – visualizing spin

?
?

F I G U R E 69 Equivalence of exchange and rotation
in space-time.

Ref. 91

Vol. II, page 72

The connection between rotation and antiparticles may be the most astonishing conclusion from the experiments showing the existence of spin. So far, we have seen that
rotation requires the existence of spin, that spin appears when relativity is introduced
into quantum theory, and that relativity requires antimatter. Taking these three statements together, the conclusion of the title is not surprising any more: rotation requires
antiparticles. Interestingly, there is a simple argument making the same point with the
belt model, if it is extended from space alone to full space-time.
To learn how to think in space-time, let us take a particle and reduce it to two short
tails, so that the particle is a short line segment. When moving in a 2+1 dimensional
* Obviously, the exact structure of the electron still remains unclear at this point. Any angular momentum
? is given classically by ? = Θ?; however, neither the moment of inertia Θ, connected to the rotation radius
and electron mass, nor the angular velocity ? are known at this point. We have to wait quite a while, until
the final part of our adventure, to find out more.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Rotation requires antiparticles

Motion Mountain – The Adventure of Physics

Page 131

Since we deduced spin, like Wigner, from rotation invariance, this conclusion is not a
surprise. In addition, the belt model of a spin 1/2 particle tells us that such a particle
can rotate continuously without any hindrance. Also the magnetic moment then gets its
correct value. In short, we are allowed to maintain that spin is rotation about an axis,
without any contradiction to observations, even for spin 1/2.
In summary, the belt model shows that also spin 1/2 is rotation, as long as we assume
that only the buckle can be observed, not the belt(s), and that elementary charge is not
continuously distributed in space.*
Since permutation properties and spin properties of fermions are so well described
by the belt model, we could be led to the conclusion that these properties might really
be consequence of such belt-like connections between particles and the outside world.
Maybe for some reason we only observe the belt buckles, not the belts themselves. In the
final part of this walk we will discover whether this idea is correct.

6 rotations and statistics – visualizing spin

t

t

x

t

x

139

t

x

t

x

x

F I G U R E 70 Belts in space-time: rotation and antiparticles.

Challenge 113 ny

When a sword is approaching dangerously, we can stop it with a second sword. Many old
films use such scenes. When a laser beam is approaching, it is impossible to fend it off
with a second beam, despite all science fiction films showing so. Banging two laser beams
against each other is impossible. The above explanation of the spin–statistics theorem
shows why.
The electrons in the swords are fermions and obey the Pauli exclusion principle. Fermions make matter impenetrable. On the other hand, the photons in laser beams are

Challenge 114 ny

* Obviously, the full argument would need to check the full spin 1/2 model of Figure 65 in four-dimensional
space-time. But doing this is not an easy task; there is no good visualization yet.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Why is fencing with laser beams impossible?

Motion Mountain – The Adventure of Physics

space-time, the particle is described by a ribbon. Playing around with ribbons in spacetime, instead of belts in space, provides many interesting conclusions. For example, Figure 69 shows that wrapping a rubber ribbon around the fingers can show, again, that a
rotation of a body by 2π in presence of a second one is the same as exchanging the positions of the two bodies.* Both sides of the hand transform the same initial condition, at
one edge of the hand, to the same final condition at the other edge. We have thus successfully extended a known result from space to space-time: rotation and exchange are
equivalent.
If you think that Figure 69 is not a satisfying explanation, you are right. A more satisfying explanation must include a smooth sequence of steps realizing the equivalence
between rotation and exchange. This is shown in Figure 70. We assume that each particle
is described by a segment; in the figure, the two segments lie horizontally. The leftmost
diagram shows two particles: one at rest and one being rotated by 2π. The deformation
of the ribbons shows that this process is equivalent to the exchange in position of two
particles, which is shown in the rightmost diagram.
But the essential point is made by the intermediate diagrams. We note that the sequence showing the equivalence between rotation and exchange requires the use of a
loop. But such a loop in space-time describes the appearance of a particle–antiparticle
pair! In other words, without antiparticles, the equivalence of rotation and exchange
would not hold. In short, rotation in space-time requires the existence of antiparticles.

140

6 rotations and statistics – visualizing spin

J=0

J = 1/2

J=1

F I G U R E 71 Some visualizations of
spin representations.

Under rotations, integer spin particles behave differently from half-integer particles. Integer spin particles do not show the strange sign changes under rotations by 2π. In
the belt imagery, integer spin particles need no attached strings. In particular, a spin
0 particle obviously corresponds to a sphere. Models for other important spin values are
shown in Figure 71. Exploring their properties in the same way as above, we arrive at the
full spin–statistics theorem:
⊳ Exchange and rotation of objects are similar processes.
⊳ Objects of half-integer spin are fermions. They obey the Pauli exclusion
principle.
⊳ Objects of integer spin are bosons.
Challenge 115 e

You might prove by yourself that this suffices to show the following rule:
⊳ Composites of bosons, as well as composites of an even number of fermions (at low energy), are bosons; composites of an uneven number of fermions are fermions.*

Challenge 116 s

* This rule implies that spin 1 and higher can also be achieved with tails; can you find such a representation?
Note that composite fermions can be bosons only up to that energy at which the composition breaks
down. Otherwise, by packing fermions into bosons, we could have fermions in the same state.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Spin, statistics and composition

Motion Mountain – The Adventure of Physics

Vol. I, page 98

bosons. Two bosons can be in the same state; bosons allow interpenetration. Matter is
impenetrable because at the fundamental level it is composed of fermions. Radiation is
composed of bosons; light beams can cross each other. The distinction between fermions and bosons thus explains why objects can be touched while images cannot. In the
first part of our mountain ascent we started by noting this difference; now we know its
origin.

6 rotations and statistics – visualizing spin

Challenge 117 s

141

These connections express basic characteristics of the three-dimensional world in which
we live. To which class of particles do tennis balls, people, trees, mountains and all other
macroscopic objects belong?
The size and densit y of mat ter
The three spatial dimensions have many consequences for physical systems. We know
that all matter is made of fermions, such as electrons, protons and neutrons. The exclusion principle has an interesting consequence for systems made of ? identical fermions;
such systems obey the following expression for momentum ? and size ?:
Δ? Δ? ≳ ?1/3 ℏ .

Challenge 118 e
Ref. 92

Page 101

It is important in the use of lasers in precision experiments. The relation limits how close
a system can get to a pure sine wave; indeed for a pure sine wave, the indeterminacy
product would be zero.
For fermions, where the maximum number in the same state is 1, the number–phase
uncertainty relation reduces to a total uncertainty on the phase. In other words, we find
– again – that we cannot have fermion beams that behave as waves. There are no classical
fermion waves, no coherent fermion beams, in nature.
A summary on spin and indistinguishabilit y
The quantum of action ℏ implies that physical systems are made of two types of indistinguishable quantum particles: bosons and fermions. The two possible exchange behaviours
are related to the particle spin value, because exchange is related to rotation. The connec-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 48

Can you derive it? This extended indeterminacy relation provides a simple way to estimate
the spatial size of matter systems. In particular, the extended indeterminacy relation implies that the average energy per quanton increases with quanton density. Can you show
this?
The extended indeterminacy relation implies that matter systems whose extension is
due to electrons – thus all condensed matter systems – essentially have similar matter and
energy densities. The extended indeterminacy relation also implies that nuclei, which are
composed of protons and neutrons, all have essentially the same matter density.
For bosons, the components of radiation, there is no extended indeterminacy relation,
as the number of components ? in a particular quantum state does not have any effect
or limits. The indeterminacy relation thus does not limit the power density of laser light;
and indeed, the power density of laser beams varies much more than the matter density
of solids.
The indeterminacy relation highlights a further difference between matter and radiation. As we saw above, a system of ? identical bosons, such as a laser beam, obeys
an indeterminacy between the number and the phase which is easily derived from the
energy–time indeterminacy relation. The number–phase relation can be written, approximately, as
Δ? Δ? ≳ 1 .
(68)

Motion Mountain – The Adventure of Physics

Challenge 119 e

(67)

142

6 rotations and statistics – visualizing spin

tion between spin and rotation implies that antiparticles exist. It also implies that spin is
intrinsically a three-dimensional phenomenon.
Experiments show that radiation is made of elementary particles that behave as bosons. Bosons have integer spin. Two or more bosons, such as two photons, can share the
same state. This sharing makes laser light possible.
Experiments show that matter is made of elementary particles that behave as fermions. Fermions have half-integer spin. They obey Pauli’s exclusion principle: two fermions cannot be in the same state. The exclusion principle between electrons explains the
structure and (partly) the size of atoms, as well as the chemical behaviour of atoms, as we
will find out later on. Together with the electrostatic repulsion of electrons, the exclusion
principle explains the incompressibility of matter and its lack of impenetrability.
Fermions make matter ‘hard’, bosons allow light beams to cross.
Limits and open questions of quantum statistics

Ref. 93

⊳ In 3 + 1 space-time dimensions, only fermions and bosons exist.

Challenge 120 s

Vol. VI, page 172

Can you show that this result implies that the ghosts appearing in Scottish tales do not
exist?
From a different viewpoint, the belt model of spin 1/2 invites to study the behaviour of
braids, open links and knots. (In mathematics, braids and open links are made of strands
extending to infinity.) This fascinating part of mathematical physics has become important with in modern unified theories, which all state that particles, especially at high energies, are not point-like, but extended entities. The quest is to understand what happens
to permutation symmetry in a unified theory of nature. A glimpse of the difficulties appears already above: how can Figures 60, 65 and 70 be reconciled and combined? We will
settle this issue in the final part of our mountain ascent.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. V, page 107

Motion Mountain – The Adventure of Physics

Page 137

The topic of quantum particle statistics remains a research field in theoretical and experimental physics. In particular, researchers have searched and still are searching for
generalizations of the possible exchange behaviours of particles.
In two spatial dimensions, the effect of a particle exchange on the wave function is
a continuous phase, in contrast to three dimensions, where the result is a sign. Twodimensional quantum objects are therefore called anyons because they can have ‘any’
spin. Anyons appear as quasi-particles in various experiments in solid state physics, because the set-up is often effectively two-dimensional. The fractional quantum Hall effect,
perhaps the most interesting discovery of modern experimental physics, has pushed anyons onto the stage of modern research.
Other theorists generalized the concept of fermions in other ways, introducing parafermions, parabosons, plektons and other hypothetical concepts. Oscar Greenberg has
spent most of his professional life on this issue. His conclusion is:

Chapter 7

SU PE R P O SI T ION S A N D
PROBA BI L I T I E S – QUA N T UM
T H E ORY W I T HOU T I DE OLO G Y

“

Ref. 94

”

** It is equivalent, and often conceptually clearer, to say that the state is described by a complete set of
commuting operators. In fact, the discussion of states is somewhat simplified in the Heisenberg picture.
However, here we study the issue in the Schrödinger picture only, i.e., using wave functions.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

hy is this famous physical issue arousing such strong emotions? In particular,
ho is brainwashed, Gell-Mann, the discoverer of the quarks, or most of the
orld’s physicists working on quantum theory who follow Niels Bohr’s opinion?
In the twentieth century, quantum mechanics has thrown many in disarray. We have a
simple aim: we want to understand quantum theory. Quantum mechanics is unfamiliar
for two reasons: it allows superpositions and it leads to probabilities. In this chapter we
explore and clarify these two topics – until we understand quantum theory.
Probabilities appear whenever an aspect of a microscopic system is measured. The
quantum of action, the smallest change value found in nature, leads to the appearance of
probabilities in measurements.
Superpositions appear because the quantum of action radically changed the two most
basic concepts of classical physics: state and system. The state is not defined and described any more by the specific values taken by position and momentum, but by the
specific wave function ‘taken’ by the position and momentum operators.** In addition, in classical physics a system was described and defined as a set of permanent aspects of nature; permanence was defined as negligible interaction with the environment.
Quantum mechanics shows that these definitions have to be modified.
Clarifying the origin of superpositions and probabilities, as well as the concepts of system and state, will help us to avoid getting lost on our way to the top of Motion Mountain. Indeed, quite a number of researchers have lost their way since quantum theory
appeared, including important physicists like Murray Gell-Mann and Steven Weinberg.

Motion Mountain – The Adventure of Physics

W

The fact that an adequate philosophical
presentation has been so long delayed is no
doubt caused by the fact that Niels Bohr
brainwashed a whole generation of theorists
into thinking that the job was done fifty years
ago.
Murray Gell-Mann

144

7 superpositions and probabilities

Every such `artistic impression’ is wrong.
Challenge 121 s

(Why?)

F I G U R E 72 An artist’s
impression of a macroscopic
superposition is impossible –
because such superpositions
are not found in our
environment.

Why are people either dead or alive?

The object of discussion are linear superpositions of the type ? = ??? + ??? , where ??
and ?? are macroscopically distinct states of the system under discussion, and where ?
and ? are some complex coefficients. States are called macroscopically distinct when each
state corresponds to a different macroscopic situation, i.e., when the two states can be
distinguished using the concepts or measurement methods of classical physics. In particular, this means that the physical action necessary to transform one state into the other
must be much larger than ℏ. For example, two different positions of a body composed of
a large number of atoms are macroscopically distinct. The state of a cat that is living and
of the same cat when it is dead also differ by many quanta of action.
A ‘strange’ situation is thus a superposition of macroscopically distinct states. Let
us work out the essence of such macroscopic superpositions more clearly. Given two
macroscopically distinct states ?? and ?? , any superposition of the type ? = ??? + ??? is
called a pure state. Since the states ?? and ?? can interfere, one also talks about a (phase)
coherent superposition. In the case of a superposition of macroscopically distinct states,

Ref. 95
Ref. 96

* Most what can be said about this topic has been said by three important researchers: Niels Bohr, one of
the fathers of quantum physics, John von Neumann, who in the nineteen-thirties stressed the differences
between evolution and decoherence, and by Heinz Dieter Zeh, who in the nineteen-seventies stressed the
importance of baths and the environment in the decoherence process.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Macroscopic superpositions, coherence and incoherence

Motion Mountain – The Adventure of Physics

The evolution equation of quantum mechanics is linear in the wave function; the linearity
reflects the existence of superpositions. Superpositions imply that we can imagine and
try to construct systems where the state ? is a superposition of two radically distinct
situations, such as those of a dead and of a living cat. This famous fictional animal is
called Schrödinger’s cat after the originator of the example. Is it possible to produce it?
And how would it evolve in time? We can ask the same two questions in other situations.
For example, can we produce a superposition of a state where a car is inside a closed
garage with a state where the car is outside? What happens then?
Macroscopic superpositions are strange. Such situations are not observed in everyday
life, and only very rarely in the laboratory. The reason for this rareness is an important
aspect of what is often called the ‘interpretation’ of quantum mechanics. In fact, such
strange situations are possible, and the superposition of macroscopically distinct states
has actually been observed in a few cases, though not for cats, people or cars. To get an
idea of the constraints, let us specify the situation in more detail.*

7 quantum theory without ideology

145

the scalar product ??† ?? is obviously vanishing. In case of a coherent superposition, the
coefficient product ?∗ ? is different from zero. This fact can also be expressed with the
help of the density matrix ? of the system, defined as ? = ? ⊗ ?† . In the present case it is
given by
?pure = ? ⊗ ?† = |?|2 ?? ⊗ ??† + |?|2 ?? ⊗ ??† + ? ?∗ ?? ⊗ ??† + ?∗ ? ?? ⊗ ??†
|?|2 ? ?∗ ??†
= (?? , ?? ) ( ∗
) ( †) .
? ? |?|2
??

(69)

?mixed = |?|2 ?? ⊗ ??† + |?|2 ?? ⊗ ??†
= (?? , ?? ) (

|?|2 0
??†
)
(
)
0 |?|2 ??†

(71)

* Using the density matrix, we can rewrite the evolution equation of a quantum system:
?̇ = −??? becomes

d?
?
= − [?, ?] .
d?
ℏ

(70)

Both are completely equivalent. (The new expression is sometimes also called the von Neumann equation.)
We won’t actually do any calculations here. The expressions are given so that you recognize them when you
encounter them elsewhere.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

describes a system which possesses no phase coherence at all. (Here, ⊗ denotes the noncommutative dyadic product or tensor product which produces a tensor or matrix starting from two vectors.) Such a diagonal density matrix cannot be that of a pure state;
the density matrix describes a system which is in the state ?? with probability |?|2 and
which is in the state ?? with probability |?|2 . Such a system is said to be in a mixed state,
because its state is not known, or equivalently, is in a (phase) incoherent superposition: interference effects cannot be observed in such a situation. A system described by a mixed
state is always either in the state ?? or in the state ?? . In other words, a diagonal density matrix for macroscopically distinct states is not in contrast, but in agreement with
everyday experience.
In the picture of density matrices, the non-diagonal elements contain the difference
between normal, i.e., incoherent, and unusual or strange, i.e., coherent, superpositions.
The experimental situation is clear: for macroscopically distinct states, only diagonal
density matrices are observed in everyday life. Almost all systems in a coherent macroscopic superposition somehow lose their off-diagonal matrix elements. How does this
process of decoherence – also called disentanglement in certain settings – take place? The
density matrix itself shows the way.

Motion Mountain – The Adventure of Physics

We can then say that whenever the system is in a pure, or coherent state, then its density
matrix, or density functional, contains off-diagonal terms of the same order of magnitude
as the diagonal ones.* Such a density matrix corresponds to the above-mentioned strange
situations that we never observe in daily life. We will shortly understand why.
We now have a look at the opposite situation, a density matrix for macroscopic distinct
states with vanishing off-diagonal elements. For two states, the example

146

7 superpositions and probabilities

Decoherence is due to baths
Ref. 97
Challenge 122 ny

In thermodynamics, the density matrix ? for a large system is used for the definition of
its entropy ? – and of all its other thermodynamic quantities. These studies show that
? = −? tr (? ln ?)

(72)

where tr denotes the trace, i.e., the sum of all diagonal elements, and ? is the Boltzmann
constant. The expression is thus the quantum mechanical definition of entropy.
We now remind ourselves that a physical system with a large and constant entropy
is called a bath. In simple physical terms, a bath is a system to which we can ascribe a
temperature. More precisely,
A (physical) bath – also called a (thermodynamic) reservoir – is any large
system for which the concept of equilibrium can be applied.

Where is the bath interacting with a typical system? The bath must be outside the system
we are talking about, i.e., in its environment. Indeed, we know experimentally that a typical environment is large and characterized by a temperature. Some examples are listed
in Table 6. In short,
⊳ Any environment is a bath.
We can even go further: for every experimental situation, there is a bath interacting with
the system under study. Indeed, every system which can be observed is not isolated, as it
obviously interacts at least with the observer; and every observer by definition contains

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

How baths lead to decoherence – scat tering

Motion Mountain – The Adventure of Physics

Challenge 123 s

Experiments show that in practice, this is equivalent to the condition that a bath consists
of many interacting subsystems. For this reason, all macroscopic quantities describing
the state of a bath show small, irregular fluctuations, a property that will be of central
importance shortly.
An everyday bath is also a physical bath: indeed, a thermodynamic bath is similar to
an extremely large warm water bath, one for which the temperature does not change even
if we add some cold or warm water to it. The physical concept of bath, or reservoir, is
thus an abstraction and a generalization of the everyday concept of bath. Other examples
of physical baths are: an intense magnetic field, a large amount of gas, or a large solid.
(The meanings of ‘intense’ and ‘large’ of course depend on the system under study.)
The definition (72) of entropy tells us that the loss of off-diagonal elements corresponds to an increase in entropy. In addition, any increase in entropy of a reversible system, such as the quantum mechanical system in question, is due to an interaction with a
bath.
In short, decoherence is due to interaction with a bath. In addition, decoherence is a
process that increases entropy: decoherence is irreversible. We will now show that baths
are everywhere, that decoherence thus takes place everywhere and all the time, and that
therefore, macroscopic superpositions are (almost) never observed.

7 quantum theory without ideology

147

TA B L E 6 Common and less common baths with their main properties.

B at h t y p e

matter baths
solid, liquid
air
laboratory vacuum

300 K
300 K
50 mK

10 pm
10 pm
10 μm

1031 /m2 s
1028 /m2 s
1018 /m2 s

5800 K
300 K
2.7 K

900 nm
20 μm
2 mm

1023 /m2 s
1021 /m2 s
1017 /m2 s

Cross
Hit time
s e c t i o n 1/?? f o r
( at o m )
?
a t o m? b a l l?
10−19 m2
10−19 m2
10−19 m2

10−12 s
10−9 s
10 s

10−25 s
10−22 s
10−12 s

10−4 s
10−2 s
102 s

10−17 s
10−15 s
10−11 s

very large
very large

40 zK

>1000 K
≈ 10 MK
2.0 K

10 f m
10 f m
10 f m
3 mm

gravitational baths
gravitational radiation

5 ⋅ 1031 K

10−35 m

1 /m2 s
10−2 /m2 s
1011 /m2 s
1017 /m2 s

10−25 m2
10−25 m2
10−47 m2
10−62 m2

1025 s
1027 s
1036 s
1045 s

1012 s
1014 s
1015 s
1024 s

very large

?. Values are rough estimates. The macroscopic ball is assumed to have a 1 mm size.

Challenge 124 ny

a bath, as we will show in more detail shortly. Usually however, the most important baths
we have to take into consideration are the atmosphere around a system, the radiation or
electromagnetic fields interacting with the system, or, if the system itself is large enough
to have a temperature, those degrees of freedom of the system which are not involved in
the superposition under investigation.
Since every physical system is in contact with a bath, every density matrix of a macroscopic superposition will lose its diagonal elements eventually. At first sight, this direction of thought is not convincing. The interactions of a system with its environment can
be made extremely small by using clever experimental set-ups; that would imply that the
time for decoherence can be made extremely large. Thus we need to check how much
time a superposition of states needs to decohere. It turns out that there are two standard
ways to estimate the decoherence time: either by modelling the bath as large number of
colliding particles, or by modelling it as a continuous field.
If the bath is described as a set of particles randomly hitting the microscopic system,
it is best characterized by the effective wavelength ? eff of the particles and by the average
interval ?hit between two hits. A straightforward calculation shows that the decoherence

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

nuclear radiation baths
radioactivity
cosmic radiation
solar neutrinos
cosmic neutrinos

Motion Mountain – The Adventure of Physics

photon baths
sunlight
‘darkness’
cosmic microwaves
terrestrial radio waves
Casimir effect
Unruh radiation of Earth

T e m p e r - Wa v e - Pa r at u r e
length ticle
flux
?
?
? eff

148

7 superpositions and probabilities

time ?? is in any case smaller than this time interval, so that
?? ⩽ ?hit =

1
,
??

(73)

where ? is the flux of particles and ? the cross-section for the hit.* Typical values are
given in Table 6. We easily note that for macroscopic objects, decoherence times are extremely short. (We also note that nuclear and gravitational effects lead to large decoherence times and thus can be neglected.) Scattering leads to fast decoherence of macroscopic
systems. However, for atoms or smaller systems, the situation is different, as expected.
Microscopic systems can show long decoherence times.
We note that the quantum of action ℏ appears in the expression for the decoherence
time, as it appears in the area ?. Decoherence is a quantum process.

* The decoherence time is derived by studying the evolution of the density matrix ?(?, ?? ) of objects local? 2
ized at two points ? and ?? . One finds that the off-diagonal elements follow ?(?, ?? , ?) = ?(?, ?? , 0)e−Λ?(?−? ) ,
where the localization rate Λ is given by
(74)
Λ = ?2 ??eff
Ref. 98
Ref. 99

where ? is the wave number, ? the flux and ?eff the cross-section of the collisions, i.e., usually the size of the
macroscopic object.
One also finds the surprising result that a system hit by a particle of energy ?hit collapses the density
matrix roughly down to the de Broglie (or thermal de Broglie) wavelength of the hitting particle. Both
results together give the formula above.
** Beware of other definitions which try to make something deeper out of the concept of irreversibility,
such as claims that ‘irreversible’ means that the reversed process is not at all possible. Many so-called
‘contradictions’ between the irreversibility of processes and the reversibility of evolution equations are due
to this mistaken interpretation of the term ‘irreversible’.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

A second method to estimate the decoherence time is also common. Any interaction of
a system with a bath is described by a relaxation time ?? . The term relaxation designates
any process which leads to the return to the equilibrium state. The terms damping and
friction are also used. In the present case, the relaxation time describes the return to equilibrium of the combination bath and system. Relaxation is an example of an irreversible
evolution. A process is called irreversible if the reversed process, in which every component moves in opposite direction, is of very low probability.** For example, it is usual
that a glass of wine poured into a bowl of water colours the whole water; it is very rarely
observed that the wine and the water separate again, since the probability of all water
and wine molecules to change directions together at the same time is rather low, a state
of affairs making the happiness of wine producers and the despair of wine consumers.
Now let us simplify the description of the bath. We approximate it by a single, unspecified, scalar field which interacts with the quantum system. Due to the continuity
of space, such a field has an infinity of degrees of freedom. They are taken to model the
many degrees of freedom of the bath. The field is assumed to be in an initial state where
its degrees of freedom are excited in a way described by a temperature ?. The interaction of the system with the bath, which is at the origin of the relaxation process, can be
described by the repeated transfer of small amounts of energy ?hit until the relaxation

Motion Mountain – The Adventure of Physics

How baths lead to decoherence – relaxation

7 quantum theory without ideology

Ref. 100

process is completed.
The objects of interest in this discussion, like the mentioned cat, person or car, are
described by a mass ?. Their main characteristic is the maximum energy ?? which can
be transferred from the system to the environment. This energy describes the interactions between system and environment. The superpositions of macroscopic states we are
interested in are solutions of the Hamiltonian evolution of these systems.
The initial coherence of the superposition, so disturbingly in contrast with our everyday experience, disappears exponentially within a decoherence time ?? given by*
?? = ??

Challenge 125 ny

(77)

where ? is again the Boltzmann constant and like above, ?? is the maximum energy which
can be transferred from the system to the environment. Note that one always has ?? ⩽ ?? .
After the decoherence time ?? is elapsed, the system has evolved from the coherent to
the incoherent superposition of states, or, in other words, the density matrix has lost its
off-diagonal terms. One also says that the phase coherence of this system has been destroyed. Thus, after a time ?? , the system is found either in the state ?? or in the state ?? ,
respectively with the probability |?|2 or |?|2 , and not any more in a coherent superposition which is so much in contradiction with our daily experience. Which final state is
selected depends on the precise state of the bath, whose details were eliminated from the
calculation by taking an average over the states of its microscopic constituents.
The important result is that for all macroscopic objects, the decoherence time ?? is extremely small. In order to see this more clearly, we can study a special simplified case. A
macroscopic object of mass ?, like the mentioned cat or car, is assumed to be at the same
time in two locations separated by a distance ?, i.e., in a superposition of the two corresponding states. We further assume that the superposition is due to the object moving as
a quantum mechanical oscillator with frequency ? between the two locations; this is the
simplest possible system that shows superpositions of an object located in two different
positions. The energy of the object is then given by ?? = ??2 ?2 , and the smallest transfer
energy ?hit = ℏ? is the difference between the oscillator levels. In a macroscopic situation, this last energy is much smaller than ??, so that from the preceding expression we
get
?2hit
?2?
ℏ2
?? = ??
=
?
(78)
= ??
? 2
2?? ??
2????2
?
* This result is derived as in the above case. A system interacting with a bath always has an evolution given
by the general form
d?
?
1
∑[? ?, ??† ] + [?? , ???† ] ,
(75)
= − [?, ?] −
d?
ℏ
2?? ? ?
where ? is the density matrix, ? the Hamiltonian, ? the interaction, and ?? the characteristic time of the
interaction. Are you able to see why? Solving this equation, one finds for the elements far from the diagonal
?(?) = ?0 e−?/?0 . In other words, they disappear with a characteristic time ?? . In most situations one has a
relation of the form
?
(76)
?0 = ?? hit = ?hit
??
or some variations of it, as in the example above.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 101

?hit e?hit /?? − 1
?? e?hit /?? + 1

Motion Mountain – The Adventure of Physics

Ref. 102

149

150

Ref. 104

Ref. 105

Summary on decoherence, life and death
Our exploration showed that decoherence results from coupling to a bath in the environment. Decoherence is a quantum statistical effect, i.e., a thermodynamic effect. Decoherence follows from quantum theory, is an irreversible process, and thus occurs automatically. Above all, decoherence is a process that has been observed in experiments.
The estimates of decoherence times in everyday life told us that both the preparation
and the survival of superpositions of macroscopically different states is made impossible
by the interaction with any bath found in the environment. This is the case even if the
usual measure of this interaction, given by the friction of the motion of the system, is
very small. Even if a macroscopic system is subject to an extremely low friction, leading
to a very long relaxation time, its decoherence time is still vanishingly short. Only carefully designed microscopic systems in expensive laboratory set-ups can reach substantial
decoherence times.
Our everyday environment is full of baths. Therefore,
⊳ Coherent superpositions of macroscopically distinct states never appear in
everyday life, due to the rapid decoherence times induced by baths in the
environment.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 106

in which the frequency ? has disappeared. The quantity ? ? = ℏ/√2??? is called the
thermal de Broglie wavelength of a particle.
We note again that the quantum of action ℏ appears in the expression for the decoherence time. Decoherence is a quantum process.
It is straightforward to see that for practically all macroscopic objects the typical decoherence time ?? is extremely short. For example, setting ? = 1 g, ? = 1 mm and ? = 300 K
we get ?? /?? = 1.3⋅10−39 . Even if the interaction between the system and the environment
would be so weak that the system would have as relaxation time the age of the universe,
which is about 4 ⋅ 1017 s, the time ?? would still be shorter than 5 ⋅ 10−22 s, which is over
a million times faster than the oscillation time of a beam of light (about 2 fs for green
light). For Schrödinger’s cat, the decoherence time would be even shorter. These times
are so short that we cannot even hope to prepare the initial coherent superposition, let
alone to observe its decay or to measure its lifetime.
For microscopic systems however, the situation is different. For example, for an electron in a solid cooled to liquid helium temperature we have ? = 9.1 ⋅ 10−31 kg, and typically ? = 1 nm and ? = 4 K; we then get ?? ≈ ?? and therefore the system can stay in
a coherent superposition until it is relaxed, which confirms that for this case coherent
effects can indeed be observed if the system is kept isolated. A typical example is the behaviour of electrons in superconducting materials. We will mention a few more below.
In 1996 the first actual measurement of decoherence times was published by the Paris
team led by Serge Haroche. It confirmed the relation between the decoherence time and
the relaxation time, thus showing that the two processes have to be distinguished at microscopic scale. In the meantime, many other experiments confirmed the decoherence
process with its evolution equation, both for small and large values of ?? /?? . A particularly
beautiful experiment has been performed in 2004, where the disappearance of two-slit
interference for ?70 molecules was observed when a bath interacts with them.

Motion Mountain – The Adventure of Physics

Ref. 103

7 superpositions and probabilities

7 quantum theory without ideology

Page 155

151

Cars cannot be in and out of a garage at the same time. We cannot be dead and alive
at the same time. An illustration of a macroscopic superposition – see Figure 121 – is
impossible. In agreement with the explanation, coherent superpositions of macroscopic
states appear in some special laboratory situations.
What is a system? What is an object?
In classical physics, a system is a part of nature that can be isolated from its environment.
However, quantum mechanics tells us that isolated systems do not exist, since interactions cannot be made vanishingly small. The contradiction can be solved with the results
above: they allow us to define the concept of system with more accuracy.
⊳ A system is any part of nature that interacts incoherently with its environment.

⊳ An object is a part of nature interacting with its environment only through
baths.
In particular, we get:

In short, a microscopic or quantum mechanical system can be described by a wave function only if it interacts incoherently and weakly with its environment. (For such a system,
the energy indeterminacy Δ? is larger than the relaxation energy.) In contrast, a bath is
never isolated in the sense just given, because the evolution time of a bath, the time scale
during which its properties change, is always much larger than its relaxation time. Since
all macroscopic bodies are in contact with baths – or even contain one – they cannot be
described by a wave function. In particular, it is impossible to describe any measuring
apparatus with the help of a wave function.
We thus conclude:
⊳ A macroscopic system is a system with a decoherence time much shorter than
any other evolution time of its constituents.
Obviously, macroscopic systems also interact incoherently with their environment. Thus
cats, cars and television news speakers are all macroscopic systems.
Entanglement
One possibility is left over by the two definitions of system: what happens in the situation
in which the interactions with the environment are coherent? We will encounter some ex-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 107

⊳ A system is called microscopic or quantum mechanical and can described by
a wave function ? whenever
— it is almost isolated, with ?evol = ℏ/Δ? < ?r , and
— it is in incoherent interaction with its environment.

Motion Mountain – The Adventure of Physics

This implies:

152

7 superpositions and probabilities

amples shortly. Following the definitions, they are neither microscopic nor macroscopic
systems.

⊳ A ‘system’ in which the interaction with its environment is coherent is called
entangled.

“

[Mr. Duffy] lived a little distance away from his
body ...
James Joyce, A Painful Case

”

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Is quantum theory non-lo cal? A bit ab ou t the
Einstein–Podolsky–Rosen parad ox

Motion Mountain – The Adventure of Physics

Such ‘systems’ are not described by a wave function, and strictly speaking, they are not
systems. In these situations, when the interaction is coherent, one speaks of entanglement. For example, one says that a particle or set of particles is said to be entangled with
its environment.
Entangled, i.e., coherently interacting systems can be divided, but must be disentangled when doing so. The act of division leads to detached entities; detached entities interact incoherently. Quantum theory shows that nature is not made of detached
entities, but that it is made of detachable entities. In quantum theory, the criterion of
detachment is the incoherence of interaction. Coherent superpositions imply the surprising consequence that there are systems which, even though they look being made of
detached parts, are not. Entanglement poses a limit to detachment. All surprising properties of quantum mechanics, such as Schrödinger’s cat, are consequences of the classical
prejudice that a system made of two or more parts can obviously be detached into two
subsystems without disturbance. But coherent superpositions, or entangled systems, do
not allow detachment without disturbance. Whenever we assume to be able to detach
entangled systems, we get strange or incorrect conclusions, such as apparent faster-thanlight propagation, or, as one says today, non-local behaviour. Let us have a look at a few
typical examples.
Entangled situations are observed in many experiments. For example, when an electron and a positron annihilate into two photons, the polarisations of these two photons
are entangled, as measured already in 1949. Also when an excited atom decays in steps,
emitting two photons, the photon polarisations are entangled, as was first shown in 1966
with the help of calcium atoms. Similarly, when an unstable molecule in a singlet state,
i.e., in a spin 0 state, decays or splits into debris, the spins of the debris are entangled,
as observed in the 1970s. Also the spontaneous parametric down-conversion of photons
produces entanglement. In a non-linear optical material, an incoming photon is converted into two outgoing photons whose added energies correspond to the energy of the
incoming photon. In this case, the two outgoing photons are entangled both in their polarisation and in their direction. In 2001, the spins of two extremely cold caesium gas
samples, with millions of atoms each and located a few millimetres apart, have been entangled. Also position entanglement has been regularly observed, for example for closely
spaced ions inside ion traps.

7 quantum theory without ideology

153

space
collapse

t2

t3

screen

slit
space

Challenge 126 s
Vol. III, page 134

Ref. 108

Vol. II, page 58

Vol. II, page 284

F I G U R E 73 Quantum mechanical
motion: an electron wave function
(actually its module squared) from the
moment it passes a slit until it hits a
screen.

It is often suggested, incorrectly, that wave function collapse or quantum theory are nonlocal.* The issue needs clarification.
We start by imagining an electron hitting a screen after passing a slit. Following the
description just deduced, the collapse process proceeds schematically as depicted in Figure 73. An animation that includes another example of a collapse process – inspired by
Bohm’s thought experiment – can be seen in the lower left corners on these pages, starting at page 115. The collapse process has a surprising side: due to the shortness of the
decoherence time, during this (and any other) wave function collapse the maximum of
the wave function usually changes position faster than light. Is this reasonable?
A situation is called acausal or non-local if energy is transported faster than light.
Using Figure 73 you can determine the energy velocity involved, using the results on
signal propagation. The result is a value smaller than ?. A wave function whose maximum
moves faster than light does not automatically imply that energy moves faster than light.
In other words, quantum theory contains speeds greater than light, but no energy
speeds greater than light. In classical electrodynamics, the same happens with the scalar
and the vector potentials if the Coulomb gauge is used. We have also encountered speeds
faster than that of light in the motion of shadows and scissors, and in many other
observations. Any physicist now has two choices: he can be straight, and say that there
is no non-locality in nature; or he can be less straight, and claim there is. In the latter
* This continues a topic that we know already: we have explored a different type of non-locality, in general
relativity, earlier on.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 154

t4

Motion Mountain – The Adventure of Physics

t1

154

7 superpositions and probabilities

space

detector 2

collapse
time
F I G U R E 74 Bohm’s thought

experiment.

* David Joseph Bohm (1917–1992), was an influential physicist. He codiscovered the Aharonov–Bohm effect
and spent a large part of his later life investigating the connections between quantum physics and philosophy.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 109, Ref. 110

case, he has to claim that even classical physics is non-local. However, nobody dares to
claim this. In fact, there is a danger in this more provoking usage of the term ‘non-local’:
a small percentage of those who claim that the world is non-local after a while start to
believe that there really is faster-than-light energy transport in nature. These people become prisoners of their muddled thinking. On the other hands, muddled thinking helps
to get more easily into newspapers. In short, even though the definition of non-locality
is not unanimous, here we stick to the stricter one, and define non-locality as energy
transport faster than light.
An often cited thought experiment that shows the pitfalls of non-locality was proposed by Bohm* in the discussion around the so-called Einstein–Podolsky–Rosen
paradox. In the famous EPR paper the three authors tried to find a contradiction between
quantum mechanics and common sense. Bohm translated their rather confused paper
into a clear thought experiment that is shown schematically in Figure 74. When two
particles in a spin 0 state move apart, measuring one particle’s spin orientation implies
an immediate collapse also of the other particle’s spin, namely in the exactly opposite direction. This happens instantaneously over the whole separation distance; no speed limit
is obeyed. In other words, entanglement seems to lead to faster-than-light communication.
However, in Bohm’s experiment, no energy is transported faster than light. No nonlocality is present, despite numerous claims of the contrary by certain authors. The two

Motion Mountain – The Adventure of Physics

detector 1

7 quantum theory without ideology

Vol. III, page 137
Ref. 111

Curiosities and fun challenges ab ou t superpositions

Challenge 127 s

Some people wrongly state that atoms in a superposition of two states centred at different
positions can be photographed. (This lie is even used by some sects to attract believers.)
Why is this not true?
∗∗
In a few cases, the superposition of different macroscopic states can actually be observed
by lowering the temperature to sufficiently small values and by carefully choosing suitably small masses or distances. Two well-known examples of coherent superpositions
are those observed in gravitational wave detectors and in Josephson junctions. In the

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

entangled electrons belong to one system: assuming that they are separate only because
the wave function has two distant maxima is a conceptual mistake. In fact, no signal can
be transmitted with this method; the decoherence is a case of prediction which looks
like a signal without being one. Bohm’s experiment, like any other EPR-like experiment,
does not allow communication faster than light. We already discussed such cases in the
section on electrodynamics.
Bohm’s experiment has actually been performed. The first and most famous realization was due, in 1982, by Alain Aspect; he used photons instead of electrons. Like all
latter tests, it has fully confirmed quantum mechanics.
In fact, experiments such as the one by Aspect confirm that it is impossible to treat
either of the two particles as a system by itself; it is impossible to ascribe any physical
property, such as a spin orientation, to either of them alone. (The Heisenberg picture
would express this restriction even more clearly.) Only the two electrons together form
a physical system, because only the pair interacts incoherently with the environment.
The mentioned two examples of apparent non-locality can be dismissed with the remark that since obviously no energy flux faster than light is involved, no problems with
causality appear. Therefore the following example is more interesting. Take two identical
atoms, one in an excited state, one in the ground state, and call ? the distance that separates them. Common sense tells that if the first atom returns to its ground state emitting
a photon, the second atom can be excited only after a time ? = ?/? has been elapsed, i.e.,
after the photon has travelled to the second atom.
Surprisingly, this conclusion is wrong. The atom in its ground state has a non-zero
probability to be excited at the same moment in which the first is de-excited. This has
been shown most simply by Gerhard Hegerfeldt. The result has also been confirmed experimentally.
More careful studies show that the result depends on the type of superposition of the
two atoms at the beginning: coherent or incoherent. For incoherent superpositions, the
intuitive result is correct; the counter-intuitive result appears only for coherent superpositions. Again, a careful discussion shows that no real non-locality of energy is involved.
In summary, faster-than-light speeds in wave function collapse do not contradict the
limit on energy speed of special relativity. Collapse speeds are phase velocities. In nature,
phase velocities are unlimited; unlimited phase velocities never imply energy transport
faster than light. In addition, we recover the result that physical systems are only clearly
defined if they interact incoherently with their environment.

Motion Mountain – The Adventure of Physics

Ref. 112

155

156

Ref. 102

Ref. 113

7 superpositions and probabilities

first case, one observes a mass as heavy as 1000 kg in a superposition of states located
at different points in space: the distance between them is of the order of 10−17 m. In
the second case, in superconducting rings, superpositions of a state in which a macroscopic current of the order of 1 pA flows in clockwise direction with one where it flows
in counter-clockwise direction have been produced.
∗∗

Ref. 114

Superpositions of magnetization in up and down direction at the same time have been
observed for several materials.
∗∗

Ref. 115

∗∗
Ref. 117

In 1997, coherent atom waves were extracted from a cloud of sodium atoms.
∗∗

∗∗
Vol. V, page 47

We will discuss the relation between the environment and the decay of unstable systems
later on. The phenomenon is completely described by decoherence.
∗∗

Challenge 128 ny

Can you find a method to measure the degree of entanglement? Can you do so for a
system made of many particles?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 139

Macroscopic objects usually are in incoherent states. This is the same situation as for
light. The world is full of ‘macroscopic’, i.e., incoherent light: daylight, and all light from
lamps, from fire and from glow-worms is incoherent. Only very special and carefully
constructed sources, such as lasers or small point sources, emit coherent light. Only these
sources allow studying interference effects. In fact, the terms ‘coherent’ and ‘incoherent’
originated in optics, since for light the difference between the two, namely the capacity
to interfere, had been observed centuries before the case of matter.
Coherence and incoherence of light and of matter manifest themselves differently, because matter can stay at rest but light cannot and because matter is made of fermions,
but light is made of bosons. Coherence can be observed easily in systems composed of
bosons, such as light, sound in solids, or electron pairs in superconductors. Coherence
is less easily observed in systems of fermions, such as systems of atoms with their electron clouds. However, in both cases a decoherence time can be defined. In both cases
coherence in many particle systems is best observed if all particles are in the same state
(superconductivity, laser light) and in both cases the transition from coherent to incoherent is due to the interaction with a bath. A beam is thus incoherent if its particles arrive
randomly in time and in frequency. In everyday life, the rarity of observation of coherent
matter superpositions has the same origin as the rarity of observation of coherent light.

Motion Mountain – The Adventure of Physics

Ref. 116

Since the 1990s, the sport of finding and playing with new systems in coherent macroscopic superpositions has taken off across the world. The challenges lie in the clean
experiments necessary. Experiments with single atoms in superpositions of states are
among the most popular ones.

7 quantum theory without ideology

157

∗∗
Challenge 129 ny

The study of entanglement leads to a simple conclusion: teleportation contradicts correlation. Can you confirm the statement?
∗∗

Challenge 130 s

Are ghost images in TV sets, often due to spurious reflections, examples of interference?
∗∗

Challenge 131 d

What happens when two monochromatic electrons overlap?
∗∗

Ref. 118

Why d o probabilities and wave function collapse appear in
measurements?

Challenge 133 s

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. III, page 266

Measurements in quantum mechanics are puzzling also because they lead to statements
in which probabilities appear. For example, we speak about the probability of finding
an electron at a certain distance from the nucleus of an atom. Statements like this belong to the general type ‘when the observable ? is measured, the probability to find the
outcome ? is ?.’ In the following we will show that the probabilities in such statements
are inevitable for any measurement, because, as we will show, (1) any measurement and
any observation is a special case of decoherence or disentanglement process and (2) all
decoherence processes imply the quantum of action. (Historically, the process of measurement was studied before the more general process of decoherence. That explains in
part why the topic is so confused in many peoples’ minds.)
What is a measurement? As already mentioned earlier on, a measurement is any interaction which produces a record or a memory. (Any effect of everyday life is a record;
but this is not true in general. Can you give some examples of effects that are records and
some effects which are not?) Measurements can be performed by machines; when they
are performed by people, they are called observations. In quantum theory, the process of
measurement is not as straightforward as in classical physics. This is seen most strikingly
when a quantum system, such as a single electron, is first made to pass a diffraction slit,
or better – in order to make its wave aspect become apparent – a double slit and then
is made to hit a photographic plate, in order to make also its particle aspect appear. Experiment shows that the blackened dot, the spot where the electron has hit the screen,
cannot be determined in advance. (The same is true for photons or any other particle.)
However, for large numbers of electrons, the spatial distribution of the black dots, the
so-called diffraction pattern, can be calculated in advance with high precision.
The outcome of experiments on microscopic systems thus forces us to use probabilities for the description of microsystems. We find that the probability distribution ?(?)
of the spots on the photographic plate can be calculated from the wave function ? of the
electron at the screen surface and is given by ?(?) = |?† (?)?(?)|2 . This is in fact a special

Motion Mountain – The Adventure of Physics

Challenge 132 s

Some people say that quantum theory could be used for quantum computing, by using
coherent superpositions of wave functions. Can you give a general reason that makes
this aim very difficult – even though not impossible – even without knowing how such
a quantum computer might work, or what the so-called qubits might be?

158

7 superpositions and probabilities

ball
gravity

pegs

F I G U R E 75 A system showing probabilistic behaviour: ball
falling through an array of pegs.

case of the general first property of quantum measurements:
Motion Mountain – The Adventure of Physics

⊳ The measurement of an observable ? for a system in a state ? gives as result
one of the eigenvalues ?? , and the probability ?? to get the result ?? is given
by
?? = |??† ?|2 ,
(79)
where ?? is the eigenfunction of the operator ? corresponding to the eigenvalue ?? .*
Experiments also show a second property of quantum measurements:

These two experimental properties can also be generalized to the more general cases with
degenerate and continuous eigenvalues.
Obviously, the experimental results on the measurement process require an explanation. At first sight, the sort of probabilities encountered in quantum theory are different from the probabilities we encounter in everyday life. Take roulette, dice, the system
shown in Figure 75, pachinko machines or the direction in which a pencil on its tip
falls: all have been measured experimentally to be random (assuming no cheating by
the designer or operators) to a high degree of accuracy. These everyday systems do not
puzzle us. We unconsciously assume that the random outcome is due to the small, but
uncontrollable variations of the starting conditions or the environment every time the
experiment is repeated.**
* All linear transformations transform some special vectors, called eigenvectors (from the German word
eigen meaning ‘self’) into multiples of themselves. In other words, if ? is a transformation, ? a vector, and
?(?) = ??

(80)

where ? is a scalar, then the vector ? is called an eigenvector of ?, and ? is associated eigenvalue. The set of
all eigenvalues of a transformation ? is called the spectrum of ?.
** To get a feeling for the limitations of these unconscious assumptions, you may want to read the already

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 119

⊳ After a measurement, the observed quantum system is in the state ?? corresponding to the measured eigenvalue ?? . One also says that during the
measurement, the wave function has collapsed from ? to ?? .

7 quantum theory without ideology

159

But microscopic systems seem to be different. The two properties of quantum measurements just mentioned express what physicists observe in every experiment, even if
the initial conditions are taken to be exactly the same every time. But why then is the
position for a single electron, or most other observables of quantum systems, not predictable? In other words, what happens during the collapse of the wave function? How
long does the collapse take? In the beginning of quantum theory, there was the perception that the observed unpredictability is due to the lack of information about the state
of the particle. This lead many to search for so-called ‘hidden variables’. All these attempts were doomed to fail, however. It took some time for the scientific community to
realize that the unpredictability is not due to the lack of information about the state of
the particle, which is indeed described completely by the state vector ?.
In order to uncover the origin of probabilities, let us recall the nature of a measurement, or better, of a general observation.

Vol. III, page 264

⊳ Any observation of a system is the result of an interaction between that system and a bath in the recording apparatus.
By the way, since baths imply friction, we can also say: memory needs friction. In addition, any observation measuring a physical quantity uses an interaction depending on
that same quantity. With these seemingly trivial remarks, we can describe in more detail

Vol. I, page 127

mentioned story of those physicists who built a machine that could predict the outcome of a roulette ball
from the initial velocity imparted by the croupier.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The record can be a visual or auditive memory in our brain, or a written record on paper,
or a tape recording, or any such type of object. As explained in the previous volume, an
object is a record if it cannot have arisen or disappeared by chance. To avoid the influence
of chance, all records have to be protected as much as possible from the external world;
e.g. one typically puts archives in earthquake safe buildings with fire protection, keeps
documents in a safe, avoids brain injury as much as possible, etc.
On top of this, records have to be protected from their internal fluctuations. These
internal fluctuations are due to the many components any recording device is made of.
If the fluctuations were too large, they would make it impossible to distinguish between
the possible contents of a memory. Now, fluctuations decrease with increasing size of a
system, typically with the square root of the size. For example, if a hand writing is too
small, it is difficult to read if the paper gets brittle; if the magnetic tracks on tapes are
too small, they demagnetize and lose the stored information. In other words, a record is
rendered stable against internal fluctuations by making it of sufficient size. Every record
thus consists of many components and shows small fluctuations.
The importance of size can be expressed in another way: every system with memory,
i.e., every system capable of producing a record, contains a bath. In summary, the statement that any observation is the production of a record can be expressed more precisely
as:

Motion Mountain – The Adventure of Physics

⊳ Any observation is the production of a record.

160

7 superpositions and probabilities

the quantum
mechanical
system

H
describes
its possible
states

H int

tr

is determined
by the type
of measurement

describes its
friction, e.g.
due to heat flow

F I G U R E 76 The concepts used
in the description of
measurements.

? = ?? ?other = ∑ ?? ?? ?other .

(81)

?

Here, ?? is the aspect of the (particle or system) state that we want to measure, and ?other
represents all other degrees of freedom, i.e., those not described – spanned, in mathematical language – by the operator ? corresponding to the observable we want to measure.
The numbers ?? = |??† ?? | give the expansion of the state ?? , which is taken to be normalized, in terms of the basis ?? . For example, in a typical position measurement, the
functions ?? would be the position eigenfunctions and ?other would contain the information about the momentum, the spin and all other properties of the particle.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

the process of observation, or, as it is usually called in the quantum theory, the measurement process.
Any measurement apparatus, or detector, is characterized by two main aspects, shown
in Figure 76: the interaction it has with the microscopic system, and the bath it contains
to produce the record. Any description of the measurement process thus is the description of the evolution of the microscopic system and the detector; therefore one needs
the Hamiltonian for the particle, the interaction Hamiltonian, and the bath properties
(such as the relaxation time ?r ). The interaction specifies what is measured and the bath
realizes the memory.
We know that only classical thermodynamic systems can be irreversible; quantum
systems are not. We therefore conclude: a measurement system must be described classically: otherwise it would have no memory and would not be a measurement system: it
would not produce a record! Memory is a classical effect. (More precisely, memory is an
effect that only appears in the classical limit.) Nevertheless, let us see what happens if we
describe the measurement system quantum mechanically.
Let us call ? the observable which is measured in the experiment and its eigenfunctions ?? . We describe the quantum mechanical system under observation – often
a particle – by a state ?. The full state of the system can always be written as

Motion Mountain – The Adventure of Physics

Ref. 120

apparatus, e.g. eye, ear,
or machine, with memory,
i.e. coupled to a bath

7 quantum theory without ideology

161

(84)

?

where ?? is the (particle or system) state. After the interaction, using the just mentioned,
experimentally known characteristics of the apparatus, the combined state ?? is
?? = ∑ ?? ???other,? ?? .

(85)

?

This evolution from ?? to ?? follows from the evolution equation applied to the particle–
detector combination. Now, the combined state ?? is a superposition of macroscopically
* How does the interaction look like mathematically? From the description we just gave, we specified the
final state for every initial state. Since the two density matrices are related by
?f = ??i ?†
Challenge 134 ny

(82)

we can deduce the Hamiltonian from the matrix ?. Are you able to see how?
By the way, one can say in general that an apparatus measuring an observable ? has a system interaction
Hamiltonian depending on the pointer variable ?, and for which one has
[? + ?int , ?] = 0 .

(83)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?? = ?? ?start = ∑ ?? ?? ?other ?start ,

Motion Mountain – The Adventure of Physics

How does the system–detector interaction look like? Let us call the state of the apparatus before the measurement ?start . The measurement apparatus itself, by definition,
is a device which, when it is hit by a particle in the state ?? ?other , changes from the state
?start to the state ?? . One then says that the apparatus has measured the eigenvalue ??
corresponding to the eigenfunction ?? of the operator ?. The index ? is thus the record
of the measurement; it is called the pointer index or variable. This index tells us in which
state the microscopic system was before the interaction. The important point, taken from
our previous discussion, is that the states ?? , being records, are macroscopically distinct,
precisely in the sense of the previous section. Otherwise they would not be records, and
the interaction with the detector would not be a measurement.
Of course, during measurement, the apparatus sensitive to ?? changes the part ?other
of the particle state to some other situation ?other,? , which depends on the measurement
and on the apparatus; we do not need to specify it in the following discussion.* But let
us have an intermediate check of our reasoning. Do apparatuses as described here exist?
Yes, they do. For example, any photographic plate is a detector for the position of ionizing particles. A plate, and in general any apparatus measuring position, does this by
changing its momentum in a way depending on the measured position: the electron on
a photographic plate is stopped. In this case, ?start is a white plate, ?? would be a particle
localized at spot ?, ?? is the function describing a plate blackened at spot ? and ?other,n
describes the momentum and spin of the particle after it has hit the photographic plate
at the spot ?.
Now we are ready to look at the measurement process itself. For the moment, let us
disregard the bath in the detector, and let us just describe it with a state as well, which
we call ?start . In the time before the interaction between the particle and the detector, the
combined system (including the detector) was in the initial state ?? given simply by

162

7 superpositions and probabilities

distinct states: it is a superposition of distinct macroscopic states of the detector. In our
example ?? could correspond to a superposition of one state where a spot on the left
upper corner is blackened on an otherwise white plate with another state where a spot
on the right lower corner of the otherwise white plate is blackened. Such a situation is
never observed. Let us see why.
The density matrix ?? of the combined state ?? after the measurement given by
?? = ?? ⊗ ??† = ∑ ?? ??∗ (?? ?other,? ?? ) ⊗ (?? ?other,? ?? )† ,

(86)

?,?

?f = ∑|?? |2 (???other,? ?? ) ⊗ (???other,? ?? )†

(87)

?

Ref. 104

In other words, the collapse time is always smaller than the relaxation time of the bath.
We thus have a formula for the time the wave function takes to collapse. All experimental
measurements of the time of collapse have confirmed this result.
Why is ℏ necessary for probabilities?
At first sight, one could argue that the two properties of quantum measurements do not
contain ℏ, and thus are not consequences of quantum theory. However, this argument is
incorrect.
* Note however, that an exactly vanishing decoherence time, which would mean a strictly infinite number
of degrees of freedom of the bath or the environment, is in contradiction with the evolution equation, and
in particular with unitarity, locality and causality. It is essential in the whole argument not to confuse the
logical consequences of a extremely small decoherence time with those of an exactly vanishing decoherence
time.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

remains and has experimental relevance. As explained above, such a density matrix describes a mixed state, and the numbers ?? = |?? |2 = |??† ?? |2 give the probability of measuring the value ?? and of finding the particle in the state ?? ?other,n as well as the detector
in the state ?? . But this is precisely what the two properties of quantum measurements
state.
We therefore find that describing a measurement as an evolution of a quantum system interacting with a macroscopic detector, itself containing a bath, we can deduce the
two properties of quantum measurements, probabilistic outcomes and the collapse of the
wave function, from the quantum mechanical evolution equation. The decoherence time
?d of the previous section becomes the time of collapse for the case of a measurement; in
addition we find
?collapse = ?d < ?r .
(88)

Motion Mountain – The Adventure of Physics

contains large non-diagonal terms, i.e., terms for ? ≠ ?, whose numerical coefficients
are different from zero. Now let us take the bath back in. From the previous section we
know the effect of a bath on such a macroscopic superposition. We found that a density
matrix such as ?? decoheres extremely rapidly. We assume here that the decoherence
time is negligibly small.* After decoherence, the off-diagonal terms vanish, and only the
final, diagonal density matrix ?f , given by

7 quantum theory without ideology

Page 87
Page 31
Challenge 135 e

163

Decoherence is a quantum process, because ℏ appears in the expression of the decoherence time. Since the collapse of the wave function is based on decoherence, it is a
quantum process as well. Also probabilities are due to the quantum of action.
In addition, we have seen that the concept of wave function appears only because the
quantum of action ℏ is not zero. Wave functions, their collapse and probabilities are due
to the quantum of change ℏ.
These results recall a statement made earlier on: probabilities appear whenever an
experiment attempts to detect changes, i.e., action values, smaller than ℏ. Most puzzles
around measurement are due to such attempts. However, nature does not allow such
measurements; in every such attempt, probabilities appear.
Hidden variables

The theorem is about non-contextual variables, i.e., about hidden variables inside the
quantum mechanical system. The Kochen–Specker theorem thus states that there is no
non-contextual hidden variables model, because mathematics forbids it. This result essentially eliminates all possibilities for hidden variables, because usual quantum mechanical systems have Hilbert space dimensions larger than three.
* János Neumann (b. 1903 Budapest, d. 1957 Washington DC) influential mathematician. One of the greatest
and clearest scientific minds of the twentieth century, he settled many issues, especially in applied mathematics and quantum theory, that others still struggle with today. He then worked on the atomic and the
hydrogen bomb, on ballistic missiles, and on general defence problems. For the bomb research, he strongly
influenced the building of the earliest electronic computers, extending the ideas of Konrad Zuse. At the end
of his life, he wanted to change the weather with nuclear bombs. He died of a cancer that was due to his
exposure to nuclear radiation when watching bomb tests.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

⊳ Non-contextual hidden variables are impossible, if the Hilbert space has a
dimension equal or larger than three.

Motion Mountain – The Adventure of Physics

Ref. 121

A large number of people are not satisfied with the explanation of probabilities in the
quantum world. They long for more mystery in quantum theory. They do not like the
idea that probabilities are due to baths and to the quantum of action. The most famous
prejudice such people cultivate is the idea that the probabilities are due to some hidden
aspect of nature which is still unknown to humans. Such imagined, unknown aspects are
called hidden variables.
The beautiful thing about quantum mechanics is that it allows both conceptual and
experimental tests on whether such hidden variables exist – without the need of knowing
them. Obviously, hidden variables controlling the evolution of microscopic system would
contradict the statement that action values below ℏ cannot be detected. The smallest observable action value is the reason for the random behaviour of microscopic systems.
The smallest action thus excludes hidden variables. But let us add some more detailed
arguments.
Historically, the first, somewhat abstract argument against hidden variables was given
by John von Neumann.* An additional no-go theorem for hidden variables was published by Kochen and Specker in 1967, and independently by John Bell in 1969. The theorem states:

164

|(?1 ?1 ) + (?2 ?1 ) + (?2 ?2 ) − (?1 ?2 )| ⩽ 2 .

(89)

Here, the expressions in brackets are the averages of the measurement products over a
large number of samples. This hidden variable prediction holds independently of the
directions of the involved polarizers.
On the other hand, for the case that the polarizers 1 and 2 at position ? and the
corresponding ones at position ? are chosen with angles of π/4, quantum theory predicts
that
|(?1 ?1 ) + (?2 ?1 ) + (?2 ?2 ) − (?1 ?2 )| = 2√2 > 2 .
(90)

Ref. 123

This prediction is in complete contradiction with the hidden variable prediction.
Now, all experimental checks of Bell’s inequality have confirmed standard quantum
mechanics and falsified hidden variables. There are no exceptions.
Another measurable contradiction between quantum theory and locally realistic theories has been predicted by Greenberger, Horn and Zeilinger in systems with three entangled particles. Again, quantum theory has been confirmed in all experiments.
* Which leads to the definition: one zillion is 1023 .
** John Stewart Bell (1928–1990), theoretical physicist who worked mainly on the foundations of quantum
theory.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 136 e

We cannot avoid noting that there are no restricting theorems about contextual hidden variables, i.e., variables in the environment and in particular, in the baths contained
in it. Indeed, their necessity was shown above!
Also common sense eliminates hidden variables, without any recourse to mathematics, with a simple argument. If a quantum mechanical system had internal hidden variables, the measurement apparatus would have zillions of them.* And this would mean
that it could not work as a measurement system.
Despite all arguments, researchers have always been looking for experimental tests on
hidden variables. Most tests are based on the famed Bell’s inequality, a beautifully simple
relation published by John Bell** in the 1960s.
Can we distinguish quantum theory and locally realistic theories that use hidden variables? Bell’s starting idea is to do so by measuring the polarizations of two correlated
photons. Quantum theory says that the polarization of the photons is fixed only at the
time it is measured, whereas local realistic models – the most straightforward type of
hidden variable models – claim that the polarization is fixed already in advance by a
hidden variable. As Bell found out, experiments can be used to decide which alternative
is correct.
Imagine that the polarization is measured at two distant points ? and ?. Each observer
can measure 1 or −1 in each of his favourite direction. Let each observer choose two
directions, 1 and 2, and call their results ?1 , ?2 , ?1 and ?2 . Since the measurement results
all are either 1 or −1, the value of the specific expression (?1 + ?2 )?1 + (?2 − ?1 )?2 has
always the value ±2.
Imagine that you repeat the experiment many times, assuming that the hidden variables appear statistically. You then can deduce (a special case of) Bell’s inequality for two
hidden variables; it predicts that

Motion Mountain – The Adventure of Physics

Ref. 122

7 superpositions and probabilities

7 quantum theory without ideology

165

In summary, hidden variables do not exist. Of course, this is not really surprising. The
search for hidden variables is based on a misunderstanding of quantum mechanics or
on personal desires on how the world should be, instead of taking it as it is: there is a
smallest measurable action value, ℏ, in nature.
Summary on probabilities and determinism

“

Geometrica demonstramus quia facimus; si
physica demonstrare possemus, faceremus.
Giambattista Vico*

”

⊳ Probabilities appear whenever a process tries to distinguish between situations that differ by about one quantum of action ℏ.
⊳ The precise mechanism for the appearance of probabilities is due to the involved baths.
In short: probabilities appear whenever an experiment tries to distinguish between close
situations. In more detail:

* ‘We are able to demonstrate geometrical matters because we make them; if we could prove physical matters we would be able to make them.’ Giovanni Battista Vico (b. 1668 Napoli, d. 1744 Napoli) important
philosopher and thinker. In this famous statement he points out a fundamental distinction between mathematics and physics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

— Probabilities do not appear in measurements because the state of the quantum system is unknown or fuzzy, but because the detailed state of an interacting bath in the
environment is unknown. Quantum mechanical probabilities are of statistical origin
and are due to baths in the environment or in the measurement apparatus, in combination with the quantum of action ℏ. The probabilities are due to the large number
of degrees of freedom contained in the bath. These large numbers make the outcome
of experiments – especially those whose possible outcomes differ by about ℏ – unpredictable. If the state of the involved bath were known, the outcome of an experiment
could be predicted. The probabilities of quantum theory are due to the quantum of
action and are ‘thermodynamic’ in origin.
In other words, there are no fundamental probabilities in nature. All probabilities
in nature are due to decoherence; in particular, all probabilities are due to the statistics
of the many particles – some of which may even be virtual – that are part of the baths
in the environment. Modifying well-known words by Albert Einstein, we can agree
on the following: ‘nature does not play dice.’ Therefore we called ? the wave function
– instead of ‘probability amplitude’, as is often done. An even better term would be
state function.

Motion Mountain – The Adventure of Physics

We draw a number of conclusions which we need for the rest of our mountain ascent.
Note that these conclusions, even though in agreement with all experiments, are not yet
shared by all physicists! The whole topic is a problem for people who prefer ideology to
facts.
In everyday life, probabilities often do not appear or are not noted. Quantum theory
shows:

166

Challenge 137 s

* The opposite view is sometimes falsely attributed to Niels Bohr. The Moon is obviously in contact with
many radiation baths. Can you list a few?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 138 s

— Every observation in everyday life is a special case of decoherence. What is usually
called the ‘collapse of the wave function’ is a decoherence process due to the interaction with the baths present in the environment or in the measuring apparatus.
Because humans are warm-blooded and have memory, humans themselves are measurement apparatuses. The fact that our body temperature is 37°C is thus the reason
that we see only a single world, and no superpositions. (Actually, there are many additional reasons; can you name a few?)
— Every measurement is complete when the microscopic system has interacted with the
bath in the measuring apparatus. Quantum theory as a description of nature does not
require detectors; the evolution equation describes all examples of motion. However,
measurements do require the existence of detectors. A detector, or measurement apparatus, is a machine that records observations. Therefore, it has to include a bath,
i.e., has to be a classical, macroscopic object. In this context one speaks also of a classical apparatus. This necessity of the measurement apparatus to be classical had been
already stressed in the very early stages of quantum theory.
— All measurements, being decoherence processes that involve interactions with baths,
are irreversible processes and increase entropy.
— Every measurement, like every example of decoherence, is a special case of quantum
mechanical evolution, namely the evolution for the combination of a quantum system, a macroscopic detector and a bath. Since the evolution equation is relativistically
invariant, no causality problems appear in measurements; neither do locality problems or logical problems appear.
— Since both the evolution equation and the measurement process do not involve
quantities other than space-time, Hamiltonians, baths and wave-functions, no other
quantity plays a role in measurement. In particular, no human observer nor any consciousness is involved or necessary. Every measurement is complete when the microscopic system has interacted with the bath in the apparatus. The decoherence inherent
in every measurement takes place even if nobody is looking. This trivial consequence
is in agreement with the observations of everyday life, for example with the fact that
the Moon is orbiting the Earth even if nobody looks at it.* Similarly, a tree falling in
the middle of a forest makes noise even if nobody listens. Decoherence is independent of human observation, of the human mind and of human existence.
— In every measurement the quantum system interacts with the detector. Since there
is a minimum value for the magnitude of action, every observation influences the observed. Therefore every measurement disturbs the quantum system. Any precise description of observations must also include the description of this disturbance. In the
present section the disturbance was modelled by the change of the state of the system
from ?other to ?other,n . Without such a change of state, without a disturbance of the
quantum system, a measurement is impossible.
— Since the complete measurement process is described by quantum mechanics, unitarity is and remains the basic property of evolution. There are no non-unitary processes
in quantum mechanics.
— The description of the collapse of the wave function as a decoherence process is an

Motion Mountain – The Adventure of Physics

Vol. III, page 338

7 superpositions and probabilities

7 quantum theory without ideology

Vol. III, page 333

167

explanation exactly in the sense in which the term ‘explanation’ was defined earlier
on; it describes the relation between an observation and all the other aspects of reality,
in this case the bath in the detector or the environment. The collapse of the wave
function has been measured, calculated and explained. The collapse is not a question
of ‘interpretation’, i.e., of opinion, as unfortunately often is suggested.*
— It is not useful to speculate whether the evolution for a single quantum measurement
could be determined if the state of the environment around the system were known.
Measurements need baths. But a bath is, to an excellent approximation, irreversible
and thus cannot be described by a wave function, which behaves reversibly.**
In short:
⊳ Quantum mechanics is deterministic.
⊳ Baths are probabilistic.
⊳ Baths are probabilistic because of the quantum of action.

Ref. 125

... nobody understands quantum mechanics.

⊳ The quantum of action and decoherence are the key to understanding
quantum theory.
In fact, ℏ and decoherence allow clarifying many other issues. We explore a few interesting ones.
What is the difference bet ween space and time?
Space and time differ. Objects are localized in space but not in time. Why is this the
case? In nature, most bath–system interactions are mediated by a potential. All potentials are by definition position dependent. Therefore, every potential, being a function
of the position ?, commutes with the position observable (and thus with the interaction
Hamiltonian). The decoherence induced by baths – except if special care is taken – thus
first of all destroys the non-diagonal elements for every superposition of states centred

Ref. 124

* This implies that the so-called ‘many worlds’ interpretation is wishful thinking. The conclusion is confirmed when studying the details of this religious approach. It is a belief system, not based on facts.
** This very strong type of determinism will be very much challenged in the last part of this text, in which
it will be shown that time is not a fundamental concept, and therefore that the debate around determinism
looses most of its interest.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Nobel Prizes obviously do not prevent views distorted by ideology. The correct statement
is:

Motion Mountain – The Adventure of Physics

Page 143

In summary, there is no irrationality in quantum theory. Whoever uses quantum theory
as argument for superstitions, irrational behaviour, new age beliefs or ideologies is guilty
of disinformation. The statement by Gell-Mann at the beginning of this chapter is such
an example. Another is the following well-known, but incorrect statement by Richard
Feynman:

168

at different locations. In short, objects are localized because they interact with baths via
potentials.
For the same reason, objects also have only one spatial orientation at a time. If the
system–bath interaction is spin-dependent, the bath leads to ‘localization’ in the spin
variable. This occurs for all microscopic systems interacting with magnets. As a result,
macroscopic superpositions of magnetization are almost never observed. Since electrons,
protons and neutrons have a magnetic moment and a spin, this conclusion can even be
extended: everyday objects are never seen in superpositions of different rotation states
because their interactions with baths are spin-dependent.
As a counter-example, most systems are not localized in time, but on the contrary exist
for very long times, because practically all system–bath interactions do not commute
with time. In fact, this is the way a bath is defined to begin with. In short, objects are
permanent because they interact with baths.
Are you able to find an interaction which is momentum-dependent instead of
position-dependent? What is the consequence for macroscopic systems?
In other words, in contrast to general relativity, quantum theory produces a distinction between space and time. In fact, we can define position as the observable that commutes with interaction Hamiltonians. This distinction between space and time is due to
the properties of matter and its interactions. We could not have deduced this distinction
in general relativity.
Are we go od observers?

Challenge 140 e
Vol. VI, page 82

What relates information theory, cryptolo gy and quantum
theory?
Physics means talking about observations of nature. Like any observation, also measurements produce information. It is thus possible to translate much (but not all) of quantum
theory into the language of information theory. In particular, the existence of a smallest
change value in nature implies that the information about a physical system can never
be complete, that information transport has its limits and that information can never be
fully trusted. The details of these studies form a fascinating way to look at the microscopic world.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 126

Are humans classical apparatuses? Yes, they are. Even though several prominent physicists claim that free will and probabilities are related, a detailed investigation shows that
this in not the case. Our senses are classical machines because they obey their definition:
human senses record observations by interaction with a bath. Our brain is also a classical apparatus: the neurons are embedded in baths. Quantum probabilities do not play
a determining role in the brain.
Any observing entity, be it a machine or a human being, needs a bath and a memory
to record its observations. This means that observers have to be made of matter; an observer cannot be made of radiation. Our description of nature is thus severely biased: we
describe it from the standpoint of matter. That is a bit like describing the stars by putting
the Earth at the centre of the universe: we always put matter at the centre of our description. Can we eliminate this basic anthropomorphism? We will find out as we continue
our adventure.

Motion Mountain – The Adventure of Physics

Challenge 139 s

7 superpositions and probabilities

7 quantum theory without ideology

Ref. 127

169

Vol. VI, page 107

The quantum of action provides a limit to secure information exchange. This connection
allows us to brush aside several incorrect statements often found in the media. Stating
that ‘the universe is information’ or that ‘the universe is a computer’ is as reasonable
as saying that the universe is an observation or a chewing-gum dispenser. Any expert
of motion should beware of these and similarly fishy statements; people who use them
either deceive themselves or try to deceive others.
Does the universe have a wave function? And initial conditions?

Vol. II, page 223

The wave function of the universe is frequently invoked in discussions about quantum
theory. Various conclusions are deduced from this idea, for example on the irreversibility
of time, on the importance of initial conditions, on changes required to quantum theory
and much more. Are these arguments correct?
The first thing to clarify is the meaning of ‘universe’. As explained already, the term
can have two meanings: either the collection of all matter and radiation, or this collection
plus all of space-time. Let us also recall the meaning of ‘wave function’: it describes the
* Cryptology consists of the field of cryptography, the art of coding messages, and the field of cryptoanalysis, the art of deciphering encrypted messages. For a good introduction to cryptology, see the text
by Albrecht Beutelspacher, Jörg Schwenk & Klaus-Dieter Wolfenstätter, Moderne
Verfahren der Kryptographie, Vieweg 1995.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Is the universe a compu ter?

Motion Mountain – The Adventure of Physics

The analogy between quantum theory and information theory becomes even more
interesting when the statements are translated into the language of cryptology. Cryptology is the science of transmitting hidden messages that only the intended receiver can
decrypt. In our modern times of constant surveillance, cryptology is an important tool
to protect personal freedom.*
The quantum of action implies that messages can be sent in an (almost) safe way.
Listening to a message is a measurement process. Since there is a smallest action ℏ, we can
detect whether somebody has tried to listen to a message that we sent. To avoid a manin-the-middle attack – somebody who pretends to be the receiver and then sends a copy
of the message to the real, intended receiver – we can use entangled systems as signals
or messages to transmit the information. If the entanglement is destroyed, somebody
has listened to the message. Usually, quantum cryptologists use communication systems
based on entangled photons.
The major issue of quantum cryptology, a large modern research field, is the key distribution problem. All secure communication is based on a secret key that is used to
decrypt the message. Even if the communication channel is of the highest security – like
entangled photons – one still has to find a way to send the communication partner the
secret key necessary for the decryption of the messages. Finding such methods is the
main aspect of quantum cryptology. However, close investigation shows that all key exchange methods are limited in their security.
In short, due to the quantum of action, nature provides limits on the possibility of
sending encrypted messages. The statement of these limits is (almost) equivalent to the
statement that change in nature is limited by the quantum of action.

170

Vol. I, page 27

7 superpositions and probabilities

state of a system. The state distinguishes two otherwise identical systems; for example,
position and velocity distinguish two otherwise identical ivory balls on a billiard table.
Alternatively and equivalently, the state describes changes in time.
Does the universe have a state? If we take the wider meaning of universe, it does not.
Talking about the state of the universe is a contradiction: by definition, the concept of
state, defined as the non-permanent aspects of an object, is applicable only to parts of
the universe.
We then can take the narrower sense of ‘universe’ – the sum of all matter and radiation only – and ask the question again. To determine the state of all matter and radiation, we need a possibility to measure it: we need an environment. But the environment
of matter and radiation is space-time only; initial conditions cannot be determined since
we need measurements to do this, and thus an apparatus. An apparatus is a material system with a bath attached to it; however, there is no such system outside the universe.
In short, quantum theory does not allow for measurements of the universe.

Ref. 128

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Beware of anybody who claims to know something about the wave function of the universe. Just ask him Wheeler’s question: If you know the wave function of the universe,
why aren’t you rich?
Despite this conclusion, several famous physicists have proposed evolution equations for the wave function of the universe. (The best-known is, ironically, the Wheeler–
DeWitt equation.) It seems a silly point, but not one prediction of these equations has
been compared to experiment; the arguments just given even make this impossible in
principle. Exploring such equations, so interesting it may seem at first sight, must therefore be avoided if we want to complete our adventure and avoid getting lost in false beliefs.
There are many additional twists to this story. One is that space-time itself, even
without matter, might be a bath. This speculation will be shown to be correct in the last
volume of this adventure. The result seems to allow speaking of the wave function of the
universe. But then again, it turns out that time is undefined at the scales where space-time
is an effective bath; this again implies that the concept of state is not applicable there.
A lack of ‘state’ for the universe is a strong statement. It also implies a lack of initial conditions! The arguments are precisely the same. This is a tough result. We are so
used to think that the universe has initial conditions that we never question the term.
(Even in this text the mistake might appear every now and then.) But there are no initial
conditions for the universe.
We can retain as summary, valid even in the light of the latest research: The universe
is not a system, has no wave function and no initial conditions – independently of what
is meant by ‘universe’.

Motion Mountain – The Adventure of Physics

⊳ The universe has no state.

Chapter 8

C OL OU R S A N D OT H E R
I N T E R AC T ION S BET W E E N L IG H T
A N D M AT T E R

”

Cato

tones and all other objects have colours. Why? In other words, what is the
pecific way in which charged quantum particles that are found inside
tones and inside all other objects interact with electromagnetic fields? In this
chapter, we first give an overview of the various ways that colours in nature result from
the quantum of action, i.e., from the interaction between matter quantons and photons.
Then we explore the simplest such system: we show how the quantum of action leads
to the various colours produced by hydrogen atoms. After this, we discover that the
interaction between matter and radiation leads to other surprising effects, especially
when special relativity is taken into account.

Ref. 130

Ref. 129
Challenge 141 s

Quantum theory explains all colours in nature. Indeed, all the colours that we observe are
due to electrically charged particles. More precisely, colours are due to the interactions
of charged particles with photons. All colours are thus quantum effects.
So far, we have explored the motion of quantons that are described by mass only. Now
we study the motion of particles that are electrically charged. The charged particles at
the basis of most colours are electrons and nuclei, including their composites, from ions,
atoms and molecules to fluids and solids. Many colour issues are still topic of research.
For example, until recently it was unclear why exactly asphalt is black. The exact structure
of the chemical compounds, the asphaltenes, that produce the very dark brown colour
was unknown. Only recent research has settled this question. In fact, the development
of new colourants and colour effects is an important part of modern industry.
An overview of the specific mechanisms that generate colour is given in the following
table. The table includes all colours that appear in everyday life. (Can you find one that
is missing?)

** ‘Know the subject and the words will follow.’ Marcus Porcius Cato, (234–149 bce) or Cato the elder,
Roman politician famous for his speeches and his integrity.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

The causes of colour

Motion Mountain – The Adventure of Physics

S

“

Rem tene; verba sequentur.**

172

8 colours and more

TA B L E 7 Causes of colour.

Colour type

Example

D e ta i l s

Aurora, triboluminescence
in scotch tape,
crystalloluminescence in
strontium bromate

Lightning, arcs, sparks,
coloured fireworks, most
coloured flames, some
electroluminescence

In air, blue and red colours are
due to atomic and molecular
energy levels of nitrogen,
whereas green, yellow, orange
colours are due to oxygen
(aurora © Jan Curtis)
Colour lines are due to energy
levels of highly excited atoms
(flames of K, Cu, Cs, B, Ca
© Philip Evans)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

2. Atomic gas excitations
Red neon lamp, blue argon Colours are due to transitions
lamp, UV mercury lamp,
between atomic energy levels
yellow sodium street
(gas discharges © Pslawinski)
lamps, most gas lasers,
metal vapour lasers, some
fluorescence

Motion Mountain – The Adventure of Physics

Class I: Colours due to simple excitations
1. Incandescence and free charge radiation
Carbon arc lamp, hot steel, Colours are due to continuous
lightbulb wire, most stars, spectrum emitted by all hot
magma, lava, hot melts
matter; colour sequence,
given by Wien’s rule, is black,
red, orange, yellow, white,
blue-white (molten lead and
silver © Graela)
Wood fire, candle
Wood and wax flames are
yellow due to incandescence if
carbon-rich and oxygen-poor
White fireworks, flashlamp, Due to metals burning to
sparklers
oxide at high temperature,
such as magnesium, zinc,
iron, aluminium or zirconium
(sparkler © Sarah Domingos)
Nuclear reactors,
Due to fast free charges:
synchroton light sources,
Vavilov–Čerenkov radiation is
free electron lasers
due to speed of particle larger
than the speed of light in
matter, Bremsstrahlung is due
to the deceleration of charged
particles (nuclear reactor core
under water, courtesy NASA)

8 colours and more

173

TA B L E 7 Causes of colour (continued).

Colour type

Example

D e ta i l s

3. Vibrations and rotations of molecules
Bluish water, blue ice when Colours are due to quantized
clear, violet iodine,
levels of rotation and
red-brown bromine,
vibrations in molecules (blue
yellow-green chlorine, red iceberg © Marc Shandro)
flames from CN or
blue-green flames from
CH, some gas lasers, blue
ozone leading to blue and
grey evening sky

5. Transition metal impurities
Ruby, emerald, alexandrite, Electronic states of transition
perovskites, corresponding metal ions are excited by light
lasers
and thus absorb specific
wavelengths (ruby on calcite
from Mogok, Myanmar,
picture width 3 cm, © Rob
Lavinsky)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

4. Transition metal compounds
Green malachite
Colours are due to electronic
Cu2 CO3 (OH)2 , blue cobalt states of the ions; phosphors
oxide, blue azurite
are used in cathodes tubes for
Cu3 (CO3 )2 (OH)2 , red to
TV/computer displays and on
brown hematite Fe2 O3 ,
fluorescent lamp tubes (green
green MnO, white
malachite on yellow kasolite, a
Mn(OH)2 , brown
uranium mineral, picture
manganite, chrome green width 5 mm, found in
Cr2 O3 , green
Kolwezi, Zaire/Congo,
praesodymium, pink
© Stephan Wolfsried,
europium and yellow
television shadow mask photo
samarium compounds,
© Planemad)
piezochromic and
thermochromic
Cr2 O3 − Al2 O3 UV and
electron phosphors,
scintillation, some
fluorescence, some lasers

Motion Mountain – The Adventure of Physics

Class II: Colours due to ligand field effects

174

8 colours and more

TA B L E 7 Causes of colour (continued).

Colour type

Example

D e ta i l s

Class III: Colours due to molecular orbitals

Bioluminescence is due to
excited molecules, generally
called luciferines (angler fish,
length 4.5 cm, © Steve
Haddock)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Glow-worms, some
bacteria and fungi, most
deep-sea fish, octopi,
jellyfish, and other
deep-sea animals

Colours are due to conjugated
π-bonds, i.e. to alternating
single and double bonds in
molecules; floral pigments are
almost all anthocyanins,
betalains or carotenes; used in
colourants for foods and
cosmetics, in textile dyes, in
electrochromic displays, in
inks for colour printers, in
photosensitizers (narcissus
© Thomas Lüthi, blood on
finger © Ian Humes, berries
© Nathan Wall, hair courtesy
dusdin)

Motion Mountain – The Adventure of Physics

6. Organic compounds
Red haemoglobin in blood,
blue blood haemocyanin,
green chlorophyll in plants,
yellow or orange carotenes
in carrots, flowers and
yellow autumn leaves, red
or purple anthocyanins in
berries, flowers and red
autumn leaves, blue indigo,
red lycopene in tomatoes,
red meat from
iron-containing
myoglobin, brown
glucosamine in crust of
baked food, brown tannins,
black eumelanin in human
skin, hair and eye,
iron-rich variation
pheomelanin in redheads,
black melanin also in cut
apples and bananas as well
as in movable sacks in
chameleons, brown-black
asphalt, some fluorescence,
chemiluminescence,
phosphorescence,
halochromism,
electrochromism and
thermochromism, dye
lasers

8 colours and more

175

TA B L E 7 Causes of colour (continued).

Colour type

D e ta i l s

7. Inorganic charge transfer
Blue sapphire, blue lapis
lazuli, green amazonite,
brown-black magnetite
Fe3 O4 and most other iron
minerals (colouring basalt
black, beer bottles brown,
quartz sand yellow, and
many other rocks with
brown or red tones), black
graphite, purple
permanganate, orange
potassium dichromate,
yellow molybdates, red
hematite Fe2 O3 , some
fluorescence

Light induces change of
position of an electron from
one atom to another; for
example, in blue sapphire the
transition is between Ti and
Fe impurities; many paint
pigments use charge transfer
colours; fluorescent analytical
reagents are used in molecular
medicine and biology
(magnetite found in Laach,
Germany, picture width
10 mm, © Stephan Wolfsried,
sand desert Evelien
Willemsen)

Class IV: Colours due to energy band effects
Colours in reflection and in
transmission are due to
transitions of electrons
between overlapping bands
(saxophone © Selmer)

9. Pure semiconductor bands
Silicon, GaAs, black galena Colours are due to electron
PbS, red cinnabar HgS,
transitions between separate
cadmium yellow CdS,
bands; colour series is black,
black CdSe, red CdSx Se1−x , red, orange, yellow,
white ZnO, orange
white/colourless; some used
vermillion HgS, colourless as pigments (zinc oxide
diamond, black to gold
courtesy Walkerma)
piezochromic SmS
10. Doped semiconductor bands
Colours are due to transitions
Blue, yellow, green and
between dopants and
black diamond; LEDs;
semiconductor lasers; solar semiconductor bands
cells; ZnS and Znx Cd1−x S (e.g. blue diamond: boron
based and other phosphors accepters, black diamond:
nitrogen donors) (quantum
dots © Andrey Rogach)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

8. Metallic bands
Gold (green in
transmission), pyrite, iron,
brass, alloys, silver, copper,
ruby glass

Motion Mountain – The Adventure of Physics

Example

176

8 colours and more

TA B L E 7 Causes of colour (continued).

Colour type

Example

Colours are due to colour
centres, i.e. to electrons or to
holes bound at crystal
vacancies; colour centres are
usually created by radiation
(amethyst © Rob Lavinsky)

The photochromic colouring
is due to colour centres
formed by the UV light of the
Sun

Class V: Colours due to physical and geometrical optics
12. Dispersive refraction and polarization

13. Scattering
Blue sky, blue colouring of
distant mountains, red
sunset; colour
intensification by
pollution; blue quartz

Blue light is scattered more
than red light by Rayleigh
scattering, when scatterers
(molecules, dust) are smaller
than the wavelength of light
(Tokyo sunset © Altus
Plunkett, blue quartz © David
Lynch)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Cut diamond, cut zirconia, Spectral decomposition
halos and sun dogs formed (sparkle or ‘fire’ of
by ice crystals in the air
gemstones) is due to
dispersion in crystals
(zirconia photo © Gregory
Phillips)
Rainbow
Colours of primary and
secondary bow are due to
dispersion in water droplets
Green flash
dispersion in the atmosphere
shifts the sun colours

Motion Mountain – The Adventure of Physics

11. Colour centres
Amethyst, smoky quartz,
fluorite, green diamonds,
blue, yellow and brown
topaz, brown salt, purple
colour of irradiated glass
containing Mn2+ ,
lyoluminescence, some
fluorescence, F-centre
lasers
Some light-dependent
sunglasses

D e ta i l s

8 colours and more

177

TA B L E 7 Causes of colour (continued).

Colour type

D e ta i l s

White colour of hair, milk,
beer foam, clouds, fog,
cigarette smoke coming out
of lungs, snow, whipped
cream, shampoo, stars in
gemstones
Blue human skin colour in
cold weather, blue and
green eyes in humans, blue
monkey skin, blue turkey
necks, most blue fish, blue
reptiles, blue cigarette
smoke
Ruby glass

The white colour is due to
wavelength-independent Mie
scattering, i.e. scattering at
particles larger than the
wavelength of light (snow
man © Andreas Kostner)
Tyndall blue colours are due
to scattering on small particles
in front of a dark background
(blue poison frog Dendrobates
azureus © Lee Hancock)

14. Interference (without diffraction)
Nacre, oil films, soap
Thin film interference
bubbles, coatings on
produces a standard colour
camera lenses, eyes of cats sequence that allows precise
in the dark, wings of flies
thickness determination
and dragonflies, fish scales, (abalone shell © Anne Elliot)
some snakes, pearls,
tempering colours of steel
Polarization colours of thin Colours are due to
layers of birefringent
interference, as shown by the
crystals or thicker layers of dependence on layer
stressed polymers
thickness (photoelasticity
courtesy Nevit Dilmen)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Nonlinearities, Raman
effect, potassium
dihydrogen phosphate
(KDP)

The red colour of Murano
glass is due to scattering by
tiny colloidal gold particles
included in the glass in
combination with the metallic
band structure of gold (ruby
glass © murano-glass-shop.it)
Frequency-shifting scattering,
second harmonic generation
and other nonlinearities of
certain materials change the
colour of light impinging with
high intensities (800 nm to
400 nm frequency doubling
ring laser © Jeff Sherman)

Motion Mountain – The Adventure of Physics

Example

178

8 colours and more

TA B L E 7 Causes of colour (continued).

Colour type

Example

D e ta i l s

Supernumerary rainbows Due to interference, as shown
(see page 103 in volume III) by the dependence on drop
size
Iridescent beetles,
Due to scattering at small
butterflies and bird
structures or at nanoparticles,
feathers, iridescent colours as shown by the angular
on banknotes and on cars dependence of the colour
(mallard duck © Simon
Griffith)

Class VI: Colours due to eye limitations
Fechner colours, as on lite.bu.
edu/vision/applets/Color/
Benham/Benham.html

Benham’s wheel or top

Colours are due to different
speed response of different
photoreceptors

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Cholesteric liquid crystals, Colours are due to diffraction
certain beetles
and interference in internal
material layers (liquid crystal
colours © Ingo Dierking)

Motion Mountain – The Adventure of Physics

15. Diffraction (with interference)
Opal
Colours are due to the tiny
spheres included in the water
inside the opal; colours can
change if the opal dries out
(polished Brazilian opal
© Opalsnopals)
Aureole, glory, corona
Colours are due to diffraction
at the tiny mist droplets
(aeroplane condensation
cloud iridescence © Franz
Kerschbaum)
Diffraction gratings, CDs, Colours are due to diffraction
vinyl records, some beetles and interference at tiny,
and snakes
regular pits (CD illuminated
by flashlamp © Alfons
Reichert)
Photonic crystals
A modern research topic

8 colours and more

179

TA B L E 7 Causes of colour (continued).

Colour type

Example

Internal colour production when Phosphenes
eyes are stimulated

Ref. 132

Red light can be seen as
green
Protan, deutan or tritan

Colours fascinate. Fascination always also means business; indeed, a large part of the
chemical industry is dedicated to synthesizing colourants for paints, inks, clothes, food
and cosmetics. Also evolution uses the fascination of colours for its own business, namely
propagating life. The specialists in this domain are the flowering plants. The chemistry
of colour production in plants is extremely involved and at least as interesting as the
production of colours in factories. Practically all flower colourants, from white, yellow,
orange, red to blue, are from three chemical classes: the carotenoids, the anthocyanins
(flavonoids) and the betalains. These colourants are stored in petals inside dedicated containers, the vacuoles. There are many good review articles providing the details.
Even though colours are common in plants and animals, most higher animals do not
produce many colourants themselves. For example, humans produce only one colourant: melanin. (Hemoglobin, which colours blood red, is not a dedicated colourant, but
transports the oxygen from the lungs through the body. Also the pink myoglobin in the
muscles is not a dedicated colourant.) Many higher animals, such as birds, need to eat
the colourants that are so characteristic for their appearance. The yellow colour of legs
of pigeons is an example. It has been shown that the connection between colour and
nutrition is regularly used by potential mates to judge from the body colours whether a
proposing partner is sufficiently healthy, and thus sufficiently attractive.
Above all, the previous table distinguished six main classes among the causes of colours. The study of the first class, the colours of incandescence, led Max Planck to discover
the quantum of action. In the meantime, research has confirmed that in each class, all
colours are due to the quantum of action ℏ. The relation between the quantum of action
and the material properties of atoms, molecules, liquids and solids are so well known
that colourants can now be designed on the computer.
In summary, an exploration of the causes of colours found in nature confirms that all
colours are due to quantum effects. We show this by exploring the simplest example: the
colours of atomic gas excitations.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 131

Haidinger’s brush
Appearing and
disappearing colours

Occur through pressure
(rubbing, sneeze), or with
electric or magnetic fields
See page 113 in volume III
Effects are due to
combinations of brain
processing and eye limitations
Observable with adaptive
optics, if red light is focused
on a green-sensitive cone
Each type limits colour
perception in a different way
Motion Mountain – The Adventure of Physics

Polarization colours
Colour illusions, as on www.psy.
ritsumei.ac.jp/~akitaoka/color9e.
html
False colour output of eye, as
described on page 199 in volume
III
Colour-blind or ‘daltonic’
person, see page 209 in volume
III, with reduced colour spectrum

D e ta i l s

180

8 colours and more

lines (© Nigel Sharp, NOAO, FTS, NSO, KPNO, AURA, NSF).

Near the beginning of the eighteenth century, Bavarian instrument-maker Joseph
Fraunhofer* and the English physicist William Wollaston noted that the rainbow lacks
certain colours. These colours appear as black lines when the rainbow is spread out in
sufficient breadth. Figure 77 shows the lines in detail; they are called Fraunhofer lines
today. In 1860, Gustav Kirchhoff and Robert Bunsen showed that the colours missing in
the rainbow were exactly those colours that certain elements emit when heated. In this
way they managed to show that sodium, calcium, barium, nickel, magnesium, zinc, copper and iron are present in the Sun. Looking at the rainbow thus tells us what the Sun is
made of.

Vol. II, page 312

* Joseph Fraunhofer (b. 1787 Straubing, d. 1826 Munich), having been orphaned at the age of 11, learned lenspolishing. He taught himself optics from books. He entered an optical company at the age of 19, ensuring the
success of the business by producing the best available lenses, telescopes, micrometers, optical gratings and
optical systems of his time. He invented the spectroscope and the heliometer. He discovered and counted
476 lines in the spectrum of the Sun; these lines are now named after him. (Today, Fraunhofer lines are still
used as measurement standards: the second and the metre are defined in terms of them.) Physicists from all
over the world would buy their equipment from him, visit him, and ask for copies of his publications. Even
after his death, his instruments remained unsurpassed for generations. With his telescopes, in 1837 Bessel
was able to make the first measurement of parallax of a star, and in 1846 Johann Gottfried Galle discovered
Neptune. Fraunhofer became a professor in 1819. He died young, from the consequences of the years spent
working with lead and glass powder.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Using the rainb ow to determine what stars are made of

Motion Mountain – The Adventure of Physics

F I G U R E 77 The spectrum of daylight: a stacked image of an extended rainbow, showing its Fraunhofer

8 colours and more

181

F I G U R E 78 A
low-pressure
hydrogen discharge
in a 20 cm long glass
tube (© Jürgen Bauer
at www.
smart-elements.com).

The simplest colours to study are the sharp colour lines emitted or absorbed by single
atoms. Single atoms are found in gases. The simplest atom to study is that of hydrogen.
As shown in Figure 78, hot hydrogen gas emits light. The light consists of a handful of

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

What determines the colours of atoms?

Motion Mountain – The Adventure of Physics

Vol. III, page 125

Of the 476 Fraunhofer lines that Kirchhoff and Bunsen observed, 13 did not correspond to any known element. In 1868, Jules Janssen and Joseph Lockyer independently
predicted that these unknown lines were from an unknown element. The element was
eventually found on Earth, in an uranium mineral called cleveite, in 1895. The new element was called helium, from the Greek word ἥλιος ‘helios’ – Sun.
In 1925, using an equation developed by Saha and Langmuir, the young physicist
Cecilia Payne (b. 1900 Wendover, England, d. 1979 Cambridge, Massachusetts) taught
the world how to deduce the mass percentage of each element from the light spectrum
of a star. She did so in her brilliant PhD thesis. Above all, she found that hydrogen and
helium were the two most abundant elements in the Sun, in stars, and thus in the whole
universe. This went completely against the ideas of the time, but is now common knowledge. Payne had completed the study of physics in Cambridge, UK, but had not received
a degree there because she was a woman. So she left for the United States, where the situation was somewhat better, and where she worked on her PhD thesis; eventually, she
became professor at Harvard University, and later head of its astronomy department.
Above all, Payne became an important role model for many female scientists.
Despite being the second most common element in the universe, helium is rare on
Earth because it is a light noble gas that does not form chemical compounds. Helium
atoms on Earth thus rise in the atmosphere and finally escape into space.
Understanding the colour lines produced by each element had started to become interesting already before the discovery of helium; but afterwards the interest increased
further, thanks to the increasing number of applications of colour knowledge in chemistry, physics, technology, crystallography, biology and lasers. Colours are big business,
as the fashion industry, the media and the advertising business show.
In summary, colours are specific mixtures of light frequencies. Light is an electromagnetic wave and is emitted by moving charges. For a physicist, colours thus result from
the interaction of charged matter with the electromagnetic field. Now, sharp colour lines
cannot be explained by classical electrodynamics. We need quantum theory to explain
them.

182

8 colours and more

sharp spectral lines that are shown on the left of Figure 79. Already in 1885, the Swiss
schoolteacher Johann Balmer (1828–1898) had discovered that the wavelengths of visible
hydrogen lines obey the formula:
1
1
1
= ?( − 2)
??
4 ?

for ? = 3, 4, 5, ... .

(91)

Careful measurements, which included the hydrogen’s spectral lines in the infrared and
in the ultraviolet, allowed Johannes Rydberg (1854–1919) to generalize this formula to:
1
1
1
= ?( 2 − 2) ,
? ??
?
?

Challenge 143 e

Page 81
Challenge 144 e

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 142 s

where ? and ? > ? are positive integers, and the so-called Rydberg constant ? has the
value 10.97 μm−1 ; easier to remember, the inverse value is 1/? = 91.16 nm. All the colour
lines emitted by hydrogen satisfy this simple formula. Classical physics cannot explain
this result at all. Thus, quantum theory has a clearly defined challenge here: to explain
the formula and the value of ?.
Incidentally, the transition ? 21 for hydrogen is called the Lyman-alpha line. Its
wavelength, 121.6 nm, lies in the ultraviolet. It is easily observed with telescopes, since
most of the visible stars consist of excited hydrogen. The Lyman-alpha line is routinely
used to determine the speed of distant stars or galaxies, since the Doppler effect changes
the wavelength when the speed is large. The record in 2004 was a galaxy with a Lymanalpha line shifted to 1337 nm. Can you calculate the speed with which it is moving away
from the Earth?
From the start, it was clear that the colours of hydrogen are due to the motion of its
electron. (Why?) The first way to deduce Balmer’s formula from the quantum of action
was found by Niels Bohr in 1903. Bohr understood that in contrast to planets circling
the Sun, the electron moving around the proton has only a discrete number of possible
motion states: the angular momentum of the electron is quantized. Assuming that the
angular momentum of the electron is an integer multiple of ℏ directly yields Balmer’s
formula and explains the numerical value of the Rydberg constant ?. This calculation is
so famous that it is found in many secondary school books. The result also strengthened
Bohr’s decision to dedicate his life to the exploration of the structure of the atom.
Twenty years time later, in 1926, Erwin Schrödinger solved his equation of motion for
an electron moving in the electrostatic potential ?(?) = ?2 /4π?0 ? of a point-like proton.
By doing so, Schrödinger reproduced Bohr’s result, deduced Balmer’s formula and became famous in the world of physics. However, this important calculation is long and
complex. It can be simplified.
In order to understand hydrogen colours, it is not necessary to solve an equation of
motion for the electron; it is sufficient to compare the energies of the initial and final
states of the electron. This can be done most easily by noting that a specific form of the
action must be a multiple of ℏ/2. This approach, a generalization of Bohr’s explanation,
was developed by Einstein, Brillouin and Keller, and is now named EBK quantization. It

Motion Mountain – The Adventure of Physics

Ref. 133

(92)

8 colours and more

183

Hydrogen: spectral lines and
energy levels

Continuum of ionized states
n=

8

Energy

954.597 nm
nm

n=3

700
n=2

656.2852 nm
656.272 nm

3D5/2
3P3/2 , 3D3/2

3D5/2
3P3/2 , 3D3/2

3S1/2 , 3P1/2

3S1/2
3P1/2
2P3/2

F=2
F=1

2S1/2
2P1/2

F=0

1S1/2

F=1

2P3/2
2S1/2 , 2P1/2

650

F=1

F=1
F=0

600

500
486.133 nm
n=1

450
434.047 nm

virtual particle
nonrelativistic relativistic
(Bohr) levels
(Sommerfeld- levels (with
Dirac) levels
Lamb shift)
(fine structure)

400

F=0
nuclear levels
at higher scale
(hyperfine
structure)

F I G U R E 79 Atomic hydrogen: the visible spectrum of hydrogen (NASA) and its calculated energy levels,
in four approximations of increasing precision. Can you associate the visible lines to the correct level
transitions?

Ref. 134

relies on the fact that the action ? of any quantum system obeys
?=

?
1
∮ d?? ?? = (?? + ? ) ℏ
2π
4

(93)

for every coordinate ?? and its conjugate momentum ?? . The expression reflects the similarity between angular momentum and action. Here, ?? can be zero or any positive integer,
and ?? is the so-called Maslov index, an even integer, which in the case of atoms has the
value 2 for the radial and azimuthal coordinates ? and ?, and 0 for the rotation angle ?.
The integral is to be taken along a full orbit. In simple words, the action ? is a half-integer
multiple of the quantum of action. This result can be used to calculate the energy levels of
periodic quantum systems. Let us do so for hydrogen atoms.
Any rotational motion in a spherical potential ?(?) is characterized by a constant
energy ? and constant angular momenta ? and ? ?. Therefore the conjugate momenta

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

410.174 nm
397.007 nm

1S1/2

Motion Mountain – The Adventure of Physics

550

184

Challenge 145 ny

8 colours and more

for the coordinates ?, ? and ? are
?? = √2?(? − ?(?)) −
?? = √?2 −
?? = ? ? .

?2
?2

?2?
sin2 ?
(94)

Using these expressions in equation (93) and setting ? = ?? + ?? + ?? + 1, we get* the
result
??4
1
?ℎ?
?2 ??2
2.19 aJ
13.6 eV
?? = − 2
=
−
=
−
≈− 2 ≈−
.
(97)
2
2
2
2
? 2(4π?0 ) ℏ
?
2?
?
?2

Challenge 148 s

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 180

Motion Mountain – The Adventure of Physics

Challenge 147 e

These energy levels ??, the non-relativistic Bohr levels, are shown in Figure 79. Using the
idea that a hydrogen atom emits a single photon when its electron changes from state ??
to ?? , we get exactly the formula deduced by Balmer and Rydberg from observations!
The match between observation and calculation is about four digits. For (almost) the first
time ever, a material property, the colour of hydrogen atoms, had been explained from
a fundamental principle of nature. Key to this explanation was the quantum of action ℏ.
(This whole discussion assumes that the electrons in hydrogen atoms that emit light are
in eigenstates. Can you argue why this is the case?)
In short, the quantum of action implies that only certain specific energy values for
an electron are allowed inside an atom. The lowest energy level, for ? = 1, is called the
ground state. Its energy value 2.19 aJ is the ionization energy of hydrogen; if that energy is
added to the ground state, the electron is no longer bound to the nucleus. The ionization
energy thus plays the same role for electrons around atoms as does the escape velocity, or
better, the escape energy, for satellites or rockets shot from planets.
In the same way that the quantum of action determines the colours of the hydrogen
atom, it determines the colours of all other atoms. All Fraunhofer lines, whether observed
in the infrared, visible or ultraviolet, are due to the quantum of action. In fact, every
colour in nature is due to a mixture of colour lines, so that all colours, also those of
solids and liquids, are determined by the quantum of action.

8 colours and more

185

Motion Mountain – The Adventure of Physics

The shape of atoms
Free atoms are spherical. Atoms in external fields are deformed. Whatever the situation,
the shape of atoms is due to the shape of the wave function. The simplest case is the
Challenge 146 ny

* The calculation is straightforward. After insertion of ?(?) = ?/4π?0 ? into equation (94) one needs to
perform the (tricky) integration. Using the general result
?
d?
1
∮ √??2 + 2?? − ? = −√? +
√−?
2π
?

(95)

one gets
1
?2 √ ?
.
(?? + ) ℏ + ? = ?ℏ =
2
4π?0 −2?
This leads to the energy formula (97).

(96)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 80 The figure shows the calculated and the measured nodal structure of the hydrogen atom in
a weak external electric field, magnified by an electrostatic lens. The patterns are two-dimensional
interference shadows of the wave functions. Left column: how the wave function is projected from the
atoms to the macroscopic screen; central column: the measured nodal structure; right column:
comparison of the measured (solid) and calculated (dashed) electron densities. (© Aneta Stodolna/APS,
from Ref. 138).

186

Ref. 138

8 colours and more

The size of atoms
The calculation of the hydrogen energy levels also yields the effective radius of the electron orbits. It is given by

Page 188, page 196

Ref. 135

Page 21

ℏ2 4π?0
ℏ
=
= ?2 ?0 ≈ ?2 52.918 937 pm , with ? = 1, 2, 3, ...
2
?e ?
?? ??

(98)

We again see that, in contrast to classical physics, quantum theory allows only certain
specific orbits around the nucleus. (For more details about the fine-structure constant ?,
see below.) To be more precise, these radii are the average sizes of the electron clouds
surrounding the nucleus. The smallest orbital radius value, 53 pm for ? = 1, is called the
Bohr radius, and is denoted by ?0 .
In a gas of hydrogen atoms, most atoms are in the ground state described by ?1 = ?0
and ?1 . On the other hand, quantum theory implies that a hydrogen atom excited to the
level ? = 500 is about 12 μm in size: larger than many bacteria! Such blown-up atoms,
usually called Rydberg atoms, have indeed been observed in the laboratory, although they
are extremely sensitive to perturbations.
In short, the quantum of action determines the size of atoms. The result thus confirms
the prediction by Arthur Erich Haas from 1910. In other words
⊳ The quantum of action determines the size of all things.
In 1915, Arnold Sommerfeld understood that the analogy of electron motion with
orbital gravitational motion could be continued in two ways. First of all, electrons can
move, on average, on ellipses instead of circles. The quantization of angular momentum
then implies that only selected eccentricities are possible. The higher the angular momentum, the larger the number of possibilities: the first few are shown in Figure 81. The

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?? = ?2

Motion Mountain – The Adventure of Physics

hydrogen atom. Its wave functions – more precisely, the eigenfunctions for the first few
energy levels – are illustrated on the right hand side of Figure 81. These functions had
been calculated by Erwin Schrödinger already in 1926 and are found in all textbooks. We
do not perform the calculation here, and just show the results.
The square of the wave function is the probability density of the electron. This density
quickly decreases with increasing distance from the nucleus. Like for a real cloud, the
density is never zero, even at large distances. We could thus argue that all atoms have
infinite size; in practice however, chemical bonds or the arrangement of atoms in solids
show that it is much more appropriate to imagine atoms as clouds of finite size.
Surprisingly, the first measurement of the wave function of an atom dates only from
the year 2013; it was performed with a clever photoionization technique by Aneta Stodolna and her team. The beautiful experimental result is shown in Figure 80. The figures
confirm that wave functions, in contrast to probability densities, have nodes, i.e. lines –
or better, surfaces – where their value is zero.
In summary, all experiments confirm that the electron in the hydrogen atom forms
wave functions in exactly the way that is predicted by quantum theory. In particular, the
shape of atoms is found to agree with the calculation from quantum mechanics.

8 colours and more

0
nucleus
(not to scale)

n=2, l=1

0.2

187

0.4

0.6

0.8

1 nm

n=1, l=0

n=2, l=0

n=3, l=2

n=3, l=1

n=3, l=0

highest eccentricity corresponds to the minimum value ? = 0 of the so-called azimuthal
quantum number, whereas the case ? = ? − 1 correspond to circular orbits. Furthermore,
the ellipses can have different orientations in space.
The second point Sommerfeld noted was that the speeds of the electron in hydrogen
are – somewhat – relativistic: the speed values are not negligible compared to the speed
of light. Indeed, the orbital frequency of electrons in hydrogen is
1 ?4 ?e
1 ?? ?2 ?2 6.7 PHz
≈
?? = 3 2 3 = 3
? 4?0 ℎ
?
ℎ
?3

(99)

and the electron speed is
?? =

Ref. 136

?? 2.2 Mm/s 0.007 ?
1 ?2
=
≈
≈
.
? 4π?0 ℏ
?
?
?

(100)

As expected, the further the electron’s orbit is from the nucleus, the more slowly it moves.
This result can also be checked by experiment: exchanging the electron for a muon allows
us to measure the time dilation of its lifetime. Measurements are in excellent agreement
with the calculations.
In short, Sommerfeld noted that Bohr’s calculation did not take into account relativistic effects. And indeed, high-precision measurements show slight differences between
the Bohr’s non-relativistical energy levels and the measured ones. The calculation must
be improved.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 137

Motion Mountain – The Adventure of Physics

F I G U R E 81 The imagined, but not existing and thus false electron orbits of the Bohr–Sommerfeld
model of the hydrogen atom (left) and the correct description, using the probability density of the
electron in the various states (right) (© Wikimedia).

188

8 colours and more

R elativistic hydro gen

Ref. 134

Measuring atomic energy levels is possible with a much higher precision than measuring
wave functions. In particular, energy level measurements allow to observe relativistic
effects.
Also in the relativistic case, the EBK action has to be a multiple of ℏ/2. From the relativistic expression for the kinetic energy of the electron in a hydrogen atom
? + ?2 ? = √?2 ?2 + ?2 ?4 −

Challenge 149 e

(101)

we get the expression
??2 = 2?? (1 +

2??2
?
?
)
+
(1 + 2 ) .
2
2? ?
4π?0 ?
??

(102)

We now introduce, for convenience, the so-called fine-structure constant, as ? =
?2 /(4π?0 ℏ?) = √4πℏ?/?? ≈ 1/137.036. (? is a dimensionless constant; ? = 10.97 μm−1
is the Rydberg constant.) The radial EBK action then implies that
?2 ?

??? + ?2 ? =
√1 +

.

?2
(?−?− 12 +√(?+ 12 )2 −?2 )

(103)

2

Page 183

Page 196

Page 183

It reproduces the hydrogen spectrum to an extremely high accuracy. If we compare the
result with the non-relativistic one, we note that each non-relativistic level ? is split in
? different levels. This splitting is illustrated in Figure 79. In precision experiments, the
splitting of the lines of the hydrogen spectrum is visible as the so-called fine structure. The
magnitude of the fine structure depends on ?, a fundamental constant of nature. Since
Arnold Sommerfeld discovered the importance of this fundamental constant in this context, the name he chose, the fine-structure constant, has been taken over across the world.
The fine-structure constant describes the strength of the electromagnetic interaction; the
fine-structure constant is the electromagnetic coupling constant.
Modern high-precision experiments show additional effects that modify the colours
of atomic hydrogen. They are also illustrated in Figure 79. Virtual-particle effects and the
coupling of the proton spin give additional corrections. But that is still not all: isotope
effects, Doppler shifts and level shifts due to environmental electric or magnetic fields

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

This result, first found by Arnold Sommerfeld in 1915, is correct for point-like, i.e., nonrotating electrons. In reality, the electron has spin 1/2; the correct relativistic energy
levels thus appear when we set ? = ? ± 1/2 in the above formula. The result can be approximated by
?
?2
?
3
??? = − 2 (1 + 2 (
(104)
− ) + ...) .
1
?
?
?+ 2 4

Motion Mountain – The Adventure of Physics

Challenge 150 e

?2
4π?0 ?

8 colours and more

189

F I G U R E 82 Paul Dirac (1902–1984)

Vol. V, page 125

also influence the hydrogen spectrum. The final effect on the hydrogen spectrum, the
famous Lamb shift, will be a topic later on.

“

Page 106

The equation was more intelligent than I was.
Paul Dirac about his equation, repeating
a statement made by Heinrich Hertz.

”

?Dirac = ?? + ? ⋅ ? .

(105)

The quantities ? and the three components (?1 , ?2 , ?3 ) = ? turn out to be complex 4 × 4
matrices.
In Dirac’s representation, the position operator ? is not the position of a particle, but
has additional terms; its velocity operator has only the eigenvalues plus or minus the
velocity of light; the velocity operator is not simply related to the momentum operator;
* Paul Adrien Maurice Dirac (b. 1902 Bristol, d. 1984 Tallahassee), bilingual physicist, studied electrotechnics in Bristol, then went to Cambridge, where he later became a professor, holding the chair that Newton
had once held. In the years from 1925 to 1933 he published a stream of papers, of which several were worth a
Nobel Prize; he received it in 1933. Dirac unified special relativity and quantum theory, predicted antimatter, worked on spin and statistics, predicted magnetic monopoles, speculated on the law of large numbers,
and more besides. His introversion, friendliness and shyness, and his deep insights into nature, combined
with a dedication to beauty in theoretical physics, made him a legend all over the world during his lifetime.
For the latter half of his life he tried, unsuccessfully, to find an alternative to quantum electrodynamics, of
which he was the founder, as he was repelled by the problems of infinities. He died in Florida, where he
lived and worked after his retirement from Cambridge.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

What is the evolution equation for the wave function in the case that relativity, spin and
interactions with the electromagnetic field are taken into account? We could try to generalize the representation of relativistic motion given by Foldy and Wouthuysen to the
case of particles with electromagnetic interactions. Unfortunately, this is not a simple
matter. The simple identity between the classical and quantum-mechanical descriptions
is lost if electromagnetism is included.
Charged quantum particles are best described by another, equivalent representation
of the Hamiltonian, which was discovered much earlier, in 1926, by the British physicist
Paul Dirac.* Dirac found a neat trick to take the square root appearing in the relativistic
energy operator. In Dirac’s representation, the Hamilton operator is given by

Motion Mountain – The Adventure of Physics

R elativistic wave equations – again

190

8 colours and more

? → ? − ?? ,

Ref. 139

(106)

that treats electromagnetic momentum like particle momentum. With this prescription,
Dirac’s Hamiltonian describes the motion of charged particles interacting with an electromagnetic field ?. The minimal coupling substitution is not possible in the Foldy–
Wouthuysen Hamiltonian. In the Dirac representation, particles are pure, point-like,
structureless electric charges; in the Foldy–Wouthuysen representation they acquire a
charge radius and a magnetic-moment interaction. (We will come to the reasons below,
in the section on QED.)
In more detail, the simplest description of an electron (or any other elementary, stable,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. III, page 86

the equation of motion contains the famous ‘Zitterbewegung’ term; orbital angular momentum and spin are not separate constants of motion.
So why use this horrible Hamiltonian? Because only the Dirac Hamiltonian can easily
be used for charged particles. Indeed, it is transformed to the Hamiltonian coupled to the
electromagnetic field by the so-called minimal coupling, i.e., by the substitution

Motion Mountain – The Adventure of Physics

F I G U R E 83 The famous
Zitterbewegung: the
superposition of positive and
negative energy states leads to
an oscillation around a mean
vale. Colour indicates phase;
two coloured curves are shown,
as the Dirac equation in one
dimension has only two
components (not four); the grey
curve is the probability density.
(QuickTime film © Bernd Thaller)

8 colours and more

191

electrically-charged particle of spin 1/2) is given by the action ? and Lagrangian
? = ∫ LQED ?4 ? where
/ − ?2 ?) ? −
LQED = ? (?ℏ?D

(107)
1
? ???
4?0 ??

and

/ ? = ?? (∂? − ??? ? )
D

The QED Lagrangian implies that the wave function of a charged particle in a potential
follows the Dirac equation:
?ℏ?? (∂? − ??? ? )? = ??? .

Challenge 151 e

Challenge 152 e

(108)

The many indices should not make us forget that this equation simply states that the
eigenvalue of the energy–momentum operator is the rest mass (times the speed of light
?). In other words, the equation states that the wave ? moves with a phase velocity ?.
The wave function ? has four complex components. Two describe the motion of
particles, and two the motion of antiparticles. Each type of particle needs two complex
components, because the equation describes spin and particle density. Spin is a rotation,
and a rotation requires three real parameters. Spin and density thus require four real
parameters; they can be combined into two complex numbers, both for particles and for
antiparticles.
Each of the four components of the wave function of a relativistic spinning particle
follows the relativistic Schrödinger–Klein–Gordon equation. This means that the relativistic energy–momentum relation is followed by each component separately.
The relativistic wave function ? has the important property that a rotation by 2π
changes its sign. Only a rotation by 4π leaves the wave function unchanged. This is the
typical behaviour of spin 1/2 particles. For this reason, the four-component wave function of a spin 1/2 particle is called a spinor.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Get ting a first feeling for the Dirac equation

Motion Mountain – The Adventure of Physics

The first, matter term in the Lagrangian leads to the Dirac equation: it describes how
elementary, charged, spin 1/2 particles are moved by electromagnetic fields. The second,
radiation term leads to Maxwell’s equations, and describes how electromagnetic fields
are moved by the charged particle wave function. Together with a few calculating tricks,
these equations describe what is usually called quantum electrodynamics, or QED for
short.
As far as is known today, the relativistic description of the motion of charged matter and electromagnetic fields given the QED Lagrangian (107) is perfect: no differences
between theory and experiment have ever been found, despite intensive searches and
despite a high reward for anybody who would find one. All known predictions fully agree
with all measurements. In the most spectacular cases, the correspondence between theory and measurement extends to more than thirteen digits. But even more interesting
than the precision of QED are certain of its features that are missing in classical electrodynamics. Let’s have a quick tour.

192

8 colours and more

‘Antimatter’ is now a household term. Interestingly, the concept appeared before there
was any experimental evidence for it. The relativistic expression for the energy ? of an
electron with charge ? in the field of a charge ? is
?+

??
= √?2 ?4 + ?2 ?2 .
4π?0 ?

(109)

This expression also allows solutions with negative energy and opposite charge −?, if the
negative root is used. Quantum theory shows that this is a general property, and these
solutions correspond to what is called antimatter.
Indeed, the antimatter companion of the electron was predicted in the 1920s by Paul
Dirac from his equation (108), which is based on the above relativistic energy relation
(109). Unaware of this prediction, Carl Anderson discovered the antielectron in 1932,
and called it the positron. (The correct name would have been ‘positon’, without the ‘r’.
This correct form is used in the French language.) Anderson was studying cosmic rays,
and noticed that some ‘electrons’ were turning the wrong way in the magnetic field he
had applied to his apparatus. He checked his apparatus thoroughly, and finally deduced
that he had found a particle with the same mass as the electron but with positive electric
charge.
The existence of positrons has many strange implications. Already in 1928, before their
discovery, the Swedish theorist Oskar Klein had pointed out that Dirac’s equation for

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Antimat ter

Motion Mountain – The Adventure of Physics

F I G U R E 84 Klein’s paradox: the
motion of a relativistic wave
function that encounters a very
steep potential. Part of the
wave function is transmitted;
this part is antimatter, as the
larger lower component shows.
(QuickTime film © Bernd Thaller)

8 colours and more

Ref. 140

193

electrons makes a strange prediction: when an electron hits a sufficiently steep potential
wall, the reflection coefficient is larger than unity. Such a wall will reflect more than is
thrown at it. In addition, a large part of the wave function is transmitted through the
wall. In 1935, after the discovery of the positron, Werner Heisenberg and Hans Euler
explained the paradox. They found that the Dirac equation predicts that whenever an
electric field exceeds the critical value of
?c =

(110)

Virtual particles

Page 64

Despite what was said so far, action values smaller than the smallest action value do
have a role to play. We have already encountered one example: in a collision between
two electrons, there is an exchange of virtual photons. We learned that the exchanged
virtual photon cannot be observed. Indeed, the action ? for this exchange obeys
?⩽ℏ.

(111)

In short, virtual particles appear only as mediators in interactions. They cannot be observed. Virtual particles, in contrast to ordinary, real particles, do not obey the relation
?2 − ?2 ?2 = ?2 ?4 . For example, the kinetic energy can be negative. Indeed, virtual
particles are the opposite of ‘free’ or real particles. They may be observed in a vacuum if
the measurement time is very short. They are intrinsically short-lived.
Virtual photons are the cause for electrostatic potentials, for magnetic fields, for
the Casimir effect, for spontaneous emission, for the van der Waals force, and for the
Lamb shift in atoms. A more detailed treatment shows that in every situation with virtual photons there are also, with even lower probability, virtual electrons and virtual

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

the vacuum will spontaneously generate electron–positron pairs, which are then separated by the field. As a result, the original field is reduced. This so-called vacuum polarization is the reason for the reflection coefficient greater than unity found by Klein. Indeed,
steep potentials correspond to high electric fields.
Vacuum polarization shows that, in contrast to everyday life, the number of particles
is not a constant in the microscopic domain. Only the difference between particle number
and antiparticle number turns out to be conserved. Vacuum polarization thus limits our
possibility to count particles in nature!
Vacuum polarization is a weak effect. It has been only observed in particle collisions
of high energy. In those case, the effect even increases the fine-structure constant! Later
on we will describe truly gigantic examples of vacuum polarization that are postulated
around charged black holes.
Of course, the generation of electron–positron pairs is not a creation out of nothing,
but a transformation of energy into matter. Such processes are part of every relativistic
description of nature. Unfortunately, physicists have a habit of calling this transformation
‘pair creation’, thus confusing the issue somewhat. The transformation is described by
quantum field theory, which we will explore in the next volume.

Motion Mountain – The Adventure of Physics

Vol. V, page 153

?e ?2 ?2e ?3
=
= 1.3 EV/m ,
?? e
?ℏ

194

8 colours and more

positrons.
Massive virtual particles are essential for vacuum polarization, for the limit in the
number of the elements, for black-hole radiation and for Unruh radiation. Massive virtual particles also play a role in the strong interaction, where they hold the nucleons
together in nuclei, and in weak nuclear interaction, where they explain why beta decay
happens and why the Sun shines.
In particular, virtual particle–antiparticle pairs of matter and virtual radiation
particles together form what we call the vacuum. In addition, virtual radiation particles
form what are usually called static fields. Virtual particles are needed for a full description of all interactions. In particular, virtual particles are responsible for every decay
process.
Curiosities and fun challenges ab ou t colour and atoms

On modern high-precision measurements of the hydrogen spectra, listen to the undisputed master of the field: enjoy the 2012 talk by Theodor Hänsch, who has devoted a
large part of his life to the topic, at www.mediatheque.lindau-nobel.org.
∗∗
Ref. 142

The hydrogen atom bears many fascinating aspects. In 2015, Friedmann and Hagen
showed that a well-known formula for π = 3.14159265... can be extracted from the colour spectrum. Quantum mechanics and colours are beautiful subjects indeed.
∗∗

Ref. 136

If atoms contain orbiting electrons, the rotation of the Earth, via the Coriolis acceleration, should have an effect on their motion, and thus on the colour of atoms. This beautiful prediction is due to Mark Silverman; the effect is so small, however, that it has not
yet been measured.
∗∗

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

∗∗

Motion Mountain – The Adventure of Physics

Ref. 141

Where is the sea bluest? Sea water, like fresh water, is blue because it absorbs red and
green light. The absorption is due to a vibrational band of the water molecule that is due
to a combination of symmetric and asymmetric molecular stretches. The absorption is
weak, but noticeable. At 700 nm (red), the 1/? absorption length of water is 1 m.
Sea water can also be of bright colour if the sea floor reflects light. In addition, sea water can be green, if it contains small particles that scatter or absorb blue light. Most often,
these particles are soil or plankton. (Satellites can determine plankton content from the
‘greenness’ of the sea.) Thus the sea is especially blue if it is deep, quiet and cold; in that
case, the ground is distant, soil is not mixed into the water, and the plankton content is
low. The Sargasso Sea is 5 km deep, quiet and cold for most of the year. It is often called
the bluest of the Earth’s waters.
Lakes can also be blue if they contain small mineral particles. The particles scatter
light and lead to a blue colour for reasons similar to the blue colour of the sky. Such blue
lakes are found in many places on Earth.

8 colours and more

Ref. 143

195

Light is diffracted by material gratings. Can matter be diffracted by light gratings? Surprisingly, it actually can, as predicted by Dirac and Kapitza in 1937. This was accomplished
for the first time in 1986, using atoms. For free electrons, the feat is more difficult; the
clearest confirmation came in 2001, when new laser technology was used to perform a
beautiful measurement of the typical diffraction maxima for electrons diffracted by a
light grating.
∗∗

Ref. 136

Light is totally reflected when it is directed to a dense material at a large enough angle
so that it cannot enter the material. A group of Russian physicists have shown that if the
dense material is excited, the intensity of the totally-reflected beam can be amplified. It
is unclear whether this will ever lead to applications.
∗∗

Ref. 144

∗∗
Ref. 145
Challenge 153 s

Given two mirrors and a few photons, it is possible to capture an atom and keep it floating
between the two mirrors. This feat, one of several ways to isolate single atoms, is now
standard practice in laboratories. Can you imagine how it is done?

Ref. 146

An example of modern research is the study of hollow atoms, i.e., atoms missing a number of inner electrons. They have been discovered in 1990 by J.P. Briand and his group.
They appear when a completely ionized atom, i.e., one without any electrons, is brought
in contact with a metal. The acquired electrons then orbit on the outside, leaving the
inner shells empty, in stark contrast with usual atoms. Such hollow atoms can also be
formed by intense laser irradiation.
∗∗
Relativistic quantum effects can be seen with the unaided eye. The two most important
ones concern gold and mercury. The yellow colour of gold – which has atomic number
79 – is due to the transition energy between 5d and 6s electrons, which absorbs blue
light. Without relativistic effects, this transition would lie in the ultraviolet, similar to
the transition between 4d and 5s electrons for silver, and gold would be colourless. The
yellow colour of gold is thus a relativistic effect.
Mercury – which has atomic number 80 – has a filled 6s shell. Due to the same relativistic effects that appear in gold, these shells are contracted and do not like to form
bonds. For this reason, mercury is still liquid a room temperature, in contrast to all other
metals. Relativity is thus the reason that mercury is liquid, and that thermometers work.
∗∗

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

∗∗

Motion Mountain – The Adventure of Physics

Vol. I, page 343

The ways people handle single atoms with electromagnetic fields provide many beautiful
examples of modern applied technologies. Nowadays it is possible to levitate, to trap, to
excite, to photograph, to deexcite and to move single atoms just by shining light onto
them. In 1997, the Nobel Prize in Physics has been awarded to the originators of the field,
Steven Chu, Claude Cohen-Tannoudji and William Philips.

196

Challenge 154 s

8 colours and more

Is phosphorus phosphorescent?
∗∗

Challenge 155 ny

It is possible to detect the passage of a single photon through an apparatus without absorbing it. How would you do this?
Material properties

Ref. 147

The great physicist Wolfgang Pauli used to say that after his death, the first thing he
would ask god would be to explain Sommerfeld’s fine-structure constant. (Others used
to comment that after god will have explained it to him, he will think a little, and then
snap: ‘Wrong!’)
The fine-structure constant, introduced by Arnold Sommerfeld, is the dimensionless
constant of nature whose value is measured to be
?=

Ref. 148

?2
1
≈
≈ 0.007 297 352 5376(50) .
4π?0 ℏ? 137.035 999 679(94)

(112)

This number first appeared in explanations of the fine structure of atomic colour spectra;
hence its strange name. Sommerfeld was the first to understand its general importance.
It is central to quantum electrodynamics for several reasons.
First of all, the fine-structure constant describes the strength of electromagnetism.
The number ? results from the interaction of two electric charges ?. Writing Coulomb’s
relation for the force ? between two electrons as
?=?

ℏ?
?2

(113)

it becomes clear that the fine-structure constant describes the strength of electromagnet-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

A tough challenge: the strength of electromagnetism

Motion Mountain – The Adventure of Physics

Like the size of hydrogen atoms, also the size of all other atoms is fixed by the quantum
of action. Indeed, the quantum of action determines to a large degree the interactions
among electrons. By doing so, the quantum of change determines all the interactions
between atoms in everyday matter; therefore it determines all other material properties.
The elasticity, the plasticity, the brittleness, the magnetic and electric properties of materials are equally fixed by the quantum of action. Only ℏ makes electronics possible! We
will study some examples of material properties in the next volume. Various details of
the general connection between ℏ and material properties are still a subject of research,
though none is in contradiction with the quantum of action. Material research is among
the most important fields of modern science, and most advances in our standard of living
result from it. We will explore some aspects in the next volume.
In summary, materials science has confirmed that quantum physics is also the correct
description of all materials; quantum physics has confirmed that all material properties
of everyday life are of electromagnetic origin; and quantum physics has confirmed that
all material properties of everyday life are due to interactions that involve electrons.

8 colours and more

Ref. 149

A summary on colours and materials
In summary, the quantum of action ℏ – together with the interaction between electromagnetic fields and the electrons inside atoms, molecules, liquids and solids – determines the size, the shape, the colour and the material properties of all things around us.
The quantum of action determines mechanical properties such as hardness or elasticity,
magnetic properties, thermal properties such as heat capacity or heat of condensation,
optical properties such as transparency, and electrical properties such as metallic shine.
In addition, the quantum of action determines all chemical and biological aspects of
matter. This connection is the topic of the next volume.
The strength of the electromagnetic interaction is described by the fine-structure constant ? ≈ 1/137.036. Its value is yet unexplained.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 157 s

ism. A higher value for the fine-structure constant ? would mean a stronger attraction
or repulsion between charged bodies. Thus the value of ? determines the sizes of atoms,
and indeed of all things, as well as all colours in nature.
Secondly, it is only because the fine-structure constant ? is so small that we are able
to talk about particles at all. Indeed, only because the fine-structure constant is much
smaller than 1 it is possible to distinguish particles from each other. If the number ?
were near to or larger than 1, particles would interact so strongly that it would not be
possible to observe them separately or to talk about particles at all.
This leads on to the third reason for the importance of the fine-structure constant.
Since it is a dimensionless number, it implies some yet-unknown mechanism that fixes
its value. Uncovering this mechanism is one of the challenges remaining in our adventure. As long as the mechanism remains unknown – as was the case in 2016 – we do not
understand the colour and size of a single thing around us!
Small changes in the strength of electromagnetic attraction between electrons and
protons would have numerous important consequences. Can you describe what would
happen to the size of people, to the colour of objects, to the colour of the Sun, or to the
workings of computers, if the strength were to double? And what if it were to gradually
drop to half its usual value?
Since the 1920s, explaining the value of ? has been seen as one of the toughest challenges facing modern physics. That is the reason for Pauli’s fantasy. In 1946, during his
Nobel Prize lecture, he repeated the statement that a theory that does not determine
this number cannot be complete. Since that time, physicists seem to have fallen into two
classes: those who did not dare to take on the challenge, and those who had no clue. This
fascinating story still awaits us.
The problem of the fine-structure constant is so deep that it leads many astray. For
example, it is sometimes claimed that it is impossible to change physical units in such a
way that ℏ, ? and ? are all equal to 1 at the same time, because to do so would change the
number ? = 1/137.036...(1). Can you explain why the argument is wrong?

Motion Mountain – The Adventure of Physics

Challenge 156 s

197

Chapter 9

QUA N T UM PH YSIC S I N A N U T SH E L L

C

⊳ In nature, actions or changes smaller than ℏ = 1.054 571 800(13) ⋅ 10−34 Js
are not observed.

Physical results of quantum theory
The existence of a smallest action value in nature leads directly to the main lesson we
learned about motion in the quantum part of our adventure:
⊳ If it moves, it is made of quantons, or quantum particles.
This statement applies to every physical system, thus to all objects and to all images, i.e.,
to all matter and radiation. Moving stuff is made of quantons. Stones, water waves, light,
sound waves, earthquakes, gelatine and everything else we can interact with is made of
quantum particles.
In our exploration of relativity we discovered that also horizons and the vacuum can
move. If all moving entities are made of quantum particles, what does this imply for
horizons and empty space? We can argue that no fundamental problems are expected
for horizons, because one way to describe horizons is as an extreme state of matter. But
the details for the quantum aspects of vacuum are not simple; they will be the topic of
the last part of this adventure.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

This smallest action value, the quantum of action, leads to all the strange observations
made in the microscopic domain, such as the wave behaviour of matter, indeterminacy
relations, decoherence, randomness in measurements, indistinguishability, quantization
of angular momentum, tunnelling, pair creation, decay and particle reactions.
The essence of quantum theory is thus the lack of infinitely small change. The mathematics of quantum theory is abstract and involved, though. Was this part of our walk
worth the effort? It was: the results are profound, and the accuracy of the description is
complete. We first give an overview of these results and then turn to the questions that
are still left open.

Motion Mountain – The Adventure of Physics

ompared to classical physics, quantum theory is definitely more
omplex. The basic idea however, is simple: in nature there is a smallest
hange, or a smallest action with the value ℏ = 1.1 ⋅ 10−34 Js. More precisely,
all of quantum theory can be resumed in one sentence:

9 quantum physics in a nutshell

Vol. II, page 293
Vol. II, page 321

Page 128

Earlier in our adventure we asked: what are matter, radiation and interactions? Now
we know: they all are composites of elementary quantum particles. In particular, interactions are exchanges of elementary quantum particles.
An elementary quantum particle is a countable entity that is smaller than its own
Compton wavelength. All elementary particles are described by energy–momentum,
mass, spin, C, P and T parity. However, as we will see in the next volume, this is not
yet the complete list of particle properties. About the intrinsic properties of quantum
particles, i.e., those that do not depend on the observer, quantum theory makes a simple
statement:
⊳ In nature, all intrinsic properties of quantons, or quantum particles – with
the exception of mass – such as spin, electric charge, strong charge, parities
etc., appear as integer multiples of a basic unit. Since all physical systems are
made of quantons, in composed systems all intrinsic properties – with the
exception of mass* – either add or multiply.
In summary, all moving entities are made of quantum particles described by discrete intrinsic properties. To see how deep this result is, you can apply it to all those moving
entities for which it is usually forgotten, such as ghosts, spirits, angels, nymphs, daemons, devils, gods, goddesses and souls. You can check yourself what happens when
their particle nature is taken into account.

R esults on the motion of quantum particles

”

Quantons, or quantum particles, differ from everyday particles: quantum particles interfere: they behave like a mixture of particles and waves. This property follows directly
from the existence of ℏ, the smallest possible action in nature. From the existence of ℏ,
quantum theory deduces all its statements about quantum particle motion. We summarize the main ones.
There is no rest in nature. All objects obey the indeterminacy relation, which states
that the indeterminacies in position ? and momentum ? follow
Δ?Δ? ⩾ ℏ/2 with ℏ = 1.1 ⋅ 10−34 Js

(114)

and making rest an impossibility. The state of quantum particles is defined by the same
observables as in classical physics, with the difference that observables do not commute.
Classical physics appears in the limit that the Planck constant ℏ can effectively be set to
zero.
Quantum theory introduces a probabilistic element into motion. Probabilities result
from the quantum of action through the interactions with the baths that are part of the

* More precisely, together with mass, also mixing angles are not quantized. These properties are defined in
the next volume.
** ‘Offenses of gods are care of the gods.’

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

“

Deorum injuriae diis curae.**
Tiberius, as reported by Tacitus.

Motion Mountain – The Adventure of Physics

Challenge 158 e

199

200

9 quantum physics in a nutshell

environment of every physical system. Equivalently, probabilities result in every experiment that tries to induce a change that is smaller than the quantum of action.
Quantum particles behave like waves. The associated de Broglie wavelength ? is given
by the momentum ? through
ℎ 2πℏ
?= =
(115)
?
?

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

both in the case of matter and of radiation. This relation is the origin of the wave behaviour of light and matter. The light particles are called photons; their observation is now
standard practice. Quantum theory states that particle waves, like all waves, interfere, refract, disperse, dampen, can be dampened and can be polarized. This applies to photons,
electrons, atoms and molecules. All waves being made of quantum particles, all waves
can be seen, touched and moved. Light for example, can be ‘seen’ in photon-photon
scattering in vacuum at high energies, can be ‘touched’ using the Compton effect, and
can be ‘moved’ by gravitational bending. Matter particles, such as molecules or atoms,
can be seen in electron microscopes and can be touched and moved with atomic force
microscopes. The interference and diffraction of wave particles is observed daily in the
electron microscope.
Matter waves can be imagined as clouds that rotate locally. In the limit of negligible
cloud size, quantum particles can be imagined as rotating little arrows. Equivalently,
quantons have a phase.
Particles cannot be enclosed forever. Even though matter is impenetrable, quantum
theory shows that tight boxes or insurmountable obstacles do not exist. Enclosure is
never forever. Waiting long enough always allows us to overcome any boundary, since
there is a finite probability to overcome any obstacle. This process is called tunnelling
when seen from the spatial point of view and is called decay when seen from the temporal
point of view. Tunnelling explains the working of television tubes as well as radioactive
decay.
All particles and all particle beams can be rotated. Particles possess an intrinsic angular momentum called spin, specifying their behaviour under rotations. Bosons have
integer spin, fermions have half integer spin. An even number of bound fermions or any
number of bound bosons yield a composite boson; an odd number of bound fermions
yield a low-energy fermion. Solids are impenetrable because of the fermion character of
its electrons in the atoms.
Identical particles are indistinguishable. Radiation is made of indistinguishable
particles called bosons, matter of fermions. Under exchange of two fermions at space-like
separations, the wave function changes sign, whereas for two bosons the wave function
remains unchanged. All other properties of quantum particles are the same as for
classical particles, namely countability, interaction, mass, charge, angular momentum,
energy, momentum, position, as well as impenetrability for matter and penetrability for
radiation. Perfect copying machines do not exist.
In collisions, particles interact locally, through the exchange of other particles. When
matter particles collide, they interact through the exchange of virtual bosons, i.e., offshell bosons. Motion change is thus due to particle exchange. Exchange bosons of even
spin mediate only attractive interactions. Exchange bosons of odd spin mediate repulsive
interactions as well.

9 quantum physics in a nutshell

Achievements in accuracy and precision
Apart from the conceptual changes, quantum theory improved the accuracy of predictions from the few – if any – digits common in classical mechanics to the full number of
digits – sometimes thirteen – that can be measured today. The limited precision is usually not given by the inaccuracy of theory, it is given by the measurement accuracy. In
other words, the agreement is only limited by the amount of money the experimenter is
willing to spend. Table 8 shows this in more detail.
TA B L E 8 Selected comparisons between classical physics, quantum theory and experiment.

O b s e r va b l e

Clas Prediction of
sical
qua ntu m
predic - theory?
tion

Measurement

Cost
estim at e

(1 ± 10−2 ) ℏ/2

10 k€

Simple motion of bodies
Indeterminacy

0

Δ?Δ? ⩾ ℏ/2

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 153

The properties of collisions imply the non-conservation of particle number. In collisions, particles can appear – i.e., can be ‘created’ – or disappear – i.e., can be
‘annihilated’. This is valid both for bosons and for fermions.
The properties of collisions imply the existence of antiparticles, which are regularly
observed in experiments. Elementary fermions, in contrast to many elementary bosons,
differ from their antiparticles; they can be created and annihilated only in pairs. Elementary fermions have non-vanishing mass and move slower than light.
Particles can decay and be transformed. Detailed investigations show that collisions
imply the non-conservation of particle type. In collisions, selected particles can change
their intrinsic properties. This observation will be detailed in the next volume. Equivalently, the quantum of action implies that things break and living beings die.
Images, made of radiation, are described by the same observables as matter: position,
phase, speed, mass, momentum etc. – though their values and relations differ. Images
can only be localized with a precision of the wavelength ? of the radiation producing
them.
The appearance of Planck’s constant ℏ implies that length scales and time scales exist
in nature. Quantum theory introduces a fundamental jitter in every example of motion.
Thus the infinitely small is eliminated. In this way, lower limits to structural dimensions
and to many other measurable quantities appear. In particular, quantum theory shows
that it is impossible that on the electrons in an atom small creatures live in the same way
that humans live on the Earth circling the Sun. Quantum theory shows the impossibility
of Lilliput.
Clocks and metre bars have finite precision, due to the existence of a smallest action
and due to their interactions with baths. On the other hand, all measurement apparatuses
must contain baths, since otherwise they would not be able to record results.
Quantum effects leave no room for cold fusion, astrology, teleportation, telekinesis,
supernatural phenomena, creation out of nothing, multiple universes, or faster than light
phenomena – the EPR paradox notwithstanding.

Motion Mountain – The Adventure of Physics

Ref. 150

201

202

9 quantum physics in a nutshell

O b s e r va b l e

Clas Prediction of
sical
qua ntu m
predic - theory?
tion

Measurement

Cost
estim at e

Matter wavelength
Compton wavelength
Pair creation rate
Radiative decay time in
hydrogen
Smallest angular
momentum
Casimir effect/pressure

none
none
0
none

?? = 2πℏ
? c = ℎ/?e ?
??
? ∼ 1/?3

(1 ± 10−2 ) ℏ
(1 ± 10−3 ) ?
agrees
(1 ± 10−2 )

10 k€
20 k€
100 k€
5 k€

0

ℏ/2

(1 ± 10−6 ) ℏ/2

10 k€

0

? = (π2 ℏ?)/(240?4 )

(1 ± 10−3 )

30 k€

? max = ℎ?/(4.956 ??)
Δ? = 1057.86(1) MHz
?∞ = ?e ??2 /2ℎ
? = π2 ?4 /60ℏ3 ?2

(1 ± 10−4 ) Δ?
(1 ± 10−6 ) Δ?
(1 ± 10−9 ) ?∞
(1 ± 3 ⋅ 10−8 ) ?

10 k€
50 k€
50 k€
20 k€

? = ? max ?

(1 ± 10−5 ) ?

20 k€

1.34
from QED: finite
2.002 319 304 365(7)

1 k€
50 M€
30 M€

11 659 1827(63) ⋅ 10−11

within a few %
agrees
2.002 319 304
361 53(53)
11 659 2080(60) ⋅ 10−11

∞
4 463 302 542(620) Hz

> 1020 a
4 463 302 765(53) Hz

1€
1 M€

from QED

within 10−3

20 k€

Colours of objects

Composite matter properties
Atom lifetime
≈ 1 μs
Muonium hyperfine
none
splitting
Molecular size and shape none
Page 214

?. All these predictions are calculated from the basic physical constants given in Appendix A.

We notice that the predicted values do not differ from the measured ones. If we remember
that classical physics does not allow us to calculate any of the measured values, we get an
idea of the progress quantum physics has achieved. This advance in understanding is due
to the introduction of the quantum of action ℏ. Equivalently, we can state: no description
of nature without the quantum of action is complete.
In summary, quantum theory is precise and accurate. In the microscopic domain
quantum theory is in perfect correspondence with nature; despite prospects of fame and
riches, despite the largest number of researchers ever, no contradiction between observation and theory has been found yet. On the other hand, explaining the measured value

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

100 M€

Motion Mountain – The Adventure of Physics

Spectrum of hot objects diverges
Lamb shift
none
Rydberg constant
none
Stefan–Boltzmann
none
constant
Wien’s displacement
none
constant
Refractive index of water none
Photon-photon scattering 0
Electron gyromagnetic 1 or 2
ratio
Muon anomalous
0
magnetic moment

9 quantum physics in a nutshell

203

of the fine-structure constant, ? = 1/137.035 999 074(44), remains an open problem of
the electromagnetic interaction.
Is quantum theory magic?
Studying nature is like experiencing magic. Nature often looks different from what it is.
During magic we are fooled – but only if we forget our own limitations. Once we start
to see ourselves as part of the game, we start to understand the tricks. That is the fun of
magic. The same happens in quantum motion.
∗∗
Nature seems irreversible, even though it isn’t. We never remember the future. We are
fooled because we are macroscopic.
∗∗

∗∗
There are no clocks in nature. We are fooled by those of everyday life because we are
surrounded by a huge number of particles.
∗∗

∗∗
Objects seem distinguishable, even though the statistical properties of their components
show that they are not. We are fooled because we live at low energies.
∗∗
Matter seems continuous, even though it isn’t. We are fooled because of the limitations
of our senses.
∗∗
Motion seems deterministic in the classical sense, even though it is random. We are fooled
again because we are macroscopic.
∗∗
In short, our human condition permanently fools us. The answer to the title question is:
classical physics is like magic, and the tricks are uncovered by quantum theory. That is
its main attraction.
Q uantum theory is exact, bu t can d o more
We can summarize this part of our adventure with a simple statement:

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Motion often seems to disappear, even though it is eternal. We are fooled again, because
our senses cannot experience the microscopic domain.

Motion Mountain – The Adventure of Physics

Nature seems decoherent, even though it isn’t. We are fooled again because we are macroscopic.

204

9 quantum physics in a nutshell

⊳ Quantum physics is the description of matter and radiation without the
concept of infinitely small.

Challenge 159 e

Motion Mountain – The Adventure of Physics

Page 15

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

All change in nature, in fact, everything is described by finite quantities, and
above all, by the smallest change possible in nature, the quantum of action ℏ =
1.054 571 800(13) ⋅ 10−34 Js.
All experiments, without exception, show that the quantum of action ℏ is the smallest
observable change. The description of nature with the quantum of action is thus exact
and final. The smallest measurable action ℏ, like the maximum energy speed ?, is a fundamental property of nature. One could also call both of them fundamental truths.
Since quantum theory follows logically and completely from the smallest measurable
action ℏ, the simplest way – and the only way – to disprove quantum theory is to find an
observation that contradicts the smallest change value ℏ. Try it!
Even though we have deduced a fundamental property of nature, if we turn back to the
start of our exploration of quantum theory, we cannot hide a certain disappointment. We
know that classical physics cannot explain life. Searching for the details of microscopic
motion, we encountered so many interesting aspects that we have not yet achieved the
explanation of life. For example, we know what determines the speed of electrons in
atoms, but we do not know what determines the running speed of an athlete. In fact,
we have not even discussed the properties of any solid or liquid, let alone those of more
complex structures like living beings.
In other terms, after this introduction into quantum theory, we must still connect
quantum processes to our everyday world. Therefore, the topic of the next volume will
be the exploration of the motion of and inside living things – and of the motion inside all
kind of matter, from solids to stars, using the quantum of action as a foundation. After
that, we will explore the motion of empty space.

Appendix A

U N I T S , M E A SU R E M E N T S A N D
C ON STA N T S

M

All SI units are built from seven base units, whose official definitions, translated from
French into English, are given below, together with the dates of their formulation:
‘The second is the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state of the caesium 133
atom.’ (1967)*
‘The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.’ (1983)*
‘The kilogram is the unit of mass; it is equal to the mass of the international prototype
of the kilogram.’ (1901)*
‘The ampere is that constant current which, if maintained in two straight parallel
conductors of infinite length, of negligible circular cross-section, and placed 1 metre
apart in vacuum, would produce between these conductors a force equal to 2 ⋅ 10−7 newton per metre of length.’ (1948)*
‘The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.’ (1967)*
‘The mole is the amount of substance of a system which contains as many elementary
entities as there are atoms in 0.012 kilogram of carbon 12.’ (1971)*

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

SI units

Motion Mountain – The Adventure of Physics

Ref. 151

easurements are comparisons with standards. Standards are based on units.
any different systems of units have been used throughout the world.
ost of these standards confer power to the organization in charge of them.
Such power can be misused; this is the case today, for example in the computer industry, and was so in the distant past. The solution is the same in both cases: organize
an independent and global standard. For measurement units, this happened in the
eighteenth century: in order to avoid misuse by authoritarian institutions, to eliminate
problems with differing, changing and irreproducible standards, and – this is not a joke
– to simplify tax collection and to make it more just, a group of scientists, politicians
and economists agreed on a set of units. It is called the Système International d’Unités,
abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’.
The units are maintained by an international organization, the ‘Conférence Générale
des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale
des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM). All
originated in the times just before the French revolution.

206

a units, measurements and constants

‘The candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 ⋅ 1012 hertz and has a radiant intensity in that
direction of (1/683) watt per steradian.’ (1979)*
We note that both time and length units are defined as certain properties of a standard
example of motion, namely light. In other words, also the Conférence Générale des Poids
et Mesures makes the point that the observation of motion is a prerequisite for the definition and construction of time and space. Motion is the fundament of every observation
and of all measurement. By the way, the use of light in the definitions had been proposed
already in 1827 by Jacques Babinet.**
From these basic units, all other units are defined by multiplication and division. Thus,
all SI units have the following properties:

* The respective symbols are s, m, kg, A, K, mol and cd. The international prototype of the kilogram is
a platinum–iridium cylinder kept at the BIPM in Sèvres, in France. For more details on the levels of the
caesium atom, consult a book on atomic physics. The Celsius scale of temperature ? is defined as: ?/°C =
?/K − 273.15; note the small difference with the number appearing in the definition of the kelvin. SI also
states: ‘When the mole is used, the elementary entities must be specified and may be atoms, molecules,
ions, electrons, other particles, or specified groups of such particles.’ In the definition of the mole, it is
understood that the carbon 12 atoms are unbound, at rest and in their ground state. In the definition of the
candela, the frequency of the light corresponds to 555.5 nm, i.e., green colour, around the wavelength to
which the eye is most sensitive.
** Jacques Babinet (1794–1874), French physicist who published important work in optics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. I, page 100
Ref. 152

Motion Mountain – The Adventure of Physics

SI units form a system with state-of-the-art precision: all units are defined with a precision that is higher than the precision of commonly used measurements. Moreover, the
precision of the definitions is regularly being improved. The present relative uncertainty
of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilogram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and
for the candela 10−3 .
SI units form an absolute system: all units are defined in such a way that they can
be reproduced in every suitably equipped laboratory, independently, and with high precision. This avoids as much as possible any misuse by the standard-setting organization.
(The kilogram, still defined with the help of an artefact, is the last exception to this requirement; extensive research is under way to eliminate this artefact from the definition
– an international race that will take a few more years. There are two approaches: counting particles, or fixing ℏ. The former can be achieved in crystals, e.g., crystals made of
pure silicon, the latter using any formula where ℏ appears, such as the formula for the de
Broglie wavelength or that of the Josephson effect.)
SI units form a practical system: the base units are quantities of everyday magnitude.
Frequently used units have standard names and abbreviations. The complete list includes
the seven base units just given, the supplementary units, the derived units and the admitted units.
The supplementary SI units are two: the unit for (plane) angle, defined as the ratio
of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the
subtended area to the square of the radius, the unit is the steradian (sr).
The derived units with special names, in their official English spelling, i.e., without
capital letters and accents, are:

a units, measurements and constants

Name

A bbre v iat i o n

Name

A b b r e v i at i o n

hertz
pascal
watt
volt
ohm
weber
henry
lumen
becquerel
sievert

Hz = 1/s
Pa = N/m2 = kg/m s2
W = kg m2 /s3
V = kg m2 /As3
Ω = V/A = kg m2 /A2 s3
Wb = Vs = kg m2 /As2
H = Vs/A = kg m2 /A2 s2
lm = cd sr
Bq = 1/s
Sv = J/kg = m2 /s2

newton
joule
coulomb
farad
siemens
tesla
degree Celsius
lux
gray
katal

N = kg m/s2
J = Nm = kg m2 /s2
C = As
F = As/V = A2 s4 /kg m2
S = 1/Ω
T = Wb/m2 = kg/As2 = kg/Cs
°C (see definition of kelvin)
lx = lm/m2 = cd sr/m2
Gy = J/kg = m2 /s2
kat = mol/s

We note that in all definitions of units, the kilogram only appears to the powers of 1,
0 and −1. Can you try to formulate the reason?
The admitted non-SI units are minute, hour, day (for time), degree 1° = π/180 rad,
minute 1 ? = π/10 800 rad, second 1 ?? = π/648 000 rad (for angles), litre, and tonne. All
other units are to be avoided.
All SI units are made more practical by the introduction of standard names and abbreviations for the powers of ten, the so-called prefixes:*
Power Name

Power Name

101
102
103
106
109
1012
1015

10−1
10−2
10−3
10−6
10−9
10−12
10−15

1018 Exa
1021 Zetta
1024 Yotta
unofficial:
1027 Xenta
1030 Wekta
1033 Vendekta
1036 Udekta

deca da
hecto h
kilo k
Mega M
Giga G
Tera T
Peta P

deci
centi
milli
micro
nano
pico
femto

d
c
m
μ
n
p
f

Power Name
E
Z
Y

10−18
10−21
10−24

atto
zepto
yocto

a
z
y

xenno
weko
vendeko
udeko

x
w
v
u

Ref. 153

X
W
V
U

10−27
10−30
10−33
10−36

SI units form a complete system: they cover in a systematic way the full set of observables of physics. Moreover, they fix the units of measurement for all other sciences
as well.

Challenge 161 e

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar
to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words ἑξάκις and
πεντάκις for ‘six times’ and ‘five times’, the unofficial ones to sound similar to the Greek words for nine,
ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten
‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from
nanus ‘dwarf’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from μικρός ‘small’,
deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from μέγας
‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’).
Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my
car’s fuel consumption was two tenths of a square millimetre.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Power Name

Motion Mountain – The Adventure of Physics

Challenge 160 s

207

208

a units, measurements and constants

SI units form a universal system: they can be used in trade, in industry, in commerce,
at home, in education and in research. They could even be used by extraterrestrial civilizations, if they existed.
SI units form a self-consistent system: the product or quotient of two SI units is also
an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used
for every unit.

Page 8

The meaning of measurement
Challenge 162 e

Planck ’ s natural units
Since the exact form of many equations depends on the system of units used, theoretical
physicists often use unit systems optimized for producing simple equations. The chosen
units and the values of the constants of nature are related. In microscopic physics, the
system of Planck’s natural units is frequently used. They are defined by setting ? = 1, ℏ =
* Apart from international units, there are also provincial units. Most provincial units still in use are of
Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about
1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is defined as
exactly 1852 m. The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to
weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit
of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like
the system in which all units start with ‘f’, and which uses furlong/fortnight as its unit of velocity – are now
officially defined as multiples of SI units.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Every measurement is a comparison with a standard. Therefore, any measurement requires matter to realize the standard (even for a speed standard), and radiation to achieve
the comparison. The concept of measurement thus assumes that matter and radiation exist and can be clearly separated from each other.
Every measurement is a comparison. Measuring thus implies that space and time exist, and that they differ from each other.
Every measurement produces a measurement result. Therefore, every measurement
implies the storage of the result. The process of measurement thus implies that the situation before and after the measurement can be distinguished. In other terms, every measurement is an irreversible process.
Every measurement is a process. Thus every measurement takes a certain amount of
time and a certain amount of space.
All these properties of measurements are simple but important. Beware of anybody
who denies them.

Motion Mountain – The Adventure of Physics

The SI units are not the only possible set that could fulfil all these requirements, but
they are the only existing system that does so.* In the near future, the BIPM plans to
redefine the SI units using the physics cube diagram shown in Figure 1. This will be realized by fixing, in addition to the values of ? and ?cd , also the values of ℏ, ?, ? and ?A .
The proposed values are ℎ = 6.626 069 57 ⋅ 10−34 Js, ? = 1.602 176 565 ⋅ 10−19 C, ? =
1.380 648 8 ⋅ 10−23 J/K and ?A = 6.022 141 29 ⋅ 1023 1/mol. The definition of the second
will be retained, in order to avoid the low precision of all known measurements of ?. The
details of this future, new SI are presented on www.bipm.org/en/measurement-units/
new-si/ and www.bipm.org/utils/common/pdf/si-brochure-draft-2016.pdf.

a units, measurements and constants

Challenge 163 e

209

1, ? = 1, ? = 1, ?0 = 1/4π and ?0 = 4π. Planck units are thus defined from combinations
of fundamental constants; those corresponding to the fundamental SI units are given in
Table 10.* The table is also useful for converting equations written in natural units back
to SI units: just substitute every quantity ? by ?/?Pl.
TA B L E 10 Planck’s (uncorrected) natural units.

Name

Definition

Va l u e

Basic units
=

√ℏ?/?3

=

1.616 0(12) ⋅ 10−35 m

the Planck time

?Pl

=

√ℏ?/?5

=

5.390 6(40) ⋅ 10−44 s

the Planck mass

?Pl

=

√ℏ?/?

=

21.767(16) μg

the Planck current

?Pl

=

√4π?0 ?6 /?

=

3.479 3(22) ⋅ 1025 A

the Planck temperature

?Pl

=

√ℏ?5 /??2

=

1.417 1(91) ⋅ 1032 K

?Pl
? Pl
?aPl
?ePl

=
=
=
=

?
ℏ
ℏ
?

=
=
=
=

0.3 Gm/s
1.1 ⋅ 10−34 Js
1.1 ⋅ 10−34 Js
13.8 yJ/K

the Planck mass density

?Pl

=

?5 /?2 ℏ

=

5.2 ⋅ 1096 kg/m3

the Planck energy

?Pl

=

√ℏ?5 /?

=

2.0 GJ = 1.2 ⋅ 1028 eV

the Planck momentum

?Pl

=

√ℏ?3 /?

=

6.5 Ns

Trivial units
the Planck velocity
the Planck angular momentum
the Planck action
the Planck entropy
Composed units

Page 196

5

the Planck power
the Planck force
the Planck pressure

?Pl
?Pl
?Pl

=
=
=

? /?
?4 /?
?7 /?ℏ

=
=
=

3.6 ⋅ 1052 W
1.2 ⋅ 1044 N
4.6 ⋅ 10113 Pa

the Planck acceleration

?Pl

=

√?7 /ℏ?

=

5.6 ⋅ 1051 m/s2

the Planck frequency

?Pl

=

√?5 /ℏ?

=

1.9 ⋅ 1043 Hz

the Planck electric charge

?Pl

=

√4π?0 ?ℏ

=

1.9 aC = 11.7 e

the Planck voltage

?Pl

=

√?4 /4π?0 ?

=

1.0 ⋅ 1027 V

the Planck resistance

?Pl

=

1/4π?0 ?

=

30.0 Ω

=

1.8 ⋅ 10−45 F

the Planck capacitance

?Pl

=

4π?0 √ℏ?/?3

the Planck inductance

? Pl

=

(1/4π?0 )√ℏ?/?7

=

1.6 ⋅ 10−42 H

the Planck electric field

?Pl

=

√?7 /4π?0 ℏ?2

=

6.5 ⋅ 1061 V/m

* The natural units ?Pl given here are those commonly used today, i.e., those defined using the constant
ℏ, and not, as Planck originally did, by using the constant ℎ = 2πℏ. The electromagnetic units can also be
defined with other factors than 4π?0 in the expressions: for example, using 4π?0 ?, with the fine-structure
constant ?, gives ?Pl = ?. For the explanation of the numbers between brackets, see below.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?Pl

Motion Mountain – The Adventure of Physics

the Planck length

210

a units, measurements and constants

TA B L E 10 (Continued) Planck’s (uncorrected) natural units.

Vol. VI, page 36

Definition

the Planck magnetic flux density

?Pl

=

√?5 /4π?0 ℏ?2

Va l u e
=

2.2 ⋅ 1053 T

The natural units are important for another reason: whenever a quantity is sloppily called
‘infinitely small (or large)’, the correct expression is ‘as small (or as large) as the corresponding corrected Planck unit’. As explained throughout the text, and especially in the
final part, this substitution is possible because almost all Planck units provide, within
a correction factor of order 1, the extremal value for the corresponding observable –
some an upper and some a lower limit. Unfortunately, these correction factors are not
yet widely known. The exact extremal value for each observable in nature is obtained
when ? is substituted by 4? and 4π?0 by 4π?0 ? in all Planck quantities. These extremal
values, or corrected Planck units, are the true natural units. To exceed the extremal values
is possible only for some extensive quantities. (Can you find out which ones?)
Other unit systems

1/[?2 ] = [?]2 = [?] = [?] = [?electric] ,
1/[?] = [?] = [?] = [?] = [?] = [?] = [?] = [?] = [?] ,
1 = [?] = [?] = [?] = [?] = [?action] = [?entropy ] = ℏ = ? = ? = [?] , (116)
[?] = 1/[?] = [?] = [?] = [?] and
[?]2 =1/[?]2 = [?] = [?]
where we write [?] for the unit of quantity ?. Using the same unit for time, capacitance
and inductance is not to everybody’s taste, however, and therefore electricians do not
use this system.**

Ref. 154

* Other definitions for the proportionality constants in electrodynamics lead to the Gaussian unit system
often used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and
the electromagnetic unit system, among others.
** In the list, ? is length, ? energy, ? force, ?electric the electric and ? the magnetic field, ? mass, ? momentum,
? acceleration, ? frequency, ? electric current, ? voltage, ? temperature, ? speed, ? charge, ? resistance, ?
power, ? the gravitational constant.
The web page www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert
various units into each other.
Researchers in general relativity often use another system, in which the Schwarzschild radius ?s =
2??/?2 is used to measure masses, by setting ? = ? = 1. In this case, mass and length have the same
dimension, and ℏ has the dimension of an area.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

A central aim of research in high-energy physics is the calculation of the strengths of
all interactions; therefore it is not practical to set the gravitational constant ? to unity,
as in the Planck system of units. For this reason, high-energy physicists often only set
? = ℏ = ? = 1 and ?0 = 1/?0 = 4π,* leaving only the gravitational constant ? in the
equations.
In this system, only one fundamental unit exists, but its choice is free. Often a standard length is chosen as the fundamental unit, length being the archetype of a measured
quantity. The most important physical observables are then related by

Motion Mountain – The Adventure of Physics

Challenge 164 s

Name

a units, measurements and constants

211

Often, in order to get an impression of the energies needed to observe an effect under study, a standard energy is chosen as fundamental unit. In particle physics the most
common energy unit is the electron volt eV, defined as the kinetic energy acquired by
an electron when accelerated by an electrical potential difference of 1 volt (‘proton volt’
would be a better name). Therefore one has 1 eV = 1.6 ⋅ 10−19 J, or roughly
1 eV ≈

Challenge 165 e

Ref. 156

aJ

(117)

Curiosities and fun challenges ab ou t units
Ref. 157

The Planck length is roughly the de Broglie wavelength ? B = ℎ/?? of a man walking
comfortably (? = 80 kg, ? = 0.5 m/s); this motion is therefore aptly called the ‘Planck
stroll.’
∗∗
The Planck mass is equal to the mass of about 1019 protons. This is roughly the mass of
a human embryo at about ten days of age.
∗∗

Ref. 158

The most precisely measured quantities in nature are the frequencies of certain millisecond pulsars, the frequency of certain narrow atomic transitions, and the Rydberg
constant of atomic hydrogen, which can all be measured as precisely as the second is
defined. The caesium transition that defines the second has a finite linewidth that limits
the achievable precision: the limit is about 14 digits.
∗∗
The most precise clock ever built, using microwaves, had a stability of 10−16 during a

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

which is easily remembered. The simplification ? = ℏ = 1 yields ? = 6.9 ⋅ 10−57 eV−2 and
allows one to use the unit eV also for mass, momentum, temperature, frequency, time
and length, with the respective correspondences 1 eV ≡ 1.8 ⋅ 10−36 kg ≡ 5.4 ⋅ 10−28 Ns
≡ 242 THz ≡ 11.6 kK and 1 eV−1 ≡ 4.1 fs ≡ 1.2 μm.
To get some feeling for the unit eV, the following relations are useful. Room temperature, usually taken as 20°C or 293 K, corresponds to a kinetic energy per particle of
0.025 eV or 4.0 zJ. The highest particle energy measured so far belongs to a cosmic ray
with an energy of 3 ⋅ 1020 eV or 48 J. Down here on the Earth, an accelerator able to produce an energy of about 105 GeV or 17 nJ for electrons and antielectrons has been built,
and one able to produce an energy of 14 TeV or 2.2 μJ for protons will be finished soon.
Both are owned by CERN in Geneva and have a circumference of 27 km.
The lowest temperature measured up to now is 280 pK, in a system of rhodium
nuclei held inside a special cooling system. The interior of that cryostat may even be
the coolest point in the whole universe. The kinetic energy per particle corresponding to that temperature is also the smallest ever measured: it corresponds to 24 feV or
3.8 vJ = 3.8 ⋅ 10−33 J. For isolated particles, the record seems to be for neutrons: kinetic
energies as low as 10−7 eV have been achieved, corresponding to de Broglie wavelengths
of 60 nm.

Motion Mountain – The Adventure of Physics

Ref. 155

1
6

212

Ref. 159
Ref. 160

a units, measurements and constants

running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but
values around 10−17 seem within technological reach. The precision of clocks is limited
for short measuring times by noise, and for long measuring times by drifts, i.e., by systematic effects. The region of highest stability depends on the clock type; it usually lies
between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock
for which this region is not known yet; it certainly lies at more than 20 years, the time
elapsed at the time of writing since their discovery.
∗∗

Ref. 161
Challenge 166 s

The shortest times measured are the lifetimes of certain ‘elementary’ particles. In particular, the lifetime of certain D mesons have been measured at less than 10−23 s. Such
times are measured using a bubble chamber, where the track is photographed. Can you
estimate how long the track is? (This is a trick question – if your length cannot be observed with an optical microscope, you have made a mistake in your calculation.)

Ref. 162

The longest times encountered in nature are the lifetimes of certain radioisotopes, over
1015 years, and the lower limit of certain proton decays, over 1032 years. These times are
thus much larger than the age of the universe, estimated to be fourteen thousand million
years.
∗∗

Ref. 164

Precision and accuracy of measurements
Measurements are the basis of physics. Every measurement has an error. Errors are due
to lack of precision or to lack of accuracy. Precision means how well a result is reproduced
when the measurement is repeated; accuracy is the degree to which a measurement corresponds to the actual value.
Lack of precision is due to accidental or random errors; they are best measured by the
standard deviation, usually abbreviated ?; it is defined through
?2 =

Challenge 167 s

1 ?
∑(? − ?)̄ 2 ,
? − 1 ?=1 ?

(118)

where ?̄ is the average of the measurements ?? . (Can you imagine why ? − 1 is used in
the formula instead of ??)
For most experiments, the distribution of measurement values tends towards a normal distribution, also called Gaussian distribution, whenever the number of measure-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 163

Variations of quantities are often much easier to measure than their values. For example,
in gravitational wave detectors, the sensitivity achieved in 1992 was Δ?/? = 3 ⋅ 10−19 for
lengths of the order of 1 m. In other words, for a block of about a cubic metre of metal
it is possible to measure length changes about 3000 times smaller than a proton radius.
These set-ups are now being superseded by ring interferometers. Ring interferometers
measuring frequency differences of 10−21 have already been built; and they are still being
improved.

Motion Mountain – The Adventure of Physics

∗∗

a units, measurements and constants

213

N
number of measurements

standard deviation
full width at half maximum
(FWHM)
limit curve for a large number
of measurements: the
Gaussian distribution
x
measured values

F I G U R E 85 A precision experiment and its measurement distribution. The precision is high if the width
of the distribution is narrow; the accuracy is high if the centre of the distribution agrees with the actual
value.

ments is increased. The distribution, shown in Figure 85, is described by the expression
?(?) ≈ e−

Ref. 165

Challenge 169 e

(119)

The square ?2 of the standard deviation is also called the variance. For a Gaussian distribution of measurement values, 2.35? is the full width at half maximum.
Lack of accuracy is due to systematic errors; usually these can only be estimated. This
estimate is often added to the random errors to produce a total experimental error, sometimes also called total uncertainty. The relative error or uncertainty is the ratio between
the error and the measured value.
For example, a professional measurement will give a result such as 0.312(6) m. The
number between the parentheses is the standard deviation ?, in units of the last digits.
As above, a Gaussian distribution for the measurement results is assumed. Therefore, a
value of 0.312(6) m implies that the actual value is expected to lie
—
—
—
—
—
—
—

Challenge 170 s

.

within 1? with 68.3 % probability, thus in this example within 0.312 ± 0.006 m;
within 2? with 95.4 % probability, thus in this example within 0.312 ± 0.012 m;
within 3? with 99.73 % probability, thus in this example within 0.312 ± 0.018 m;
within 4? with 99.9937 % probability, thus in this example within 0.312 ± 0.024 m;
within 5? with 99.999 943 % probability, thus in this example within 0.312 ± 0.030 m;
within 6? with 99.999 999 80 % probability, thus within 0.312 ± 0.036 m;
within 7? with 99.999 999 999 74 % probability, thus within 0.312 ± 0.041 m.

(Do the latter numbers make sense?)
Note that standard deviations have one digit; you must be a world expert to use two,
and a fool to use more. If no standard deviation is given, a (1) is assumed. As a result,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 168 e

(?−?)̄ 2
2?2

Motion Mountain – The Adventure of Physics

x
average value

214

Challenge 171 s

a units, measurements and constants

among professionals, 1 km and 1000 m are not the same length!
What happens to the errors when two measured values ? and ? are added or subtracted? If the all measurements are independent – or uncorrelated – the standard deviation
of the sum and that of difference is given by ? = √??2 + ??2 . For both the product or ratio
of two measured and uncorrelated values ? and ?, the result is ? = √??2 + ??2 , where the
? terms are the relative standard deviations.
Assume you measure that an object moves 1 m in 3 s: what is the measured speed
value?
Limits to precision

Challenge 172 e

Ref. 166

Ref. 167
Ref. 166
Vol. V, page 261

In physics, general observations are deduced from more fundamental ones. As a consequence, many measurements can be deduced from more fundamental ones. The most
fundamental measurements are those of the physical constants.
The following tables give the world’s best values of the most important physical constants and particle properties – in SI units and in a few other common units – as published in the standard references. The values are the world averages of the best measurements made up to the present. As usual, experimental errors, including both random
and estimated systematic errors, are expressed by giving the standard deviation in the
last digits. In fact, behind each of the numbers in the following tables there is a long
story which is worth telling, but for which there is not enough room here.
In principle, all quantitative properties of matter can be calculated with quantum theory and the values of certain physical constants. For example, colour, density and elastic
properties can be predicted using the equations of the standard model of particle physics
and the values of the following basic constants.
TA B L E 11 Basic physical constants.

Q ua nt i t y

Symbol

Constants that define the SI measurement units

Va l u e i n S I u n i t s

U n c e r t. ?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Physical constants

Motion Mountain – The Adventure of Physics

Vol. VI, page 93

What are the limits to accuracy and precision? There is no way, even in principle, to
measure a length ? to a precision higher than about 61 digits, because in nature, the ratio
between the largest and the smallest measurable length is Δ?/? > ?Pl/?horizon = 10−61 .
(Is this ratio valid also for force or for volume?) In the final volume of our text, studies
of clocks and metre bars strengthen this theoretical limit.
But it is not difficult to deduce more stringent practical limits. No imaginable machine
can measure quantities with a higher precision than measuring the diameter of the Earth
within the smallest length ever measured, about 10−19 m; that is about 26 digits of precision. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits.
If, as predicted above, time measurements really achieve 17 digits of precision, then they
are nearing the practical limit, because apart from size, there is an additional practical
restriction: cost. Indeed, an additional digit in measurement precision often means an
additional digit in equipment cost.

a units, measurements and constants

215

TA B L E 11 (Continued) Basic physical constants.

U n c e r t. ?

Q ua nt i t y

Symbol

Va l u e i n S I u n i t s

Vacuum speed of light?
Vacuum permeability?

?
?0

299 792 458 m/s
0
4π ⋅ 10−7 H/m
0
= 1.256 637 061 435 ... μH/m0
8.854 187 817 620 ... pF/m 0
6.626 069 57(52) ⋅ 10−34 Js
4.4 ⋅ 10−8
1.054 571 726(47) ⋅ 10−34 Js 4.4 ⋅ 10−8

?0 = 1/?0 ?2
Vacuum permittivity?
Original Planck constant
ℎ
Reduced Planck constant,
ℏ
quantum of action
Positron charge
?
Boltzmann constant
?
Gravitational constant
?
Gravitational coupling constant? = 8π?/?4

0

CKM quark mixing matrix

|?|

Jarlskog invariant

?

PMNS neutrino mixing m.

?

0?
3.2 ⋅ 10−10

= 0.007 297 352 5698(24)
1.166 364(5) ⋅ 10−5 GeV−2
1/30.1(3)
0.231 24(24)
0.2224(19)

3.2 ⋅ 10−10
4.3 ⋅ 10−6
1 ⋅ 10−2
1.0 ⋅ 10−3
8.7 ⋅ 10−3

0.118(3)
25 ⋅ 10−3
0.97428(15) 0.2253(7)
0.00347(16)
( 0.2252(7) 0.97345(16) 0.0410(11) )
0.00862(26) 0.0403(11) 0.999152(45)
2.96(20) ⋅ 10−5
0.82
0.55
−0.15 + 0.038?
(−0.36 + 0.020? 0.70 + 0.013?
)
0.61
0.44 + 0.026? −0.45 + 0.017?
0.77

Elementary particle masses (of unknown origin)
9.109 382 91(40) ⋅ 10−31 kg
Electron mass
?e
5.485 799 0946(22) ⋅ 10−4 u
0.510 998 928(11) MeV
Muon mass
?μ
1.883 531 475(96) ⋅ 10−28 kg

Tau mass
El. neutrino mass
Muon neutrino mass

??
??e
???

0.113 428 9267(29) u
105.658 3715(35) MeV
1.776 82(16) GeV/?2
< 2 eV/?2
< 2 eV/?2

4.4 ⋅ 10−8
4.0 ⋅ 10−10
2.2 ⋅ 10−8
5.1 ⋅ 10−8
2.5 ⋅ 10−8
3.4 ⋅ 10−8

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Strong coupling constant?

= ?em (?2e ?2 )
?F /(ℏ?)3
?w (?Z ) = ?w2 /4π
sin2 ?W (??)
sin2 ?W (on shell)
= 1 − (?W /?Z )2
?s (?Z ) = ?s2 /4π

e.m. coupling constant
Fermi coupling constant? or
weak coupling constant
Weak mixing angle

3+1
1/137.035 999 074(44)

Motion Mountain – The Adventure of Physics

Fundamental constants (of unknown origin)
Number of space-time dimensions
2
Fine-structure constant? or
? = 4π?? ℏ?

0.160 217 656 5(35) aC
2.2 ⋅ 10−8
−23
1.380 6488(13) ⋅ 10 J/K 9.1 ⋅ 10−7
6.673 84(80) ⋅ 10−11 Nm2 /kg2 1.2 ⋅ 10−4
2.076 50(25) ⋅ 10−43 s2 /kg m 1.2 ⋅ 10−4

216

a units, measurements and constants

TA B L E 11 (Continued) Basic physical constants.

Symbol

Va l u e i n S I u n i t s

Tau neutrino mass
Up quark mass
Down quark mass
Strange quark mass
Charm quark mass
Bottom quark mass
Top quark mass
Photon mass
W boson!mass
Z boson!mass
Higgs mass
Gluon mass

???
?
?
?
?
?
?
γ
?±
?0
H
g1...8

< 2 eV/?2
1.8 to 3.0 MeV/?2
4.5 to 5.5 MeV/?2
95(5) MeV/?2
1.275(25) GeV/?2
4.18(17) GeV/?2
173.5(1.4) GeV/?2
< 2 ⋅ 10−54 kg
80.385(15) GeV/?2
91.1876(21) GeV/?2
126(1) GeV/?2
c. 0 MeV/?2

Composite particle masses
Proton mass

?p

Neutron mass

Page 131

Page 208

4.4 ⋅ 10−8
8.9 ⋅ 10−11
2.2 ⋅ 10−8
4.4 ⋅ 10−8
4.2 ⋅ 10−10
2.2 ⋅ 10−8
4.4 ⋅ 10−8

?. Uncertainty: standard deviation of measurement errors.
?. Only measured from to 10−19 m to 1026 m.
?. Defining constant.
?. All coupling constants depend on the 4-momentum transfer, as explained in the section on
renormalization. Fine-structure constant is the traditional name for the electromagnetic coupling constant ?em in the case of a 4-momentum transfer of ?2 = ?2e ?2 , which is the smallest
2 2
? ) ≈ 1/128.
one possible. At higher momentum transfers it has larger values, e.g., ?em (?2 = ?W
In contrast, the strong coupling constant has lover values at higher momentum transfers; e.g.,
?s (34 GeV) = 0.14(2).

Why do all these basic constants have the values they have? For any basic constant
with a dimension, such as the quantum of action ℏ, the numerical value has only historical meaning. It is 1.054 ⋅ 10−34 Js because of the SI definition of the joule and the second.
The question why the value of a dimensional constant is not larger or smaller therefore
always requires one to understand the origin of some dimensionless number giving the
ratio between the constant and the corresponding natural unit that is defined with ?, ?,
ℏ and ?. More details and the values of the natural units are given above. Understanding
the sizes of atoms, people, trees and stars, the duration of molecular and atomic processes, or the mass of nuclei and mountains, implies understanding the ratios between
these values and the corresponding natural units. The key to understanding nature is
thus the understanding of all ratios, and thus of all dimensionless constants. The quest

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Atomic mass unit

1.672 621 777(74) ⋅ 10−27 kg
1.007 276 466 812(90) u
938.272 046(21) MeV
?n
1.674 927 351(74) ⋅ 10−27 kg
1.008 664 916 00(43) u
939.565 379(21) MeV
?u = ?12 C /12 = 1 u1.660 538 921(73) yg

U n c e r t. ?

Motion Mountain – The Adventure of Physics

Q ua nt i t y

a units, measurements and constants

217

of understanding all ratios, including the fine-structure constant ? itself, is completed
only in the final volume of our adventure.
The basic constants yield the following useful high-precision observations.
TA B L E 12 Derived physical constants.

Va l u e i n S I u n i t s

U n c e r t.

Vacuum wave resistance
Avogadro’s number
Loschmidt’s number
at 273.15 K and 101 325 Pa
Faraday’s constant
Universal gas constant
Molar volume of an ideal gas
at 273.15 K and 101 325 Pa
Rydberg constant ?
Conductance quantum
Magnetic flux quantum
Josephson frequency ratio
Von Klitzing constant
Bohr magneton
Classical electron radius
Compton wavelength
of the electron
Bohr radius ?
Quantum of circulation
Specific positron charge
Cyclotron frequency
of the electron
Electron magnetic moment

?0 = √?0 /?0
?A
?L

376.730 313 461 77... Ω
6.022 141 29(27) ⋅ 1023
2.686 7805(24) ⋅ 1023

0
4.4 ⋅ 10−8
9.1 ⋅ 10−7

? = ?A ?
? = ?A ?
? = ??/?

96 485.3365(21) C/mol
8.314 4621(75) J/mol K
22.413 968(20) l/mol

2.2 ⋅ 10−8
9.1 ⋅ 10−7
9.1 ⋅ 10−7

?∞ = ?e ??2 /2ℎ
?0 = 2?2 /ℎ
?0 = ℎ/2?
2?/ℎ
ℎ/?2 = ?0 ?/2?
?B = ?ℏ/2?e
?e = ?2 /4π?0 ?e ?2
? C = ℎ/?e ?
?c = ℏ/?e ? = ?e /?
?∞ = ?e /?2
ℎ/2?e
?/?e
?c /? = ?/2π?e

10 973 731.568 539(55) m−1
77.480 917 346(25) μS
2.067 833 758(46) pWb
483.597 870(11) THz/V
25 812.807 4434(84) Ω
9.274 009 68(20) yJ/T
2.817 940 3267(27) f m
2.426 310 2389(16) pm
0.386 159 268 00(25) pm
52.917 721 092(17) pm
3.636 947 5520(24) ⋅ 10−4 m2 /s
1.758 820 088(39) ⋅ 1011 C/kg
27.992 491 10(62) GHz/T

5 ⋅ 10−12
3.2 ⋅ 10−10
2.2 ⋅ 10−8
2.2 ⋅ 10−8
3.2 ⋅ 10−10
2.2 ⋅ 10−8
9.7 ⋅ 10−10
6.5 ⋅ 10−10
6.5 ⋅ 10−10
3.2 ⋅ 10−10
6.5 ⋅ 10−10
2.2 ⋅ 10−8
2.2 ⋅ 10−8

?e
?e /?B
?e /?N
?e
?μ /?e

−9.284 764 30(21) ⋅ 10−24 J/T
−1.001 159 652 180 76(27)
−1.838 281 970 90(75) ⋅ 103
−2.002 319 304 361 53(53)
206.768 2843(52)

2.2 ⋅ 10−8
2.6 ⋅ 10−13
4.1 ⋅ 10−10
2.6 ⋅ 10−13
2.5 ⋅ 10−8

?μ
?μ

−4.490 448 07(15) ⋅ 10−26 J/T 3.4 ⋅ 10−8
−2.002 331 8418(13)

6.3 ⋅ 10−10

?p /?e
?/?p
? C,p = ℎ/?p ?
?N = ?ℏ/2?p
?p
?p /?B
?p /?N

1 836.152 672 45(75)
9.578 833 58(21) ⋅ 107 C/kg
1.321 409 856 23(94) f m
5.050 783 53(11) ⋅ 10−27 J/T
1.410 606 743(33) ⋅ 10−26 J/T
1.521 032 210(12) ⋅ 10−3
2.792 847 356(23)

4.1 ⋅ 10−10
2.2 ⋅ 10−8
7.1 ⋅ 10−10
2.2 ⋅ 10−8
2.4 ⋅ 10−8
8.1 ⋅ 10−9
8.2 ⋅ 10−9

Electron g-factor
Muon–electron mass ratio
Muon magnetic moment
muon g-factor
Proton–electron mass ratio
Specific proton charge
Proton Compton wavelength
Nuclear magneton
Proton magnetic moment

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Symbol

Motion Mountain – The Adventure of Physics

Q ua nt i t y

218

a units, measurements and constants

TA B L E 12 (Continued) Derived physical constants.

Symbol

Va l u e i n S I u n i t s

U n c e r t.

Proton gyromagnetic ratio
Proton g factor
Neutron–electron mass ratio
Neutron–proton mass ratio
Neutron Compton wavelength
Neutron magnetic moment

?p = 2?? /ℏ
?p
?n /?e
?n /?p
? C,n = ℎ/?n ?
?n
?n /?B
?n /?N
? = π2 ?4 /60ℏ3 ?2
? = ? max ?

2.675 222 005(63) ⋅ 108 Hz/T
5.585 694 713(46)
1 838.683 6605(11)
1.001 378 419 17(45)
1.319 590 9068(11) f m
−0.966 236 47(23) ⋅ 10−26 J/T
−1.041 875 63(25) ⋅ 10−3
−1.913 042 72(45)
56.703 73(21) nW/m2 K4
2.897 7721(26) mmK
58.789 254(53) GHz/K
1.602 176 565(35) ⋅ 10−19 J
1023 bit = 0.956 994 5(9) J/K
3.7 to 4.0 MJ/kg

2.4 ⋅ 10−8
8.2 ⋅ 10−9
5.8 ⋅ 10−10
4.5 ⋅ 10−10
8.2 ⋅ 10−10
2.4 ⋅ 10−7
2.4 ⋅ 10−7
2.4 ⋅ 10−7
3.6 ⋅ 10−6
9.1 ⋅ 10−7
9.1 ⋅ 10−7
2.2 ⋅ 10−8
9.1 ⋅ 10−7
4 ⋅ 10−2

Stefan–Boltzmann constant
Wien’s displacement constant

Electron volt
eV
Bits to entropy conversion const. ? ln 2
TNT energy content
?. For infinite mass of the nucleus.

Some useful properties of our local environment are given in the following table.
TA B L E 13 Astronomical constants.

Symbol

Tropical year 1900 ?
Tropical year 1994
Mean sidereal day
Average distance Earth–Sun ?
Astronomical unit ?
Light year, based on Julian year ?
Parsec
Earth’s mass
Geocentric gravitational constant
Earth’s gravitational length
Earth’s equatorial radius ?

?
?
?

Earth’s polar radius ?
Equator–pole distance ?
Earth’s flattening ?
Earth’s av. density
Earth’s age
Earth’s normal gravity
Earth’s standard atmospher. pressure

Va l u e

31 556 925.974 7 s
31 556 925.2 s
23ℎ 56? 4.090 53??
149 597 870.691(30) km
AU
149 597 870 691 m
al
9.460 730 472 5808 Pm
pc
30.856 775 806 Pm = 3.261 634 al
?♁
5.973(1) ⋅ 1024 kg
??
3.986 004 418(8) ⋅ 1014 m3 /s2
?♁ = 2??/?2 8.870 056 078(16) mm
?♁eq
6378.1366(1) km
?♁p
6356.752(1) km
?♁
?♁
?♁
?
?0

10 001.966 km (average)
1/298.25642(1)
5.5 Mg/m3
4.50(4) Ga = 142(2) Ps
9.806 65 m/s2
101 325 Pa

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Q ua nt it y

Motion Mountain – The Adventure of Physics

Q ua nt i t y

a units, measurements and constants

219

TA B L E 13 (Continued) Astronomical constants.

Q ua nt it y

Symbol

Va l u e

Moon’s radius
Moon’s radius
Moon’s mass
Moon’s mean distance ?
Moon’s distance at perigee ?

?v
?h
?
?

Sun’s average density
Sun’s average distance
Sun’s age
Solar velocity
around centre of galaxy
Solar velocity
against cosmic background
Sun’s surface gravity
Sun’s lower photospheric pressure
Distance to Milky Way’s centre
Milky Way’s age
Milky Way’s size
Milky Way’s mass

?⊙
AU
?⊙
?⊙g
?⊙b

370.6(5) km/s

?⊙
?⊙

274 m/s2
15 kPa
8.0(5) kpc = 26.1(1.6) kal
13.6 Ga
c. 1021 m or 100 kal
1012 solar masses, c. 2 ⋅ 1042 kg

Moon’s angular size ?

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?
?
?
?
?
?
?
?
?
?⊙
2??⊙ /?2
??⊙
?⊙
?⊙

Moon’s distance at apogee ?

Motion Mountain – The Adventure of Physics

Moon’s average density
Moon’s surface gravity
Moon’s atmospheric pressure
Jupiter’s mass
Jupiter’s radius, equatorial
Jupiter’s radius, polar
Jupiter’s average distance from Sun
Jupiter’s surface gravity
Jupiter’s atmospheric pressure
Sun’s mass
Sun’s gravitational length
Heliocentric gravitational constant
Sun’s luminosity
Solar equatorial radius
Sun’s angular size

1738 km in direction of Earth
1737.4 km in other two directions
7.35 ⋅ 1022 kg
384 401 km
typically 363 Mm, historical minimum
359 861 km
typically 404 Mm, historical maximum
406 720 km
average 0.5181° = 31.08 ? , minimum
0.49°, maximum 0.55°
3.3 Mg/m3
1.62 m/s2
from 10−10 Pa (night) to 10−7 Pa (day)
1.90 ⋅ 1027 kg
71.398 Mm
67.1(1) Mm
778 412 020 km
24.9 m/s2
from 20 kPa to 200 kPa
1.988 43(3) ⋅ 1030 kg
2.953 250 08(5) km
132.712 440 018(8) ⋅ 1018 m3 /s2
384.6 YW
695.98(7) Mm
0.53∘ average; minimum on fourth of July
(aphelion) 1888 ?? , maximum on fourth of
January (perihelion) 1952 ??
1.4 Mg/m3
149 597 870.691(30) km
4.6 Ga
220(20) km/s

220

a units, measurements and constants

TA B L E 13 (Continued) Astronomical constants.

Challenge 173 s
Ref. 168

Symbol

Va l u e

Most distant galaxy cluster known

SXDF-XCLJ 9.6 ⋅ 109 al
0218-0510

Some properties of nature at large are listed in the following table. (If you want a challenge, can you determine whether any property of the universe itself is listed?)
TA B L E 14 Cosmological constants.

Q ua nt it y

Symbol

Va l u e

Cosmological constant
Λ
c. 1 ⋅ 10−52 m−2
Age of the universe ?
?0
4.333(53) ⋅ 1017 s = 13.8(0.1) ⋅ 109 a
(determined from space-time, via expansion, using general relativity)
Age of the universe ?
?0
over 3.5(4) ⋅ 1017 s = 11.5(1.5) ⋅ 109 a
(determined from matter, via galaxies and stars, using quantum theory)
?0
2.3(2) ⋅ 10−18 s−1 = 0.73(4) ⋅ 10−10 a−1
Hubble parameter ?
= ℎ0 ⋅ 100 km/s Mpc = ℎ0 ⋅ 1.0227 ⋅ 10−10 a−1
Reduced Hubble parameter ?
ℎ0
0.71(4)
̈ 0 /?02 −0.66(10)
?0 = −(?/?)
Deceleration parameter ?
Universe’s horizon distance ?
?0 = 3??0
40.0(6) ⋅ 1026 m = 13.0(2) Gpc
Universe’s topology
trivial up to 1026 m

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?. Defining constant, from vernal equinox to vernal equinox; it was once used to define the
second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less,
corresponding to a slowdown of roughly 0.2 ms/a. (Watch out: why?) There is even an empirical
formula for the change of the length of the year over time.
?. The truly amazing precision in the average distance Earth–Sun of only 30 m results from time
averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty
years. Note that the International Astronomical Union distinguishes the average distance Earth–
Sun from the astronomical unit itself; the latter is defined as a fixed and exact length. Also the
light year is a unit defined as an exact number by the IAU. For more details, see www.iau.org/
public/measuring.
?. The shape of the Earth is described most precisely with the World Geodetic System. The last
edition dates from 1984. For an extensive presentation of its background and its details, see the
www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii
and the flattening given here are those for the ‘mean tide system’. They differ from those of the
‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself.
?. Measured centre to centre. To find the precise position of the Moon at a given date, see
the www.fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page www.
fourmilab.ch/solar/solar.html and the other pages on the same site.
?. Angles are defined as follows: 1 degree = 1∘ = π/180 rad, 1 (first) minute = 1 ? = 1°/60, 1 second
(minute) = 1 ?? = 1 ? /60. The ancient units ‘third minute’ and ‘fourth minute’, each 1/60th of the
preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in
modern English.)

Motion Mountain – The Adventure of Physics

Challenge 174 s

Q ua nt it y

a units, measurements and constants

221

TA B L E 14 (Continued) Cosmological constants.

Q ua nt it y

Symbol

Number of space dimensions
Critical density
of the universe
(Total) density parameter ?
Baryon density parameter ?
Cold dark matter density parameter ?
Neutrino density parameter ?
Dark energy density parameter ?
Dark energy state parameter
Baryon mass
Baryon number density
Luminous matter density
Stars in the universe
Baryons in the universe
Microwave background temperature ?
Photons in the universe
Photon energy density
Photon number density
Density perturbation amplitude
Gravity wave amplitude
Mass fluctuations on 8 Mpc
Scalar index
Running of scalar index

3, for distances up to 1026 m
?c =
ℎ20 ⋅ 1.878 82(24) ⋅ 10−26 kg/m3
= 0.95(12) ⋅ 10−26 kg/m3
Ω0 = ?0 /?c
1.02(2)
ΩB0 = ?B0 /?c
0.044(4)
ΩCDM0 = ?CDM0 /?c 0.23(4)
Ω?0 = ??0 /?c
0.001 to 0.05
ΩX0 = ?X0 /?c
0.73(4)
? = ?X /?X
−1.0(2)
?b
1.67 ⋅ 10−27 kg
0.25(1) /m3
3.8(2) ⋅ 10−28 kg/m3
?s
1022±1
?b
1081±1
?0
2.725(1) K
??
1089
?? = π2 ?4 /15?04
4.6 ⋅ 10−31 kg/m3
410.89 /cm3 or 400 /cm3 (?0 /2.7 K)3
√?
5.6(1.5) ⋅ 10−6
√?
< 0.71√?
?8
0.84(4)
?
0.93(3)
d?/d ln ?
−0.03(2)

Planck length

?Pl = √ℏ?/?3

1.62 ⋅ 10−35 m

Planck time

?Pl = √ℏ?/?5

5.39 ⋅ 10−44 s

?Pl = √ℏ?/?

21.8 μg

?0 /?Pl
?0 = (?0 /?Pl )3 ⋅
(?0 /?Pl )
?

8.7(2.8) ⋅ 1060
10244±1

Instants in history
Space-time points
inside the horizon ?
Mass inside horizon

Vol. II, page 231

1054±1 kg

?. The index 0 indicates present-day values.
?. The radiation originated when the universe was 380 000 years old and had a temperature of
about 3000 K; the fluctuations Δ?0 which led to galaxy formation are today about 16 ± 4 μK =
6(2) ⋅ 10−6 ?0 .

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

?

3?02 /8π?

Motion Mountain – The Adventure of Physics

Planck mass

Va l u e

222

a units, measurements and constants

Useful numbers

Ref. 169

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959
0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923
0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255
2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772
3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

If the number π is normal, i.e., if all digits and digit combinations in its decimal expansion
appear with the same limiting frequency, then every text ever written or yet to be written,
as well as every word ever spoken or yet to be spoken, can be found coded in its sequence.
The property of normality has not yet been proven, although it is suspected to hold.
Does this mean that all wisdom is encoded in the simple circle? No. The property is
nothing special: it also applies to the number 0.123456789101112131415161718192021...
and many others. Can you specify a few examples?
By the way, in the graph of the exponential function e? , the point (0, 1) is the only
point with two rational coordinates. If you imagine painting in blue all points on the
plane with two rational coordinates, the plane would look quite bluish. Nevertheless, the
graph goes through only one of these points and manages to avoid all the others.

Motion Mountain – The Adventure of Physics

Challenge 175 s

π
e
γ
ln 2
ln 10
√10

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Appendix B

N UM BE R S A N D V E C TOR SPAC E S

Ref. 170

M

”

Numbers as mathematical structures
Challenge 176 ny

“

A person who can solve ?2 − 92?2 = 1 in less
than a year is a mathematician.
Brahmagupta (b. 598 Sindh, d. 668) (implied:
solve in integers)

”

Children know: numbers are entities that can be added and multiplied. Mathematicians
are more discerning. Any mathematical system with the same basic properties as the natural numbers is called a semi-ring. Any mathematical system with the same basic properties as the integers is called a ring. (The terms are due to David Hilbert. Both structures
can also be finite rather than infinite.) More precisely, a ring (?, +, ⋅) is a set ? of ele-

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

athematical concepts can all be expressed in terms of ‘sets’ and ‘relations.’
any fundamental concepts were presented in the last chapter. Why does
athematics, given this simple basis, grow into a passion for certain people? How
can sets and relations become the center of a person’s life? The mathematical appendices
present a few more advanced concepts as simply and vividly as possible, for all those who
want to understand and to smell the passion for mathematics.
Unfortunately, the passion for mathematics is not easy to spot, because like many
other professions, also mathematicians hide their passions. In mathematics, this is done
through formalism and apparent detachment from intuition. Good mathematical teaching however, puts intuition at the beginning. In this appendix we shall introduce the
simplest algebraic structures. The appendix in the next volume will present some more
involved algebraic structures and then the most important topological structures; the
third basic type of mathematical structures, order structures, are not so important in
physics – with one exception: the definition of the real numbers contains an order structure.
Mathematicians are concerned not only with the exploration of concepts, but also
with their classification. Whenever a new mathematical concept is introduced, mathematicians try to classify all the possible cases and types. This has been achieved most spectacularly for the different types of numbers, for finite simple groups and for many types
of spaces and manifolds.

Motion Mountain – The Adventure of Physics

Vol. III, page 284

“

A mathematician is a machine that transforms
coffee into theorems.
Paul Erdős (b. 1913 Budapest, d. 1996 Warsaw)

224

b numbers and vector spaces

ments with two binary operations, called addition and multiplication, usually written +
and ⋅ (the latter may simply be understood, thus without explicit notation), for which the
following properties hold for all elements ?, ?, ? ∈ ?:
— ? is a commutative group with respect to addition, i.e.
? + ? ∈ ?, ? + ? = ? + ?, ? + 0 = ?, ? + (−?) = ? − ? = 0 and ? + (? + ?) = (? + ?) + ?;
— ? is closed under multiplication, i.e., ?? ∈ ?;
— multiplication is associative, i.e., ?(??) = (??)?;
— distributivity holds, i.e., ?(? + ?) = ?? + ?? and (? + ?)? = ?? + ??.
Many authors add the axiom
— a multiplicative unit exists, i.e., 1? = ?1 = ?.

— a multiplicative identity 1, such that all elements ? obey 1? = ?;
— at least one element different from zero; and most importantly
— a (multiplicative) inverse ?−1 for every element ? ≠ 0.

* A set is mathematically complete if physicists call it continuous. More precisely, a set of numbers is complete
if every non-empty subset that is bounded above has a least upper bound.
A set is totally ordered if there exists a binary relation ⩽ between pairs of elements such that for all
elements ? and ?
— if ? ⩽ ? and ? ⩽ ?, then ? ⩽ ?;
— if ? ⩽ ? and ? ⩽ ?, then ? = ?;
— ? ⩽ ? or ? ⩽ ? holds.
In summary, a set is totally ordered if there is a binary relation that allows saying about any two elements
which one is the predecessor of the other in a consistent way. This is the fundamental – and also the only –
order structure used in physics.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 177 e

A ring or field is said to be commutative if the multiplication is commutative. A noncommutative field is also called a skew field. Fields can be finite or infinite. (A field or a
ring is characterized by its characteristic ?. This is the smallest number of times one has
to add 1 to itself to give zero. If there is no such number the characteristic is set to 0. ? is
always a prime number or zero.) All finite fields are commutative. In a field, all equations
of the type ?? = ? and ?? = ? (? ≠ 0) have solutions for ?; there is a unique solution
if ? ≠ 0. To sum up sloppily by focusing on the most important property, a field is a set
of elements for which, together with addition, subtraction and multiplication, a division
(by non-zero elements) is also defined. The rational numbers are the simplest field that
incorporates the integers.
The system of the real numbers is the minimal extension of the rationals which is
complete and totally ordered.* Can you show that √2 is a real, but not a rational number?

Motion Mountain – The Adventure of Physics

Defining properties such as these are called axioms. We stress that axioms are not basic
beliefs, as is often stated or implied; axioms are the basic properties used in the definition
of a concept: in this case, of a ring. With the last axiom, one also speaks of a unital ring.
A semi-ring is a set satisfying all the axioms of a ring, except that the existence of
neutral and negative elements for addition is replaced by the weaker requirement that if
? + ? = ? + ? then ? = ?. Sloppily, a semi-ring is a ring ‘without’ negative elements.
To incorporate division and define the rational numbers, we need another concept. A
number field or field K is a ring with

b numbers and vector spaces

225

imaginary axis

? = ? + ?? = ?e??

?
? = |?|
?
−?

real axis

?

? = |?|
?∗ = ? − ?? = ?e−??

Ref. 171

In classical physics and quantum theory, it is always stressed that measurement results
are and must be real numbers. But are all real numbers possible measurement results? In
other words, are all measurement results just a subset of the reals?
However, the concept of ‘number’ is not limited to these examples. It can be generalized in several ways. The simplest generalization is achieved by extending the real
numbers to manifolds of more than one dimension.

In nature, complex numbers are a useful way to describe in compact form systems and
situations that contain a phase. Complex numbers are thus useful to describe waves of
any kind.
Complex numbers form a two-dimensional manifold. A complex number is defined,
in its cartesian form, by ? = ? + ??, where ? and ? are real numbers, and ? is a new symbol, the so-called imaginary unit. Under multiplication, the generators of the complex
numbers, 1 and ?, obey
⋅
1 ?
1
1 ?
(120)
?
? −1
often summarized as ? = +√−1 . In a complex number ? = ? + ??, ? is called the real part,
and ? the complex part. They are illustrated in Figure 86.
The complex conjugate ?∗ , also written ?,̄ of a complex number ? = ? + ?? is defined
as ?∗ = ? − ??. The absolute value |?| of a complex number is defined as |?| = √??∗ =
√?∗ ? = √?2 + ?2 . It defines a norm on the vector space of the complex numbers. From
|??| = |?| |?| follows the two-squares theorem
(?12 + ?22 )(?12 + ?22 ) = (?1 ?1 − ?2 ?2 )2 + (?1 ?2 + ?2 ?1 )2

(121)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

C omplex numbers

Motion Mountain – The Adventure of Physics

Challenge 178 s

F I G U R E 86 Complex numbers are points in the
two-dimensional plane; a complex number ? and its
conjugate ?∗ can be described in cartesian form or
in polar form.

226

b numbers and vector spaces

??

?ℎ = − ???
?

?

0

?
F I G U R E 87 A property of triangles
easily provable with complex numbers.

Challenge 180 s
Page 235

Challenge 181 e

Challenge 182 e

Challenge 183 e

? ?
)
−? ?

with ?, ? ∈ ℝ .

(122)

Matrix addition and multiplication then correspond to complex addition and multiplication. In this way, complex numbers can be represented by a special type of real matrix.
What is |?| in matrix language?
The set ℂ of complex numbers with addition and multiplication as defined above
forms both a commutative two-dimensional field and a vector space over ℝ. In the field
of complex numbers, quadratic equations ??2 + ?? + ? = 0 for an unknown ? always have
two solutions (for ? ≠ 0 and counting multiplicity).
Complex numbers can be used to describe the points of a plane. A rotation around
the origin can be described by multiplication by a complex number of unit length. Other
two-dimensional quantities can also be described with complex numbers. Electrical engineers use complex numbers to describe quantities with phases, such as alternating currents or electrical fields in space.
Writing complex numbers of unit length as cos ? + ? sin ? is a useful method for remembering angle addition formulae. Since one has cos ?? + ? sin ?? = (cos ? + ? sin ?)? ,
one can easily deduce formulae such as cos 2? = cos2 ? − sin2 ? and sin 2? = 2 sin ? cos ?.
By the way, the unit complex numbers form the Lie group SO(2)=U(1).

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

(

Motion Mountain – The Adventure of Physics

Challenge 179 e

valid for all real numbers ?? , ?? . It was already known, in its version for integers, to Diophantus of Alexandria in the third century CE.
Complex numbers can also be written as ordered pairs (?, ?) of real numbers, with
their addition defined as (?, ?) + (?, ?) = (? + ?, ? + ?) and their multiplication defined
as (?, ?) ⋅ (?, ?) = (?? − ??, ?? + ??). This notation allows us to identify the complex
numbers with the points on a plane or, if we prefer, to arrows in a plane. Translating the
definition of multiplication into geometrical language allows us to rapidly prove certain
geometrical theorems, such as the one of Figure 87.
Complex numbers ? + ?? can also be represented as 2 × 2 matrices

b numbers and vector spaces

227

Every complex number can be written as
? = ?e?? .

(123)

This polar form of writing complex numbers is the reason for introducing them in the
first place. The angle ? is called the phase; the real number ? = |?| is called the absolute
value or the modulus or the magnitude. When used to describe oscillations or waves, it
makes sense to call ? the amplitude. The complex exponential function is periodic in 2π?;
in other words, we have
e1 = e1+2π? ,
(124)
which shows the property we expect from a phase angle.
If one uses the last equation twice, one may write
2

2

Challenge 184 e

Challenge 185 s

Oops, that would imply π = 0! What is wrong here?
Complex numbers can also be used to describe Euclidean plane geometry. Rotations,
translations and other isometries, but also reflections, glide reflections and scaling are
easily described by simple operations on the complex numbers that describe the coordinate of points.
By the way, there are exactly as many complex numbers as there are real numbers.
Can you show this?

Ref. 173

“

”

Love is complex: it has real and imaginary parts.
Anonymous

The positions of the points on a line can be described by real numbers. Complex numbers can be used to describe the positions of the points of a plane. It is natural to try
to generalize the idea of a number to higher-dimensional spaces. However, it turns out
that no useful number system can be defined for three-dimensional space. A new number system, the quaternions, can be constructed which corresponds the points of fourdimensional space, but only if the commutativity of multiplication is sacrificed. No useful
number system can be defined for dimensions other than 1, 2 and 4.
The quaternions were discovered by several mathematicians in the nineteenth century, among them Hamilton,* who studied them for much of his life. In fact, Maxwell’s theory of electrodynamics was formulated in terms of quaternions before threedimensional vectors were used.
Under multiplication, the quaternions ℍ form a 4-dimensional algebra over the reals

* William Rowan Hamilton (b. 1805 Dublin, d. 1865 Dunsink), child prodigy and famous mathematician,
named the quaternions after an expression from the Vulgate (Acts. 12: 4).

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Q uaternions

Vol. V, page 358

(125)

Motion Mountain – The Adventure of Physics

e1 = e1+2π? = (e1+2π? )1+2π? = e(1+2π?)(1+2π?) = e1−4π +4π? = e1−4π .

228

b numbers and vector spaces

with a basis 1, ?, ?, ? satisfying
⋅
1
?
?
?

(126)

These relations are also often written ?2 = ?2 = ?2 = −1, ?? = −?? = ?, ?? = −?? = ?,
?? = −?? = ?. The quaternions 1, ?, ?, ? are also called basic units or generators. The lack of
symmetry across the diagonal of the table shows the non-commutativity of quaternionic
multiplication. With the quaternions, the idea of a non-commutative product appeared
for the first time in mathematics. However, the multiplication of quaternions is associative. As a consequence of non-commutativity, polynomial equations in quaternions
have many more solutions than in complex numbers: just search for all solutions of the
equation ?2 + 1 = 0 to convince yourself of it.
Every quaternion ? can be written in the form
? = ?0 + ?1 ? + ?2 ? + ?3 ? = ?0 + ? = (?0 , ?1 , ?2 , ?3 ) = (?0 , ?) ,

(127)

where ?0 is called the scalar part and ? the vector part. The multiplication is thus defined
as (?, ?)(?, ?) = (??−?⋅?, ??+??+?×?). The multiplication of two general quaternions
can be written as

The conjugate quaternion ? is defined as ? = ?0 − ?, so that ?? = ? ?. The norm |?|
of a quaternion ? is defined as |?|2 = ?? = ?? = ?20 + ?21 + ?22 + ?23 = ?20 + ?2 . The
norm is multiplicative, i.e., |??| = |?| |?|.
Unlike complex numbers, every quaternion is related to its complex conjugate by
? = − 12 (? + ??? + ??? + ???) .

(129)

No relation of this type exists for complex numbers. In the language of physics, a complex
number and its conjugate are independent variables; for quaternions, this is not the case.
As a result, functions of quaternions are less useful in physics than functions of complex
variables.
The relation |??| = |?| |?| implies the four-squares theorem
(?12 + ?22 + ?32 + ?42 )(?12 + ?22 + ?32 + ?42 )
= (?1 ?1 − ?2 ?2 − ?3 ?3 − ?4 ?4 )2 + (?1 ?2 + ?2 ?1 + ?3 ?4 − ?4 ?3 )2
+ (?1 ?3 + ?3 ?1 + ?4 ?2 − ?2 ?4 )2 + (?1 ?4 + ?4 ?1 + ?2 ?3 − ?3 ?2 )2

(130)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

(?1 , ?1 , ?1 , ?1 )(?2 , ?2 , ?2 , ?2 ) = (?1 ?2 − ?1 ?2 − ?1 ?2 − ?1 ?2 , ?1 ?2 + ?1 ?2 + ?1 ?2 − ?1 ?2 ,
?1 ?2 − ?1 ?2 + ?1 ?2 + ?1 ?2 , ?1 ?2 + ?1 ?2 − ?1 ?2 + ?1 ?2 ) . (128)

Motion Mountain – The Adventure of Physics

Challenge 186 s

1
?
? ?
1
?
? ?
? −1 ? −?
? −? −1
?
?
? −? −1

b numbers and vector spaces

229

?/2
?

π − ?/2

?
?/2
?
F I G U R E 88 Combinations of rotations.

(131)

Thus, if we use pure quaternions such as ? or ? to describe positions, we can use unit
quaternions to describe rotations and to calculate coordinate changes. The concatenation
of two rotations is then given by the product of the corresponding unit quaternions.
Indeed, a rotation by an angle ? about the axis ? followed by a rotation by an angle ? about
the axis ? gives a rotation by an angle ? about the axis ?, with the values determined by
(cos ?/2, sin ?/2?) = (cos ?/2, sin ?/2?)(cos ?/2, sin ?/2?) .

(132)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

? = ???∗ .

Motion Mountain – The Adventure of Physics

Challenge 187 ny

valid for all real numbers ?? and ?? , and thus also for any set of eight integers. It was
discovered in 1748 by Leonhard Euler (1707–1783) when trying to prove that each integer
is the sum of four squares. (The latter fact was proved only in 1770, by Joseph Lagrange.)
Hamilton thought that a quaternion with zero scalar part, which he simply called
a vector (a term which he invented), could be identified with an ordinary threedimensional translation vector; but this is wrong. Such a quaternion is now called a
pure, or homogeneous, or imaginary quaternion. The product of two pure quaternions
? = (0, ?) and ? = (0, ?) is given by ?? = (−? ⋅ ?, ? × ?), where ⋅ denotes the scalar
product and × denotes the vector product. Note that any quaternion can be written as
the ratio of two pure quaternions.
In reality, a pure quaternion (0, ?) does not behave like a translation vector under
coordinate transformations; in fact, a pure quaternion represents a rotation by the angle
π or 180° around the axis defined by the direction ? = (??, ?? , ?? ).
It turns out that in three-dimensional space, a general rotation about the origin can
be described by a unit quaternion ?, also called a normed quaternion, for which |?| = 1.
Such a quaternion can be written as (cos ?/2, ? sin ?/2), where ? = (?? , ?? , ?? ) is the
normed vector describing the direction of the rotation axis and ? is the rotation angle.
Such a unit quaternion ? = (cos ?/2, ? sin ?/2) rotates a pure quaternion ? = (0, ?) into
another pure quaternion ? = (0, ?) given by

230

b numbers and vector spaces

?
?
?

?

1
?

?
palm
of right
hand

back
of right
hand

Challenge 188 e

Challenge 189 s

Page 130

Vol. V, page 369

Page 130

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Ref. 174

One way to show the result graphically is given in Figure 88. By drawing a triangle on a
unit sphere, and taking care to remember the factor 1/2 in the angles, the combination
of two rotations can be simply determined.
The interpretation of quaternions as rotations is also illustrated, in a somewhat different way, in the motion of any hand. To see this, take a green marker and write the letters
1, ?, ? and ? on your hand as shown in Figure 89. Defining the three possible 180° rotation axes as shown in the figure and taking concatenation as multiplication, the motion
of the right hand follows the same ‘laws’ as those of pure unit quaternions. (One needs
to distinguish +? and −?, and the same for the other units, by the sense of the arm twist.
And the result of a multiplication is that letter that can be read by a person facing you.)
You can show that ?2 = ?2 = ?2 = −1, that ?4 = 1, and conform all other quaternion relations. The model also shows that the rotation angle of the arm is half the rotation angle
of the corresponding quaternion. In other words, quaternions can be used to describe
the belt trick, if the multiplication ?? of two quaternions is taken to mean that rotation
? is performed after rotation ?. Quaternions, like human hands, thus behave like a spin
1/2 particle. Quaternions and spinors are isomorphic.
The reason for the half-angle behaviour of rotations can be specified more precisely
using mathematical language. The rotations in three dimensions around a point form the
‘special orthogonal group’ in three dimensions, which is called SO(3). But the motions
of a hand attached to a shoulder via an arm form a different group, isomorphic to the Lie
group SU(2). The difference is due to the appearance of half angles in the parametrization
of rotations; indeed, the above parametrizations imply that a rotation by 2π corresponds
to a multiplication by −1. Only in the twentieth century was it realized that there exist fundamental physical observables that behaves like hands attached to arms: they are
called spinors. More on spinors can be found in the section on permutation symmetry,
where belts are used as an analogy as well as arms. In short, the group SU(2) formed by

Motion Mountain – The Adventure of Physics

F I G U R E 89 The top and
back of the right hand, and
the quaternions.

b numbers and vector spaces

Ref. 175

231

?
?
( ∗
)
−? ?∗

with ?, ? ∈ ℂ

thus

? = ? + ??, ? = ? + ?? ,

(133)

or, alternatively, as

with ?, ?, ?, ? ∈ ℝ ,

(134)

where the quaternion ? then is given as ? = ? + ?? = ? + ?? + ?? + ??. Matrix addition
and multiplication then corresponds to quaternionic addition and multiplication.
The generators of the quaternions can be realized as
1 : ?0

, ? : −??1

, ? : −??2

, ? : −??3

(135)

where the ?? are the Pauli spin matrices.*

* The Pauli spin matrices are the complex Hermitean matrices
1
?0 = 1 = (
0

0
)
1

, ?1 = (

0
1

1
)
0

0
, ?2 = (
?

−?
)
0

1
, ?3 = (
0

0
)
−1

(136)

all of whose eigenvalues are ±1; they satisfy the relations [?? , ?? ]+ = 2 ??? and [?? , ?? ] = 2? ???? ?? . The linear
combinations ?± = 12 (?1 ± ?2 ) are also frequently used. By the way, another possible representation of the
quaternions is ? : ??3 , ? : ??2 , ? : ??1 .

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

? ?
? ?
−? ? −?
?
(
)
−? ?
? −?
−? −?
? ?

Motion Mountain – The Adventure of Physics

the unit quaternions is the double cover of the rotation group SO(3).
The simple representation of rotations and positions with quaternions is used by computer programmes in robotics, in astronomy and in flight simulation. In the software
used to create three-dimensional images and animations, visualization software, quaternions are often used to calculate the path taken by repeatedly reflected light rays and
thus give surfaces a realistic appearance.
The algebra of the quaternions is the only associative, non-commutative, finite-dimensional normed algebra with an identity over the field of real numbers. Quaternions
form a non-commutative field, i.e., a skew field, in which the inverse of a quaternion ?
is ?/|?|. We can therefore define division of quaternions (while being careful to distinguish ??−1 and ?−1 ?). Therefore quaternions are said to form a division algebra. In fact,
the quaternions ℍ, the complex numbers ℂ and the reals ℝ are the only three finitedimensional associative division algebras. In other words, the skew-field of quaternions
is the only finite-dimensional real associative non-commutative algebra without divisors
of zero. The centre of the quaternions, i.e., the set of quaternions that commute with all
other quaternions, is just the set of real numbers.
Quaternions can be represented as matrices of the form

232

b numbers and vector spaces

Real 4 × 4 representations are not unique, as the alternative representation
?
? −? −?
−?
? −? ?
(
)
?
?
? ?
? −? −? ?
Challenge 190 ny

Vol. III, page 77
Ref. 173
Challenge 191 s

shows. No representation of quaternions by 3 × 3 matrices is possible.
These matrices contain real and complex elements, which pose no special problems.
In contrast, when matrices with quaternionic elements are constructed, care has to be
taken, because quaternionic multiplication is not commutative, so that simple relations
such as tr?? = tr?? are not generally valid.
What can we learn from quaternions about the description of nature? First of all, we
see that binary rotations are similar to positions, and thus to translations: all are represented by 3-vectors. Are rotations the basic operations of nature? Is it possible that
translations are only ‘shadows’ of rotations? The connection between translations and
rotations is investigated in the last volume of our mountain ascent.
When Maxwell wrote down his equations of electrodynamics, he used quaternion
notation. (The now usual 3-vector notation was introduced later by Hertz and Heaviside.)
The equations can be written in various ways using quaternions. The simplest is achieved
when one keeps a distinction between √−1 and the units ?, ?, ? of the quaternions. One
then can write all of electrodynamics in a single equation:
?
?0

(138)

where ? is the generalized electromagnetic field and ? the generalized charge. These are
defined by
? = ? + √−1 ??
? = ??? + ??? + ???
? = ??? + ??? + ???

(139)

d = ? + √−1 ∂? /?
? = ?∂? + ?∂? + ?∂?
? = ? + √−1 ?/?
where the fields ? and ? and the charge distributions ? and ? have the usual meanings.
The content of equation (138) for the electromagnetic field is exactly the same as the usual
formulation.
Despite their charm and their four-dimensionality, quaternions do not seem to be
useful for the reformulation of special relativity; the main reason for this is the sign in
the expression for their norm. Therefore, relativity and space-time are usually described
using real numbers. And even if quaternions were useful, they would not provide additional insights into physics or into nature.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

d? = −

Motion Mountain – The Adventure of Physics

Vol. VI, page 172

(137)

b numbers and vector spaces

233

Octonions
In the same way that quaternions are constructed from complex numbers, octonions
can be constructed from quaternions. They were first investigated by Arthur Cayley
(1821–1895). Under multiplication, octonions (or octaves) are the elements of an eightdimensional algebra over the reals with the generators 1, ?? with ? = 1 . . . 7 satisfying
1 ?1
?2
?3
?4
?5
?6
?7
1 ?1
?2
?3
?4
?5
?6
?7
?1 −1 ?3 −?2
?5 −?4
?7 −?6
?2 −?3 −1 ?1 −?6
?7
?4 −?5
?3
?2 −?1 −1 ?7
?6 −?5 −?4
?4 −?5
?6 −?7 −1 ?1 −?2
?3
?5
?4 −?7 −?6 −?1 −1 ?3
?2
?6 −?7 −?4
?5
?2 −?3 −1 ?1
?7
?6
?5
?4 −?3 −?2 −?1 −1

(140)

(

? ?
) where ?, ? ∈ ℍ , or as real 8 × 8 matrices.
−?̄ ?̄

Matrix multiplication then gives the same result as octonionic multiplication.

(141)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

In fact, 479 other, equivalent multiplication tables are also possible. This algebra is
called the Cayley algebra; it has an identity and a unique division. The algebra is noncommutative, and also non-associative. It is, however, alternative, meaning that for all
elements ? and ?, one has ?(??) = ?2 ? and (??)? = ??2 : a property somewhat weaker
than associativity. It is the only 8-dimensional real alternative algebra without zero divisors. Because it is not associative, the set ? of all octonions does not form a field, nor
even a ring, so that the old designation of ‘Cayley numbers’ has been abandoned. The
octonions are the most general hypercomplex ‘numbers’ whose norm is multiplicative.
Its generators obey (???? )?? = ±??(?? ?? ), where the minus sign, which shows the nonassociativity, is valid for combinations of indices that are not quaternionic, such as 1-2-4.
Octonions can be represented as matrices of the form

Motion Mountain – The Adventure of Physics

⋅
1
?1
?2
?3
?4
?5
?6
?7

234

b numbers and vector spaces

The relation |??| = |?| |?| allows one to deduce the impressive eight-squares theorem
(?12 + ?22 + ?32 + ?42 + ?52 + ?62 + ?72 + ?82 )(?12 + ?22 + ?32 + ?42 + ?52 + ?62 + ?72 + ?82 )
= (?1 ?1 − ?2 ?2 − ?3 ?3 − ?4 ?4 − ?5 ?5 − ?6 ?6 − ?7 ?7 − ?8 ?8 )2
+ (?1 ?2 + ?2 ?1 + ?3 ?4 − ?4 ?3 + ?5 ?6 − ?6 ?5 + ?7 ?8 − ?8 ?7 )2
+ (?1 ?3 − ?2 ?4 + ?3 ?1 + ?4 ?2 − ?5 ?7 + ?6 ?8 + ?7 ?5 − ?8 ?6 )2
+ (?1 ?4 + ?2 ?3 − ?3 ?2 + ?4 ?1 + ?5 ?8 + ?6 ?7 − ?7 ?6 − ?8 ?5 )2
+ (?1 ?5 − ?2 ?6 + ?3 ?7 − ?4 ?8 + ?5 ?1 + ?6 ?2 − ?7 ?3 + ?8 ?4 )2
+ (?1 ?6 + ?2 ?5 − ?3 ?8 − ?4 ?7 − ?5 ?2 + ?6 ?1 + ?7 ?4 + ?8 ?3 )2
+ (?1 ?7 − ?2 ?8 − ?3 ?5 + ?4 ?6 + ?5 ?3 − ?6 ?4 + ?7 ?1 + ?8 ?2 )2
+ (?1 ?8 + ?2 ?7 + ?3 ?6 + ?4 ?5 − ?5 ?4 − ?6 ?3 − ?7 ?2 + ?8 ?1 )2

(142)

anticommutativity
exchange rule.

(143)

Using the definition
1
? × ? = (?? − ??) ,
2

Ref. 171
Challenge 192 e

Challenge 193 s

(144)

the cross products of imaginary quaternions, i.e., of quaternions of the type (0, ?), are
again imaginary, and correspond to the usual, three-dimensional vector product, thus
fulfilling (143). Interestingly, it is possible to use definition (144) for octonions as well.
In that case, the product of imaginary octonions is also imaginary, and (143) is again
satisfied. In fact, this is the only other non-trivial example of a vector product.
In summary: A vector product exists only in three and in seven dimensions. Many scholars have conjectured that this relation is connected with a possible ten-dimensionality of
nature; however, these speculations have not met with any success.
The symmetries of the forces in nature lead to a well-known question. The unit complex numbers from the Lie group U(1) and the unit quaternions the Lie group SU(2). Do
the unit octonions form the Lie group SU(3)?
Other t ypes of numbers
The process of constructing new systems of hypercomplex ‘numbers’ or real algebras by
‘doubling’ a given one can be continued ad infinitum. However, octonions, sedenions and

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

? × ? = −? × ?
(? × ?) ? = ? (? × ?)

Motion Mountain – The Adventure of Physics

valid for all real numbers ?? and ?? and thus in particular also for all integers. (There
are many variations of this expression, with different possible sign combinations.) The
theorem was discovered in 1818 by Carl Ferdinand Degen (1766–1825), and then rediscovered in 1844 by John Graves and in 1845 by Arthur Cayley. There is no generalization
to higher numbers of squares, a fact proved by Adolf Hurwitz (1859–1919) in 1898.
The octonions can be used to show that a vector product can be defined in more than
three dimensions. A vector product or cross product is an operation × satisfying

b numbers and vector spaces

Ref. 176
Challenge 194 s

Vol. III, page 292

Challenge 195 ny
Ref. 171

Ref. 172

From vector spaces to Hilbert spaces
Vector spaces, also called linear spaces, are mathematical generalizations of certain aspects of the intuitive three-dimensional space. A set of elements any two of which can
be added together and any one of which can be multiplied by a number is called a vector
space, if the result is again in the set and the usual rules of calculation hold.
More precisely, a vector space over a number field ? is a set of elements, called vectors,
for which a vector addition and a scalar multiplication is defined, such that for all vectors
?, ?, ? and for all numbers ? and ? from ? one has
(? + ?) + ? = ? + (? + ?) = ? + ? + ?
?+? = ?
(−?) + ? = ?
1? = ?
(? + ?)(? + ?) = ?? + ?? + ?? + ??

associativity of vector addition
existence of null vector
existence of negative vector
(145)
regularity of scalar multiplication
complete distributivity of scalar multiplication

If the field ?, whose elements are called scalars in this context, is taken to be the real (or

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Vol. III, page 294

all the following doublings are neither rings nor fields, but only non-associative algebras with unity. Other finite-dimensional algebras with unit element over the reals, once
called hypercomplex ‘numbers’, can also be defined: they include the so-called ‘dual
numbers’, ‘double numbers’, ‘Clifford–Lifshitz numbers’ etc. They play no role in physics.
Mathematicians have also defined number fields which have ‘one and a bit’ dimensions, such as algebraic number fields. There is also a generalization of the concept of
integers to the complex domain: the Gaussian integers, defined as ? + ??, where ? and ?
are ordinary integers. Gauss even defined what are now known as Gaussian primes. (Can
you find out how?) They are not used in the description of nature, but are important in
number theory, the exploration of the properties of integers.
Physicists used to call quantum-mechanical operators ‘q-numbers.’ But this term has
now fallen out of fashion.
Another way in which the natural numbers can be extended is to include numbers
larger than infinite. The most important such classes of transfinite number are the ordinals, the cardinals and the surreals. The ordinals are essentially an extension of the integers
beyond infinity, whereas the surreals are a continuous extension of the reals, also beyond infinity. Loosely speaking, among the transfinites, the ordinals have a similar role as
the integers have among the reals; the surreals fill in all the gaps between the ordinals,
like the reals do for integers. Interestingly, many series that diverge in ℝ converge in the
surreals. Can you find one example?
The surreals include infinitely small numbers, as do the numbers of nonstandard
analysis, also called hyperreals. In both number systems, in contrast to real numbers,
the numbers 1 and 0.999 999... (where an infinite, but hyperfinite string of nines is implied) do not coincide, but are separated by infinitely many other numbers. We explored
surreals earlier on. Nonstandard numbers can be used to define the infinitesimals used
in integration and differentiation, even at secondary school level.

Motion Mountain – The Adventure of Physics

Ref. 177

235

236

Challenge 196 s

b numbers and vector spaces

complex, or quaternionic) numbers, one speaks of a real (or complex, or quaternionic)
vector space. Vector spaces are also called linear vector spaces or simply linear spaces.
The complex numbers, the set of all real functions defined on the real line, the set of
all polynomials, the set of matrices with a given number of rows and columns, all form
vector spaces. In mathematics, a vector is thus a more general concept than in physics.
(What is the simplest possible mathematical vector space?)
In physics, the term ‘vector’ is reserved for elements of a more specialized type of
vector space, namely normed inner product spaces. To define these, we first need the
concept of a metric space.
A metric space is a set with a metric, i.e., a way to define distances between elements.
A real function ?(?, ?) between elements is called a metric if

Challenge 197 s

A non-trivial example is the following. We define a special distance ? between cities. If
the two cities lie on a line going through Paris, we use the usual distance. In all other
cases, we define the distance ? by the shortest distance from one to the other travelling
via Paris. This strange method defines a metric between all cities in France, the so-called
French railroad distance.
A normed vector space is a linear space with a norm, or ‘length’, associated to each a
vector. A norm is a non-negative number ‖?‖ defined for each vector ? with the properties

(147)

Usually there are many ways to define a norm for a given vector space. Note that a norm
can always be used to define a metric by setting
?(?, ?) = ‖? − ?‖

(148)

so that all normed spaces are also metric spaces. This is the natural distance definition
(in contrast to unnatural ones like that between French cities given above).
The norm is often defined with the help of an inner product. Indeed, the most special
class of linear spaces are the inner product spaces. These are vector spaces with an inner
product, also called scalar product ⋅ (not to be confused with the scalar multiplication!)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

‖??‖ = |?| ‖?‖ linearity of norm
‖? + ?‖ ⩽ ‖?‖ + ‖?‖ triangle inequality
‖?‖ = 0 only if ? = 0 regularity
Challenge 198 ny

(146)
Motion Mountain – The Adventure of Physics

?(?, ?) ⩾ 0 positivity of metric
?(?, ?) + ?(?, ?) ⩾ ?(?, ?) triangle inequality
?(?, ?) = 0 if and only if ? = ? regularity of metric

b numbers and vector spaces

237

which associates a number to each pair of vectors. An inner product space over ℝ satisfies
?⋅? =?⋅?
(??) ⋅ (??) = ??(? ⋅ ?)
(? + ?) ⋅ ? = ? ⋅ ? + ? ⋅ ?
? ⋅ (? + ?) = ? ⋅ ? + ? ⋅ ?
?⋅?⩾0
? ⋅ ? = 0 if and only if ? = 0

commutativity of scalar product
bilinearity of scalar product
left distributivity of scalar product
right distributivity of scalar product
positivity of scalar product
regularity of scalar product

(149)

for all vectors ?, ?, ? and all scalars ?, ?. A real inner product space of finite dimension
is also called a Euclidean vector space. The set of all velocities, the set of all positions, or
the set of all possible momenta form such spaces.
An inner product space over ℂ satisfies*

Challenge 199 s
Challenge 200 s

(150)

for all vectors ?, ?, ? and all scalars ?, ?. A complex inner product space (of finite dimension) is also called a unitary or Hermitean vector space. If the inner product space is
complete, it is called, especially in the infinite-dimensional complex case, a Hilbert space.
The space of all possible states of a quantum system forms a Hilbert space.
All inner product spaces are also metric spaces, and thus normed spaces, if the metric
is defined by
?(?, ?) = √(? − ?) ⋅ (? − ?) .
(151)
Only in the context of an inner product spaces we can speak about angles (or phase
differences) between vectors, as we are used to in physics. Of course, like in normed
spaces, inner product spaces also allows us to speak about the length of vectors and to
define a basis, the mathematical concept necessary to define a coordinate system. Which
vector spaces or inner product spaces are of importance in physics?
The dimension of a vector space is the number of linearly independent basis vectors.
Can you define these terms precisely?
A Hilbert space is a real or complex inner product space that is also a complete metric space. In other terms, in a Hilbert space, distances vary continuously and behave as
naively expected. Hilbert spaces usually, but not always, have an infinite number of dimensions.
* Two inequivalent forms of the sesquilinearity axiom exist. The other is (??) ⋅ (??) = ??(? ⋅ ?). The term
sesquilinear is derived from Latin and means for ‘one-and-a-half-linear’.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Page 224

Hermitean property
sesquilinearity of scalar product
left distributivity of scalar product
right distributivity of scalar product
positivity of scalar product
regularity of scalar product

Motion Mountain – The Adventure of Physics

?⋅?=?⋅? =?⋅?
(??) ⋅ (??) = ??(? ⋅ ?)
(? + ?) ⋅ ? = ? ⋅ ? + ? ⋅ ?
? ⋅ (? + ?) = ? ⋅ ? + ? ⋅ ?
?⋅?⩾ 0
? ⋅ ? = 0 if and only if ? = 0

238

Challenge 201 s

b numbers and vector spaces

The definition of Hilbert spaces and vector spaces assume continuous sets to start
with. If nature would not be continuous, could one still use the concepts?
Mathematical curiosities and fun challenges
Mathematics provides many counter-intuitive results. Reading a good book on the topic,
such as Bernard R . Gelbaum & John M. H. Olmsted, Theorems and Counterexamples in Mathematics, Springer, 1993, can help you sharpen your mind and make you
savour the beauty of mathematics even more.
∗∗

∗∗
Challenge 203 ny

Show that two operators ? and ? obey

(152)

for most operators ? and ?. This result is often called the Baker–Campbell–Hausdorff
formula or the BCH formula.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

1
e? e? = exp(? + ? + [?, ?]
2
1
1
+ [[?, ?], ?] − [[?, ?], ?]
12
12
1
1
− [?, [?, [?, ?]]] − [?, [?, [?, ?]]]
48
48
+ ...)

Motion Mountain – The Adventure of Physics

Challenge 202 e

It is possible to draw a curve that meets all points in a square or all points in a cube. This
is shown, for example, in the text Hans Sagan, Space Filling Curves, Springer Verlag,
1994. As a result, the distinction between one, two and three dimensions is blurred in
pure mathematics. In physics however, dimensions are clearly and well-defined; every
object in nature has three dimensions.

C HA L L E NG E H I N T S A N D S OLU T ION S

“

Never make a calculation before you know the
answer.
John Wheeler’s motto

”

Challenge 1, page 10: Do not hesitate to be demanding and strict. The next edition of the text
Challenge 2, page 16: Classical physics fails in explaining any material property, such as colour

Challenge 7, page 21: The electron charge is special to the electromagnetic interactions; it does

not take into account the nuclear interactions or gravity. It is unclear why the length defined with
the elementary charge ? should be of importance for neutral systems or for the vacuum. On the
other hand, the quantum of action ℏ is valid for all interactions and all observations.
In addition, we can argue that the two options to define a fundamental length – with the
quantum of action and with the quantum of charge – are not too different: the electron charge
is related to the quantum of action by ? = √4π?0 ??ℏ . The two length scales defined by the two
options differ only by a factor near 11.7. In fact, both scales are quantum scales.
Challenge 8, page 21: On purely dimensional grounds, the radius of an atom must be
?≈
Page 186

Vol. I, page 337

ℏ2 4π?0
,
?e ?2

(153)

which is about 53 nm. Indeed, this guess is excellent: it is just the Bohr radius.
Challenge 9, page 21: Due to the quantum of action, atoms in all people, be they giants or
dwarfs, have the same size. This implies that giants cannot exist, as was shown already by Galileo.
The argument is based on the given strength of materials; and a same strength everywhere is
equivalent to the same properties of atoms everywhere. That dwarfs cannot exist is due to a similar reason; nature is not able to make people smaller than usual (even in the womb they differ
markedly from adults) as this would require smaller atoms.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

or softness. Material properties result from nature’s interactions; they are inevitably quantum.
Explanations of material properties require, without exception, the use of particles and their
quantum properties.
Challenge 3, page 17: Classical physics allows any observable to change smoothly with time. In
classical physics, there is no minimum value for any observable physical quantity.
Challenge 4, page 19: The higher the mass, the smaller the motion fuzziness induced by the
quantum of action, because action is mass times speed times distance: For a large mass, the speed
and distance variations are small.
Challenge 5, page 20: The simplest time is √?ℏ/?5 . The numerical factor is obviously not fixed;
it is changed later on. Using 4? instead of ? the time becomes the shortest time measurable in
nature.

Motion Mountain – The Adventure of Physics

will benefit from it.

240

challenge hints and solutions

Challenge 12, page 27: A disappearance of a mass ? in a time Δ? is an action change ?2 ?Δ?.

That is much larger than ℏ for all objects of everyday life.
Challenge 14, page 29: Tunnelling of a lion would imply action values ? of the order of ? =
100 kgm2 /s ≫ ℏ. This cannot happen spontaneously.
Challenge 15, page 29: Every memory, be it human memory or an electronic computer memory,

must avoid decay. And decay can only be avoided through high walls and low tunnelling rates.
Challenge 16, page 30: Yes! Many beliefs and myths – from lottery to ghosts – are due to the

neglect of quantum effects.
Challenge 17, page 30: Perfectly continuous flow is in contrast to the fuzziness of motion induced by the quantum of action.
Challenge 18, page 30: The impossibility of following two particles along their path appears
when their mutual distance ? is smaller than their position indeterminacy due to their relative momentum ?, thus when ? < ℏ/?. Check the numbers with electrons, atoms, molecules,
bacteria, people and galaxies.
Challenge 21, page 36: In the material that forms the escapement mechanism.
Challenge 22, page 36: Growth is not proportional to light intensity or to light frequency, but

shows both intensity and frequency thresholds. These are quantum effects.
Challenge 23, page 36: All effects mentioned above, such as tunnelling, interference, decay,
transformation, non-emptiness of the vacuum, indeterminacy and randomness, are also observed in the nuclear domain.
Challenge 24, page 36: This is not evident from what was said so far, but it turns out to be cor-

Challenge 26, page 37: The total angular momentum counts, including the orbital angular mo-

mentum. The orbital angular momentum ? is given, using the radius and the linear momentum,
? = ? × ?. The total angular momentum is a multiple of ℏ.
Challenge 27, page 37: Yes, we could have!
Challenge 28, page 37: That is just the indeterminacy relation. Bohr expanded this idea to all

sort of other pairs of concepts, more in the philosophical domain, such as clarity and precision
of explanations: both cannot be high at the same time.
Challenge 29, page 39: The big bang cannot have been an event, for example.
Challenge 32, page 45: Charged photons would be deflected by electric of magnetic fields; in

particular, they would not cross undisturbed. This is not observed. Massive photons would be
deflected by masses, such as the Sun, much more than is observed.
Challenge 34, page 45: To measure momentum, we need a spatially extended measurement
device; to measure position, we need a localized measurement device.
Challenge 35, page 47: Photons are elementary because they realize the minimum action, because they cannot decay, because they cannot be deformed or split, because they have no mass,
no electric charge and no other quantum number, and because they appear in the Lagrangian of
quantum electrodynamics.
Challenge 36, page 50: The measured electric fields and photon distribution are shown in the

famous graphs reproduced in Figure 90.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

rect. In fact, there is no other option, as you will see when you try to find one.
Challenge 25, page 37: Tom Thumb is supposedly as smart as a normal human. But a brain cannot be scaled down. Fractals contradict the existence of Planck’s length, and Moore’s law contradicts the existence of atoms.

Motion Mountain – The Adventure of Physics

Challenge 19, page 30: Also photons are indistinguishable. See page 63.

challenge hints and solutions

241

Motion Mountain – The Adventure of Physics

Challenge 38, page 50: This is an unclearly posed problem. The radiation is thermal, but the

photon number depends on the volume under discussion.
Challenge 40, page 56: Radio photons can be counted using optical pumping experiments in

which atomic states are split by a small, ‘radio-wavelength’ amount, with the help of magnetic
fields. Also caesium clocks detect radio photons with optical means. The Josephons effect and
magnetic resonance imaging are additional detection methods for radio photons.
Challenge 41, page 57: To be observable to the eye, the interference fringes need to be visible for

around 0.1 s. That implies a maximum frequency difference between the two beams of around
10 Hz. This is achievable only if either a single beam is split into two or if the two beams come
from high-precision, stabilized lasers.
Challenge 42, page 62: Implicit in the arrow model is the idea that one quantum particle is de-

scribed by one arrow.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F I G U R E 90 Left, from top to bottom: the electric field and its fuzziness measured for a coherent state,
for a squeezed vacuum state, for a phase-squeezed state, for a mixed, quadrature-squeezed state and
for an amplitude-squeezed state, all with a small number of photons. Right: the corresponding photon
number distributions for the uppermost four states. (© G. Breitenbach/Macmillan, from Ref. 19)

242

challenge hints and solutions

Challenge 48, page 64: Despite a huge number of attempts and the promise of eternal fame, this

Ref. 178

is the sober conclusion.
Challenge 53, page 68: Yes, the argument is correct. In fact, more detailed discussions show that
classical electrodynamics is in contradiction with all colours observed in nature.
Challenge 57, page 73: The calculation is not easy, but not too difficult either. For an initial orientation close to the vertical, the fall time ? turns out to be
?=

(154)

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

where ? is the starting angle, and a fall through angle π is assumed. Here ?0 is the oscillation
time of the pencil for small angles. (Can you determine it?) The indeterminacy relation for the
tip of the pencil yields a minimum starting angle, because the momentum indeterminacy cannot
be made arbitrarily large. You should be able to provide an upper limit. Once this angle is known,
you can calculate the maximum time.
Challenge 58, page 74: Use the temperature to calculate the average kinetic energy, and thus the
average speed of atoms.
Challenge 59, page 74: At such low temperatures, the atoms cannot be fully distinguished; they
form a state of matter with peculiar properties, called a condensate. The condensate is not at rest
either; but due to its large mass, its fluctuations are greatly reduced, compared to those of a single
atom.
Challenge 61, page 78: Only variables whose product has the same units as physical action – Js
– can be complementary to each other.
Challenge 62, page 79: Use Δ? < ? and ? Δ? < ?.
Challenge 67, page 86: The quantum of action does not apply only to measurements, it applies
to motion itself, and in particular, to all motion. Also effects of the nuclear forces, of nuclear
particles and of nuclear radiation particles must comply to the limit. And experiments show that
they indeed do. In fact, if they did not, the quantum of action in electrodynamic situations could
be circumvented, as you can check.
Challenge 73, page 96: Outside the garage, all atoms need to form the same solid structure again.
Challenge 74, page 97: Terabyte chips would need to have small memory cells. Small cells imply
thin barriers. Thin barriers imply high probabilities for tunnelling. Tunnelling implies lack of
memory.
Challenge 80, page 108: If a particle were not elementary, its components would be bound by
an interaction. But there are no known interactions outside those of the standard model.
Challenge 81, page 109: The difficulties to see hydrogen atoms are due to their small size and
their small number of electrons. As a result, hydrogen atoms produce only weak contrasts in Xray images. For the same reasons it is difficult to image them using electrons; the Bohr radius of
hydrogen is only slightly larger than the electron Compton wavelength.
For the first time, in 2008, a research team claimed to have imaged hydrogen atoms adsorbed
on graphene with the help of a transmission electron microscope. For details, see J. C. Meyer,
C. O. Grit, M. F. Crommle & A. Zetti, Imaging and dynamics of light atoms and molecules
on graphene, Nature 454, pp. 319–322, 2008. However, it seems that the report has not been confirmed by another group yet.
More hydrogen images have appeared in recent years. You may search for olympicene on the
internet, for example. For another recent result about hydrogen imaging, see above.
Challenge 83, page 109: This is not easy! Can you use the concept of action to show that there
indeed is a fundamental difference between very similar and very different operators?

Motion Mountain – The Adventure of Physics

Page 185

1
8
? ln
2π 0 ?

challenge hints and solutions

243

Challenge 85, page 110: ? = 86 pm, thus ? = 12 eV. This compares to the actual value of 13.6 eV.

The trick for the derivation of the formula is to use ⟨? | ??2 | ?⟩ = 13 ⟨? | ?? | ?⟩, a relation valid for
states with no orbital angular momentum. It is valid for all coordinates and also for the three
momentum observables, as long as the system is non-relativistic.

Challenge 86, page 111: A quantum fluctuation would require the universe to exist already. Such

statements, regularly found in the press, are utter nonsense.
Challenge 87, page 112: Point particles cannot be marked; nearby point particles cannot be dis-

tinguished, due to the quantum of action.
Challenge 88, page 112: The solution is two gloves. In the original setting, if two men and two

women want to make love without danger, in theory they need only two condoms.
Challenge 93, page 114: The Sackur–Tetrode formula is best deduced in the following way. We

We use Stirling’s approximation ?! ≈ ?? /?? , and the definition of the entropy as ? =
∂(?? ln ?)/∂?. Inserting the definition of Λ, this gives the Sackur–Tetrode equation.
Challenge 95, page 117: To write anything about two particles on paper, we need to distinguish

them, even if the distinction is arbitrary.
Challenge 98, page 123: The idea, also called quantum money, is not compatible with the size

and lifetime requirements of actual banknotes.
Challenge 99, page 124: Twins differ in the way their intestines are folded, in the lines of their

Challenge 108, page 135: Three.
Challenge 109, page 135: Not for a mattress. This is not easy to picture.
Challenge 110, page 136: Angels can be distinguished by name, can talk and can sing; thus they

are made of a large number of fermions. In fact, many angels are human sized, so that they do
not even fit on the tip of a pin.
Challenge 116, page 140: A boson can be represented by an object glued to one infinitesimally

thin thread whose two tails reach spatial infinity.
Challenge 117, page 141: Trees, like all macroscopic objects, have a spin value that depends on

their angular momentum. Being classical objects whose phase can be observed, the spin value
is uncertain. It makes no sense to ask whether trees or other macroscopic objects are bosons or
fermions, as they are not quantons.
Challenge 120, page 142: Ghosts, like angels, can be distinguished by name, can talk and can

be seen; thus they contain fermions. However, they can pass through walls and they are transparent; thus they cannot be made of fermions, but must be images, made of bosons. That is a
contradiction.
Challenge 121, page 144: Macroscopic superpositions cannot be drawn, because observation

implies interaction with a bath, which destroys macroscopic superposition.
Challenge 123, page 146: The loss of non-diagonal elements leads to an increase in the diagonal

elements, and thus of entropy.
Challenge 126, page 153: The energy speed is given by the advancement of the outer two tails;

that speed is never larger than the speed of light.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

hands and other skin folds. Sometimes, but not always, features like black points on the skin are
mirror inverted on the two twins.

Motion Mountain – The Adventure of Physics

start with an ideal monoatomic gas of volume ?, with ? particles, and total energy ?. In phase
space, state sum ? is given by
?? 1
?=
.
(155)
?! Λ3?

244

challenge hints and solutions

Challenge 127, page 155: No, as taking a photo implies an interaction with a bath, which would

destroy the superposition. In more detail, a photograph requires illumination; illumination is
a macroscopic electromagnetic field; a macroscopic field is a bath; a bath implies decoherence;
decoherence destroys superpositions.
Challenge 130, page 157: It depends. They can be due to interference or to intensity sums. In

the case of radio the effect is clearer. If at a particular frequency the signals changes periodically
from one station to another, one has a genuine interference effect.
Challenge 131, page 157: They interfere. But this is a trick question; what is a monochromatic

electron? Does it occur in the laboratory?
Challenge 132, page 157: Such a computer requires clear phase relations between components;

such phase relations are extremely sensitive to outside disturbances. At present, they do not hold
longer than a microsecond, whereas long computer programs require minutes and hours to run.
Challenge 133, page 157: A record is an effect of a process that must be hard to reverse or undo.

Challenge 137, page 166: Any other bath also does the trick, such as the atmosphere, sound vi-

brations, electromagnetic fields, etc.
Challenge 138, page 166: The Moon is in contact with baths like the solar wind, falling meteor-

ites, the electromagnetic background radiation of the deep universe, the neutrino flux from the
Sun, cosmic radiation, etc.
Challenge 139, page 168: Spatially periodic potentials have the property. Decoherence then

leads to momentum diagonalization.
Challenge 142, page 182: The red shift value is ? = 9.9995. From the formula for the longitudinal

Doppler shift we get ?/? = ((? + 1)2 − 1)/((? + 1)2 + 1); this yields 0.984 in the present case. The
galaxy thus moves away from Earth with 98.4 % of the speed of light.
Challenge 148, page 184: Hydrogen atoms are in eigenstates for the reasons explained in the

chapter on superpositions and probabilities: in a gas, atoms are part of a bath, and thus almost
always in energy eigenstates.
Challenge 153, page 195: If several light beams are focused in the space between the mirrors,

and if the light beam frequency is properly tuned with respect to the absorption frequencies
of the atoms, atoms will experience a restoring force whenever they move away from the focus
region. By shining light beams to the focus region from 6 directions, atoms are trapped. The
technique of laser cooling is now widely used in research laboratories.
Challenge 154, page 196: No, despite its name, phosphorus is not phosphorescent, but chemo-

luminescent.
Challenge 156, page 197: This is a trick question. A change in ? requires a change in ?, ℏ, ? or ?0 .

None of these changes is possible or observable, as all our measurement apparatus are based on
these units. Speculations about change of ?, despite their frequency in the press and in scientific
journals, are idle talk.
Challenge 157, page 197: A change of physical units such that ℏ = ? = ? = 1 would change the

value of ?0 in such a way that 4π?o = 1/? ≈ 137.036.
Challenge 160, page 207: Mass is a measure of the amount of energy. The ‘square of mass’ makes

no sense.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Challenge 141, page 171: If so, let the author know.

Motion Mountain – The Adventure of Physics

The traces of a broken egg are easy to clean on a large glass plate, but hard in the wool of a sheep.
Broken teeth, torn clothes, or scratches on large surfaces are good records. Forensic scientists
know many additional examples.

challenge hints and solutions

245

Challenge 164, page 210: Planck limits can be exceeded for extensive observables for which

Vol. VI, page 110

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

many particle systems can exceed single particle limits, such as mass, momentum, energy or
electrical resistance.
Challenge 166, page 212: Do not forget the relativistic time dilation.
Challenge 167, page 212: The formula with ? − 1 is a better fit. Why?
Challenge 170, page 213: No! They are much too precise to make sense. They are only given as
an illustration for the behaviour of the Gaussian distribution. Real measurement distributions
are not Gaussian to the precision implied in these numbers.
Challenge 171, page 214: About 0.3 m/s. It is not 0.33 m/s, it is not 0.333 m/s and it is not any
longer strings of threes.
Challenge 173, page 220: The slowdown goes quadratically with time, because every new slowdown adds to the old one!
Challenge 174, page 220: No, only properties of parts of the universe are listed. The universe
itself has no properties, as shown in the last volume.
Challenge 175, page 222: The double of that number, the number made of the sequence of all
even numbers, etc.
Challenge 178, page 225: We will find out in the last volume that all measurement values have
upper and lower bounds. We will also find out that two physical measurement results cannot
differ just from, say, the 300th decimal place onwards. So indeed, all measurement results are
real numbers, but not vice versa. It needs to be stressed that for quantum theory, for relativity
and also for Galilean physics this restriction has no consequences whatsoever.
? ?
Challenge 180, page 226: |?|2 is the determinant of the matrix ? = (
).
−? ?
Challenge 185, page 227: Use Cantor’s diagonal argument, as in challenge 271.
Challenge 186, page 228: Any quaternion ? = ??+??+?? with ?2 +?2 +?2 = 1 solves the equation
?2 + 1 = 0; the purely imaginary solutions +? and −? are thus augmented by a continuous sphere
of solutions in quaternion space.
Challenge 189, page 230: Any rotation by an angle 2π is described by −1. Only a rotation by 4π
is described by +1; quaternions indeed describe spinors.
Challenge 191, page 232: Just check the result component by component. See also the mentioned reference.
Challenge 193, page 234: No. Because the unit octonions are not associative, they do not form
a group at all. Despite its superficial appeal, this line of reasoning has not led to any insight into
the nature of the fundamental interactions.
Challenge 194, page 235: For a Gaussian integer ? + ?? to be prime, the integer ?2 + ?2 must
be prime, and in addition, a condition on ? mod 3 must be satisfied; which one and why?
Challenge 196, page 236: The set that contains only the zero vector.
Challenge 197, page 236: The metric is regular, positive definite and obeys the triangle inequality.
Challenge 199, page 237: Essentially only the vector spaces listed in the appendix (or in the
book).
Challenge 200, page 237: If you cannot, blame your math teacher at secondary school, and then
look up the definitions. It is not a difficult topic.
Challenge 201, page 238: Spaces could exist approximately, as averages of non-continuous
structures. This idea is explored in modern research; an example is given in the last volume of
this series.

challenge hints and solutions
246

Motion Mountain – The Adventure of Physics

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

BI BL IO G R A PH Y

2

3

”
”

Giuseppe Fumagalli, Chi l’ha detto?, Hoepli, Milano, 1983. Cited on page 15.
The quantum of action was introduced in Max Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Preußischen Akademie der Wissenschaften, Berlin pp. 440–
480, 1899. In the paper, Planck used the letter ? for what nowadays is called ℎ. Cited on page
17.
Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. See
N. B ohr, Atomtheorie und Naturbeschreibung, Springer, 1931. On page 16 he writes: ‘No
more is it likely that the fundamental concepts of the classical theories will ever become
superfluous for the description of physical experience. The recognition of the indivisibility
of the quantum of action, and the determination of its magnitude, not only depend on an
analysis of measurements based on classical concepts, but it continues to be the application of these concepts alone that makes it possible to relate the symbolism of the quantum
theory to the data of experience.’ He also writes: ‘...the fundamental postulate of the indivisibility of the quantum of action is itself, from the classical point of view, an irrational
element which inevitably requires us to forgo a causal mode of description and which, because of the coupling between phenomena and their observation, forces us to adopt a new
mode of description designated as complementary in the sense that any given application
of classical concepts precludes the simultaneous use of other classical concepts which in
a different connection are equally necessary for the elucidation of the phenomena ...’ and
‘...the finite magnitude of the quantum of action prevents altogether a sharp distinction
being made between a phenomenon and the agency by which it is observed, a distinction
which underlies the customary concept of observation and, therefore, forms the basis of
the classical ideas of motion.’ Other statements about the indivisibility of the quantum of
action can be found in N. B ohr, Atomic Physics and Human Knowledge, Science Editions,
1961. See also Max Jammer, The Philosophy of Quantum Mechanics, Wiley, first edition,
1974, pp. 90–91. Cited on page 17.
For some of the rare modern publications emphasizing the quantum of action see
M. B. Mensky, The action uncertainty principle and quantum gravity, Physics Letters
A 162, p. 219, 1992, and M. B. Mensky, The action uncertainty principle in continuous
quantum measurements, Physics Letters A 155, pp. 229–235, 1991. Schwinger’s quantumaction principle is also used in Richard F. W. Bader, Atoms in Molecules – A Quantum
Theory, Oxford University Press, 1994.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

4

As soon as you write, no time to read remains.
Anonymous

Motion Mountain – The Adventure of Physics

1

“
“

No man but a blockhead ever wrote except for
money.
Samuel Johnson

248

bibliography

The best source for the story about the walk in the forest with Planck’s son Erwin is
Hans Roos & Armin Hermann, editors, Max Planck – Vorträge, Reden, Erinnerungen,
Springer, 2001, page 125. As the text explains, the story was told by Erwin Planck to at least
two different people. Erwin Planck himself was part of the failed 1944 plot against Hitler
and was hanged in January 1945. Cited on page 20.

6

Max B orn, Zur Quantenmechanik der Stoßvorgänge (vorläufige Mitteilung), Zeitschrift
für Physik 37, pp. 863–867, 1926, Max B orn, Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 38, pp. 803–827, 1926. Cited on page 24.
See for example the papers by Jan Hilgevoord, The uncertainty principle for energy and
time, American Journal of Physics 64, pp. 1451–1456, 1996, and by Paul Busch, On the
time–energy uncertainty reaction, parts 1 & 2, Foundations of Physics 20, pp. 1–43, 1990.
A classic is the paper by Eugene P. Wigner, On the time–energy uncertainty relation, in
Abdus Salam & Eugene P. Wigner, editors, Aspects of Quantum Theory, Cambridge
University Press, 1972. Cited on page 25.
See also the booklet by Claus Mattheck, Warum alles kaputt geht – Form und Versagen
in Natur und Technik, Forschungszentrum Karlsruhe, 2003. Cited on page 29.
R. Clifton, J. Bub & H. Halvorson, Characterizing quantum theory in terms of
information-theoretic constraints, arxiv.org/abs/quant-ph/0211089. Cited on page 36.
This way to look at cans of beans goes back to the text by Susan Hewitt & Edward Subitzky, A call for more scientific truth in product warning labels, Journal of Irreproducible Results 36, nr. 1, 1991. Cited on page 37.
J. Malik, The yields of the Hiroshima and Nagasaki nuclear explosions, Technical Report
LA-8819, Los Alamos National Laboratory, September 1985. Cited on page 38.
The quotes on motion are found in chapter VI of F. Engels, Herrn Eugen Dührings Umwälzung der Wissenschaft, Verlag für fremdsprachliche Literatur, 1946. The book is commonly called Anti-Dühring. Cited on pages 39 and 74.
Rodney Loudon, The Quantum Theory of Light, Oxford University Press, 2000. Cited
on page 40.
E. M. Brumberg & S. I. Vavilov, Izvest. Akad. Nauk. Omen Ser. 7, p. 919, 1933. Cited
on page 40.
On photon detection in the human eye, see the influential review by F. Rieke &

7

8
9
10

11
12

13
14
15

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

5

Motion Mountain – The Adventure of Physics

There is a large number of general textbooks on quantum theory. There is one for every
taste.
A well-known conceptual introduction is Jean-Marc Lév y-Leblond &
Françoise Balibar, Quantique – Rudiments, Masson, 1997, translated into English
as Quantics, North-Holland, 1990.
One of the most beautiful books is Julian Schwinger, Quantum Mechanics – Symbolism of Atomic Measurements, edited by Berthold-Georg Englert, Springer Verlag, 2001.
A modern approach with a beautiful introduction is Max Schubert & Gerhard Weber, Quantentheorie – Grundlagen und Anwendungen, Spektrum Akademischer
Verlag, 1993.
A standard beginner’s text is C. Cohen-Tannoudji, B. Diu & F. Laloë, Mécanique quantique I et II, Hermann, Paris, 1977. It is also available in several translations.
A good text is Asher Peres, Quantum Theory – Concepts and Methods, Kluwer, 1995.
For a lively approach, see Vincent Icke, The Force of Symmetry, Cambridge University Press, 1994.
New textbooks are published regularly around the world. Cited on pages 17 and 255.

bibliography

249

D. A. Baylor, Single-photon detection by rod cells of the retina, Reviews of Modern
Physics 70, pp. 1027–1036, 1998. It can be found on the internet as a pdf file. Cited on page
42.

17

E. Fischbach, H. Kloor, R. A. Langel, A. T. Y. Lui & M. Peredo, New geomagnetic limit on the photon mass and on long-range forces coexisting with electromagnetism,
Physical Review Letters 73, pp. 514–517, 1994. Cited on page 45.

18

A. H. Compton, The scattering of X-rays as particles, American Journal of Physics 29,
pp. 817–820, 1961. This is a pedagogical presentation of the discoveries he made in 1923.
Cited on page 45.

19

The reference paper on this topic is G. Breitenbach, S. Schiller & J. Mlynek,
Measurement of the quantum states of squeezed light, 387, pp. 471–475, 1997. It is available
freely at gerdbreitenbach.de/publications/nature1997.pdf. Cited on pages 48 and 241.

20

The famous paper is R. Hanbury Brown & R. Q. Twiss, Nature 178, p. 1046, 1956.
They got the idea to measure light in this way from their earlier work, which used the same
method with radio waves: R. Hanbury Brown & R. Q. Twiss, Nature 177, p. 27, 1956.
The complete discussion is given in their papers R. Hanbury Brown & R. Q. Twiss,
Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between
photons in coherent beams of radiation, Proceedings of the Royal Society A 242, pp. 300–
324, 1957, and R. Hanbury Brown & R. Q. Twiss, Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light, Proceedings
of the Royal Society A 243, pp. 291–319, 1958. Both are dowloadable for free on the internet
and are well worth reading. Cited on page 52.

21

J. Glanz, First light from a space laser, Science 269, p. 1336, 1995. Cited on page 54.

22

A. Einstein, Über einen die Erzeugung und Umwandlung des Lichtes betreffenden heuristischen Standpunkt, Annalen der Physik 17, pp. 132–184, 1905. Cited on page 55.

23

See the summary by P. W. Milonni, Answer to question 45: What (if anything) does the
photoelectric effect teach us?, American Journal of Physics 65, pp. 11–12, 1997. Cited on page
55.

24

For a detailed account, See J. J. Prentis, Poincaré ’s proof of the quantum discontinuity of nature, American Journal of Physics 63, pp. 339–350, 1995. The original papers are
Henri Poincaré, Sur la théorie des quanta, Comptes Rendus de l’Académie des Sciences (Paris) 153, pp. 1103–1108, 1911, and Henri Poincaré, Sur la théorie des quanta,
Journal de Physique (Paris) 2, pp. 5–34, 1912. Cited on page 55.

25

J. Jacobson, G. Björk, I. Chang & Y. Yamamoto, Photonic de Broglie waves,
Physical Review Letters 74, pp. 4835–4838, 1995. The first measurement was published
by E. J. S. Fonseca, C. H. Monken & S. de Pádua, Measurement of the de Broglie
wavelength of a multiphoton wave packet, Physical Review Letters 82, pp. 2868–2671, 1995.
Cited on page 55.

26

For the three-photon state, see M. W. Mitchell, J. S. Lundeen & A. M. Steinberg,
Super-resolving phase measurements with a multiphoton entangled state, Nature 429, pp. 161–
164, 2004, and for the four-photon state see, in the same edition, P. Walther, J. -W. Pan,

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F. Rieke & D. A. Baylor, Single-photon detection by rod cells of the retina, Reviews of
Modern Physics 70, pp. 1027–1036, 1998. They also mention that the eye usually works at
photon fluxes between 108 /?m2 s (sunlight) and 10−2 /?m2 s (starlight). The cones, in the
retina detect, in colour, light intensities in the uppermost seven or eight decades, whereas
the rods detect, in black and white, the lower light intensities. Cited on page 44.

Motion Mountain – The Adventure of Physics

16

250

27
28
29

30

32

33

35
36
37

38

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

34

M. Aspelmeyer, R. Ursin, S. Gasparoni & A. Zeilinger, De Broglie wavelength
of a non-local four-photon state, Nature 429, pp. 158–161, 2004. Cited on page 55.
For an introduction to squeezed light, see L. Mandel, Non-classical states of the electromagnetic field, Physica Scripta T 12, pp. 34–42, 1986. Cited on page 55.
Friedrich Herneck, Einstein und sein Weltbild: Aufsätze und Vorträge, Buchverlag Der
Morgen, 1976, page 97. Cited on page 56.
The famous quote on single-photon interference is found on page 9 of famous, beautiful
but difficult textbook P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon
Press, 1930. It is also discussed, in a somewhat confused way, in the otherwise informative
article by H. Paul, Interference between independent photons, Reviews of Modern Physics
58, pp. 209–231, 1986. Cited on pages 59 and 66.
The original papers on coherent states are three: R. J. Glauber, The quantum theory of
optical coherence, Physical Review 130, pp. 2529–2539, 1963, J. R. Klauder, Continuousrepresentation theory, I and II, Journal of Mathematical Physics 4, pp. 1055–1058, 1963, and
E. C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions
of statistical light beams, Physical Review Letters 10, p. 227, 1963. Cited on page 63.
See, for example the wonderful text Richard P. Feynman, QED – The Strange Theory of
Light and Matter, pp. 73–75, Princeton University Press, 1988, or Richard P. Feynman
& Steven Weinberg, Elementary Particles and the Laws of Physics, p. 23, Cambridge
University Press 1987. Cited on page 63.
Wolf gang Tittel, J. Brendel, H. Zbinden & N. Gisin, Violation of Bell inequalities by photons more than 10 km apart, Physical Review Letters 81, pp. 3563–3566, 26 October 1998. Cited on page 64.
N. B ohr & L. Rosenfeld, Zur Frage der Meßbarkeit der elektromagnetischen Feldgrößen, Mat.-fys. Medd. Danske Vid. Selsk. 12, p. 8, 1933. The results were later published in
English as N. B ohr & L. Rosenfeld, Field and charge measurements in quantum electrodynamics, Physical Review 78, pp. 794–798, 1950. Cited on page 65.
Misleading statements are given in the introduction and in the conclusion of the review
by H. Paul, Interference between independent photons, Review of Modern Physics 58,
pp. 209–231, 1986. However, in the bulk of the article the author in practice retracts the
statement, e.g. on page 221. Cited on page 66.
G. Magyar & L. Mandel, Interference fringes produced by superposition of two independent maser light beams, Nature 198, pp. 255–256, 1963. Cited on page 67.
R. Kidd, J. Aedini & A. Anton, Evolution of the modern photon, American Journal of
Physics 57, pp. 27–35, 1989, Cited on page 69.
The whole bunch of atoms behaves as one single molecule; one speaks of a Bose–Einstein
condensate. The first observations, worthy of a Nobel prize, were by M.H. Anderson & al., Observation of Bose–Einstein condensation in a dilute atomic vapour, Science
269, pp. 198–201, 1995, C. C. Bradley, C. A. Sackett, J. J. Tollett & R. G. Hulet,
Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions, Physical
Review Letters 75, pp. 1687–1690, 1995, K. B. Davis, M. -O. Mewes, M. R. Andrews,
N. J. van Druten, D. S. Durfee, D. M. Kurn & W. Ketterle, Bose–Einstein condensation in a gas of sodium atoms, Physical Review Letters 75, pp. 3969–3973, 1995. For a
simple introduction, see W. Ketterle, Experimental studies of Bose–Einstein condensation, Physics Today pp. 30–35, December 1999. Cited on page 74.
J. L. Costa-Krämer, N. Garcia, P. García-Mochales & P. A. Serena,
Nanowire formation in macroscopic metallic contacts: a universal property of metals, Surface

Motion Mountain – The Adventure of Physics

31

bibliography

bibliography

39

40
41

44

45

46

47
48
49

50

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

43

Science Letters 342, pp. L1144–L1152, 1995. See also J. L. Costa-Krämer, N. Garcia,
P. A. Serena, P. García-Mochales, M. Marqués & A. Correia, Conductance
quantization in nanowires formed in macroscopic contacts, Physical Review B p. 4416, 1997.
Cited on page 74.
The beautiful undergraduate experiments made possible by this discovery are desribed
in E. L. Foley, D. Candela, K. M. Martini & M. T. Tuominen, An undergraduate laboratory experiment on quantized conductance in nanocontacts, American Journal of
Physics 67, pp. 389–393, 1999. Cited on pages 74 and 75.
L. de Broglie, Ondes et quanta, Comptes rendus de l’Académie des Sciences 177,
pp. 507–510, 1923. Cited on page 76.
C. Jönsson, Interferenz von Elektronen am Doppelspalt, Zeitschrift für Physik 161,
pp. 454–474, 1961, C. Jönsson, Electron diffraction at multiple slits, American Journal of
Physics 42, pp. 4–11, 1974. Because of the charge of electons, this experiment is not easy to
perform: any parts of the set-up that are insulating get charged and distort the picture. That
is why the experient was performed much later with electrons than with atoms, neutrons
and molecules. Cited on page 77.
M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw &
A. Zeilinger, Wave–particle duality of C60 molecules, Nature 401, pp. 680–682, 14
October 1999. See also the observation for tetraphenyleprophyrin and C60 F48 by the
same team, published as L. Hackermüller & al., Wave nature of biomolecules and
fluorofullerenes, Physical Review Letters 91, p. 090408, 2003.
No phenomoenon of quantum theory has been experimentally studied as much as
quantum interference. The transition from interference to non-interference has also been
explored, as in P. Facchi, A. Mariano & S. Pascazio, Mesoscopic interference, Recent Developments in Physics 3, pp. 1–29, 2002. Cited on page 77.
G. Papini, Shadows of a maximal acceleration, arxiv.org/abs/gr-qc/0211011. Cited on page
79.
J. Perrin, Nobel Prize speech, found at www.nobel.se, and H. Nagaoka, Kinetics of a
system of particles illustrating the line and the band spectrum and the phenomena of radioactivity, Philosophical Magazine S6, 7, pp. 445–455, March 1904. Cited on page 79.
N. B ohr, On the constitution of atoms and molecules: Introduction and Part I – binding of
electrons by positive nuclei, Philosophical Magazine 26, pp. 1–25, 1913, On the constitution of
atoms and molecules: Part II – systems containing only a single nucleus, ibid., pp. 476–502,
On the constitution of atoms and molecules: Part III, ibid., pp. 857–875. Cited on page 79.
Robert H. Dicke & James P. Wittke, Introduction to Quantum Theory, AddisonWesley, Reading, Massachusetts, 1960. See also Stephen Gasiorowicz, Quantum Physics, John Wiley & Sons, 1974. Cited on page 81.
P. Carruthers & M. M. Nieto, Phase and angle variables in quantum mechanics, Review of Modern Physics 40, pp. 411–440, 1968. Cited on page 82.
The indeterminacy relation for rotational motion is well explained by W. H. Louisell,
Amplitude and phase uncertainty relations, Physics Letters 7, p. 60, 1963. Cited on page 82.
S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial &
M. Padgett, Uncertainty principle for angular position and angular momentum, New
Journal of Physics 6, p. 103, 2004. This is a freely accessible online journal. Cited on page
82.
W. Gerlach & O. Stern, Der experimentelle Nachweis des magnetischen Moments des
Silberatoms, Zeitschrift für Physik 8, p. 110, 1921. See also the pedagogical explanation by

Motion Mountain – The Adventure of Physics

42

251

252

bibliography
M. Hannout, S. Hoyt, A. Kryowonos & A. Widom, Quantum measurement and
the Stern–Gerlach experiment, American Journal of Physics 66, pp. 377–379, 1995. Cited on
page 83.

52

J. Schmiedmayer,
M. S. Chapman,
C. R. Ekstrom,
T. D. Hammond,
S. Wehinger & D. E. Pritchard, Index of refraction of various gases for sodium
matter waves, Physical Review Letters 74, p. 1043-1046, 1995. Cited on page 85.

53

The original result is due to V. de Sabbata & C. Sivaram, A minimal time and timetemperature uncertainty principle, Foundations of Physics Letters 5, pp. 183–189, 1992. Experimental details are found, for example, in G. T. Gillies & S. W. Allison, Experimental test of a time-temperature formulation of the uncertainty principle via nanoparticle
fluorescence, Foundations of Physics Letters 18, pp. 65–74, 2005. Cited on page 86.

54

Albert Einstein & Max B orn, Briefwechsel 1916 bis 1955, Rowohlt, 1969, as cited on
page 34. Cited on page 87.

55

E. Schrödinger, Quantisierung als Eigenwertproblem I, Annalen der Physik 79, pp. 361–
376, 1926, and Quantisierung als Eigenwertproblem II, Annalen der Physik 79, pp. 489–527,
1926. Cited on page 92.

56

C. G. Gray, G. Karl & V. A. Novikov, From Maupertius to Schrödinger. Quantization
of classical variational principles, American Journal of Physics 67, pp. 959–961, 1999. Cited
on page 92.

57

Y. Aharonov & D. B ohm, Significance of electromagnetic potentials in the quantum theory, Physical Review 115, pp. 485–491, 1959. Cited on page 98.

58

R. Colella, A. W. Overhauser & S. A. Werner, Observation of gravitationally induced quantum interference, Physical review Letters 34, pp. 1472–1474, 1975. Cited on page
100.

59

The trend-setting result that started this exploration was Hans-Werner Fink & al.,
Atomic resolution in lens-less low-energy electron holography, Physical Review Letters 67,
pp. 1543–1546, 1991. Cited on page 101.

60

L. Cser, Gy. Török, G. Krexner, I. Sharkov & B. Faragó, Holographic imaging
of atoms using thermal neutrons, Physical Review Letters 89, p. 175504, 2002. Cited on page
101.

61

G. E. Uhlenbeck & S. Goudsmit, Ersetzung der Hypothese vom unmechanischen
Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons,
Naturwissenschaften 13, pp. 953–954, 1925. Cited on page 104.

62

L. Thomas, The motion of the spinning electron, Nature 117, p. 514, 1926. Cited on page
105.

63

K. von Meyenn & E. Schucking, Wolfgang Pauli, Physics Today pp. 43–48, February
2001. Cited on page 105.

64

T. D. Newton & E. P. Wigner, Localized states for elementary systems, Review of Modern Physics 21, pp. 400–406, 1949. L. L. Foldy & S. A. Wouthuysen, On the Dirac
theory of spin 1/2 particles and its nonrelativistic limit, Physical Review 78, pp. 29–36, 1950.
Both are classic papers. Cited on page 106.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

J. P. Woerdman, G. Nienhuis, I. Kuščer, Is it possible to rotate an atom?, Optics Communications 93, pp. 135–144, 1992. We are talking about atoms rotating around
their centre of mass; atoms can of course rotate around other bodies, as discussed by
M. P. Silverman, Circular birefringence of an atom in uniform rotation: the classical perspective, American Journal of Physics 58, pp. 310–317, 1990. Cited on page 85.

Motion Mountain – The Adventure of Physics

51

bibliography

65
66

67

68

70

71

73

74

75

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

72

J. P. Costella & B. H. J. McKellar, The Foldy–Wouthuysen transformation, American Journal of Physics 63, pp. 1119–1121, 1995. Cited on page 106.
For an account of the first measuremnt of the g-factor of the electron, see H. R. Crane,
How we happended to measure g-2: a tale of serendipity, Physics in Perspective 2, pp. 135–140,
2000. The most interesting part is how the experimentalists had to overcome the conviction
of almost all theorists that the measurement was impossible in principle. Cited on page 107.
The ?-factors for composite nuclei are explained briefly on en.wikipedia.org/wiki/
Nuclear_magnetic_moment and measured values are found at www-nds.iaea.org. See
also H. Dehmelt, Is the electron a composite particle?, Hyperfine Interactions 81, pp. 1–3,
1993. Cited on pages 108 and 254.
The nearest anyone has come to an image of a hydrogen atom is found in A. Yazdani,
Watching an atom tunnel, Nature 409, pp. 471–472, 2001. The experiments on Bose–Einstein
condensates are also candidates for images of hydrogen atoms. The company Hitachi made
a fool of itself in 1992 by claiming in a press release that its newest electron microscope
could image hydrogen atoms. Cited on page 109.
A. M. Wolsky, Kinetic energy, size, and the uncertainty principle, American Journal of
Physics 42, pp. 760–763, 1974. Cited on page 110.
For a fascinating summary, see M. A. Cirone, G. Metikas & W. P. Schleich, Unusual bound or localized states, preprint at arxiv.org/abs/quant-ph/0102065. Cited on page
110.
See the paper by Martin Gardner, Science fiction puzzle tales, Clarkson Potter, 67,
pp. 104–105, 1981, or his book Aha! Insight, Scientific American & W.H. Freeman, 1978. Several versions are given A. Hajnal & P. Lovász, An algorithm to prevent the propagation
of certain diseases at minimum cost, in Interfaces Between Computer Science and Operations
Research, edited by J. K. Lenstra, A. H. G. Rinnooy Kan & P. Van Emde B oas,
Mathematisch Centrum, Amsterdam 1978, whereas the computer euphemism is used by
A. Orlitzky & L. Shepp, On curbing virus propagation, Technical memorandum, Bell
Labs 1989. Cited on page 112.
A complete discussion of the problem can be found in chapter 10 of Ilan Vardi, Computational Recreations in Mathematica, Addison Wesley, 1991. Cited on page 112.
On Gibbs’ paradox, see your favourite text on thermodynamics or statistical mechanics.
See also W. H. Zurek, Algorithmic randomness and physical entropy, Physical Review A
40, pp. 4731–4751, 1989. Zurek shows that the Sackur–Tetrode formula can be derived from
algorithmic entropy considerations. Cited on page 114.
S. N. B ose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik 26, pp. 178–
181, 1924. The theory was then expanded in A. Einstein, Quantentheorie des einatomigen
idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin
22, pp. 261–267, 1924, A. Einstein, Quantentheorie des einatomigen idealen Gases. Zweite
Abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin 23,
pp. 3–14, 1925, A. Einstein, Zur Quantentheorie des idealen Gases, Sitzungsberichte der
Preussischen Akademie der Wissenschaften zu Berlin 23, pp. 18–25, 1925. Cited on page
118.
C. K. Hong, Z. Y. Ou & L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Physical Review Letters 59, pp. 2044–2046,
1987. See also T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin,
A. V. Sergienko & Y. H. Shih, Can two-photon interference be considered the interference of two photons?, Physical Review Letters 77, pp. 1917–1920, 1996. Cited on page
118.

Motion Mountain – The Adventure of Physics

69

253

254

76

77

79

81

82

83
84

85

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

80

An example of such an experiment performed with electrons instead of photons is
described in E. B ocquillon, V. Freulon, J. -M. Berroir, P. Degiovanni,
B. Plaçais, A. Cavanna, Y. Jin & G. Fève, Coherence and indistinguishability of
single electrons emitted by independent sources, Science 339, pp. 1054–1057, 2013. See
also the comment C. Schönenberger, Two indistinguishable electrons interfere in an
electronic device, Science 339, pp. 1041–1042, 2013. Cited on page 119.
M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. B oiron,
C. I. Westbrook & A. Aspect, Hanbury Brown Twiss effect for ultracold quantum
gases, Science 310, p. 648, 2005, preprint at arxiv.org/abs/cond-mat/0508466. J. Viana
Gomes, A. Perrin, M. Schellekens, D. B oiron, C. I. Westbrook &
M. Belsley, Theory for a Hanbury Brown Twiss experiment with a ballistically expanding cloud of cold atoms, Physical Review A 74, p. 053607, 2006, preprint at arxiv.org/
abs/quant-ph/0606147. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen,
V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. B oiron,
A. Aspect & C. I. Westbrook, Comparison of the Hanbury Brown-Twiss effect for bosons and fermions, Nature 445, p. 402, 2007, preprint at arxiv.org/abs/cond-mat/0612278.
Cited on page 120.
The experiment is described in E. Ramberg & G. A. Snow, Experimental limit on a small
violation of the Pauli principle, Physics Letters B 238, pp. 438–441, 1990. Other experimental
tests are reviewed in O. W. Greenberg, Particles with small violations of Fermi or Bose
statistics, Physical Review D 43, pp. 4111–4120, 1991. Cited on page 122.
The original no-cloning theorem is by D. Dieks, Communication by EPR devices, Physics Letters A 92, pp. 271–272, 1982, and by W. K. Wootters & W. H. Zurek, A single
quantum cannot be cloned, Nature 299, pp. 802–803, 1982. For a discussion of photon and
multiparticle cloning, see N. Gisin & S. Massar, Optimal quantum cloning machines,
Physics Review Letters 79, pp. 2153–2156, 1997. The whole topic has been presented in detail
by V. Buzek & M. Hillery, Quantum cloning, Physics World 14, pp. 25–29, November
2001. Cited on page 122.
S. J. Wiesner, Conjugate Coding, SIGACT News, 15, pp. 78–88, 1983. This widely cited
paper was one of starting points of quantum information theory. Cited on page 123.
The most recent experimental and theoretical results on physical cloning are described
in A. Lamas-Linares, C. Simon, J. C. Howell & D. B ouwmeester, Experimental quantum cloning of single photons, Science 296, pp. 712 – 714, 2002, D. Collins
& S. Popescu, A classical analogue of entanglement, preprint arxiv.org/abs/quant-ph/
0107082, 2001, and A. Daffertshofer, A. R. Plastino & A. Plastino, Classical
no-cloning theorem, Physical Review Letters 88, p. 210601, 2002. Cited on page 123.
E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of
Mathematics 40, pp. 149–204, 1939. This famous paper summarises the work which later
brought him the Nobel Prize in Physics. Cited on pages 125 and 137.
For a full list of isotopes, see R. B. Firestone, Table of Isotopes, Eighth Edition, 1999 Update, with CDROM, John Wiley & Sons, 1999. Cited on page 127.
This is deduced from the ? − 2 measurements, as explained in his Nobel-prize talk by
Hans Dehmelt, Experiments with an isolated subatomic particle at rest, Reviews of Modern Physics 62, pp. 525–530, 1990. On this topic, see also his paper Ref. 67. No citations.
and in Hans Dehmelt, Is the electron a composite particle?, Hyperfine Interactions
81, pp. 1–3, 1993.
G. Gabrielse, H. Dehmelt & W. Kells, Observation of a relativistic, bistable hysteresis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. 537–540,

Motion Mountain – The Adventure of Physics

78

bibliography

bibliography

255

1985. No citations.

87

The belt trick has been popularized by Dirac, Feynman and many others. An example
is R. P. Feynman, The reason for antiparticles, in Elementary Particles and the Laws of
Physics: The 1986 Dirac Memorial Lectures, Cambridge University Press, 1987. The belt
trick is also explained, for example, on page 1148 in C. W. Misner, K. S. Thorne &
J. A. Wheeler, Gravitation, Freeman, 1973. It is called the scissor trick on page 43 of
volume 1 of R. Penrose & W. Rindler, Spinors and Spacetime, 1984. It is also cited and
discussed by R. Gould, Answer to question #7, American Journal of Physics 63, p. 109,
1995. Still, some physicists do not like the belt-trick image for spin 1/2 particles; for an
example, see I. Duck & E. C. G. Sudarshan, Toward an understanding of the spinstatistics theorem, American Journal of Physics 66, pp. 284–303, 1998. Cited on page 133.

88

M. V. Berry & J. M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proceedings of the Royal Society in London A 453, pp. 1771–
1790, 1997. See also the comments to this result by J. Twamley, Statistics given a spin,
Nature 389, pp. 127–128, 11 September 1997. Their newer results are M. V. Berry &
J. M. Robbins, Quantum indistinguishability: alternative constructions of the transported basis, Journal of Physics A (Letters) 33, pp. L207–L214, 2000, and M. V. Berry &
J. M. Robbins, in Spin–Statistics, eds. R. Hilborn & G. Tino, American Institute of
Physics, 2000, pp. 3–15. See also Michael Berry’s home page at www.phy.bris.ac.uk/people/
berry_mv. Cited on page 135.

89

R. W. Hartung, Pauli principle in Euclidean geometry, American Journal of Physics 47,
pp. 900–910, 1979. Cited on page 136.

90

The issue is treated in his Summa Theologica, in question 52 of the first part. The complete text, several thousand pages, can be found on the www.newadvent.org website. Also
present-day angelologists, of which there are only a few across the world, agree with Aquinas. Cited on page 136.

91

The point that spin can be seen as a rotation was already made by F. J. Belinfante,
On the spin angular momentum of mesons, Physica 6, p. 887, 1939, and taken up again by
Hans C. Ohanian, What is spin?, American Journal of Physics 54, pp. 500–505, 1986.
See also E. Duran & A. Erschow, Physikalische Zeitschrift der Sowjetunion 12, p. 466,
1937. Cited on page 138.

92

See the book by Jean-Marc Lév y-Leblond & Françoise Balibar in Ref. 4. Cited
on page 141.

93

Generalizations of bosons and fermions are reviewed in the (serious!) paper by
O. W. Greenberg, D. M. Greenberger & T. V. Greenbergest, (Para)bosons,
(para)fermions, quons and other beasts in the menagerie of particle statistics, at arxiv.org/
abs/hep-th/9306225. A newer summary is O. W. Greenberg, Theories of violation of
statistics, electronic preprint available at arxiv.org/abs/hep-th/0007054. Cited on page 142.

94

Gell-Mann wrote this for the 1976 Nobel Conference (not for the Nobel speech; he is the
only winner who never published it.) M. Gell-Mann, What are the building blocks of
matter?, in D. Huff & O. Prewitt, editors, The Nature of the Physical Universe, New
York, Wiley, 1979, p. 29. Cited on page 143.

95

See e.g. the reprints of his papers in the standard collection by John A. Wheeler & Wojciech H. Zurek, Quantum Theory and Measurement, Princeton University Press, 1983.
Cited on page 144.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

W. Pauli, The connection between spin and statistics, Physical Review 58, pp. 716– 722,
1940. Cited on page 133.

Motion Mountain – The Adventure of Physics

86

256

96
97
98

99

100

102

104

105

106

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

103

H. D. Zeh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. 69–76, 1970. Cited on page 144.
Linda Reichl, A Modern Course in Statistical Physics, Wiley, 2nd edition, 1998. An excellent introduction into thermodynamics. Cited on page 146.
E. Joos & H. D. Zeh, The emergence of classical properties through interactions with the
environment, Zeitschrift für Physik B 59, pp. 223–243, 1985. See also Erich Joos, Decoherence and the appearance of a classical world in quantum theory, Springer Verlag, 2003.
Cited on page 148.
M. Tegmark, Apparent wave function collapse caused by scattering, Foundation of Physics Letters 6, pp. 571–590, 1993, preprint at arxiv.org/abs/gr-qc/9310032. See also his paper
that shows that the brain is not a quantum computer, M. Tegmark, The importance of
quantum decoherence in brain processes, Physical Review E 61, pp. 4194–4206, 2000, preprint at arxiv.org/abs/quant-ph/9907009. Cited on page 148.
The decoherence time is bound from above by the relaxation time. See A. O. Caldeira
& A. J. Leggett, Influence of damping on quantum interference: an exactly soluble model,
Physical Review A 31, 1985, pp. 1059–1066. This is the main reference about effects of decoherence for a harmonic oscillator. The general approach to relate decoherence to the influence of the environment is due to Niels Bohr, and has been pursued in detail by Hans Dieter
Zeh. Cited on page 149.
G. Lindblad, On the generators of quantum dynamical subgroups, Communications in
Mathematical Physics 48, pp. 119–130, 1976. Cited on page 149.
Wojciech H. Zurek, Decoherence and the transition from quantum to classical, Physics
Today pp. 36–44, October 1991. An easy but somewhat confusing article. His reply to the
numerous letters of response in Physics Today, April 1993, pp. 13–15, and pp. 81–90, exposes
his ideas in a clearer way and gives a taste of the heated discussions on this topic. Cited on
pages 149 and 156.
John Bardeen, explained this regularly in the review talks he gave at the end of his life,
such as the one the author heard in Tokyo in the year 1990. Cited on page 150.
Collapse times have been measured for the first time by the group of Serge Haroche in Paris. See M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali,
C. Wunderlich, J. M. Raimond & S. Haroche, Observing the progressive decoherence of the “meter” in a quantum measurement, Physical Review Letters 77, pp. 4887–
4890, 1996. See also C. Guerlin, J. Bernu, S. Deléglise, C. Sayrin, S. Gleyzes,
S. Kuhr, M. Brune, J. -M. Raimond & S. Haroche, Progressive field-state collapse
and quantum non-demolition photon counting, Nature 448, pp. 889–893, 2007. Cited on
pages 150 and 162.
Later experiments confirming the numerical predictions from decoherence were published
by C. Monroe, D. M. Meekhof, B. E. King & D. J. Wineland, A “Schrödinger cat”
superposition state of an atom, Science 272, pp. 1131–1136, 1996, W. P. Schleich, Quantum
physics: engineering decoherence, Nature 403, pp. 256–257, 2000, C. J. Myatt, B. E. King,
Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe &
D. J. Wineland, Decoherence of quantum superpositions through coupling to engineered
reservoirs, Nature 403, pp. 269–273, 2000. See also the summary by W. T. Strunz,
G. Alber & F. Haake, Dekohärenz in offenen Quantensystemen, Physik Journal 1,
pp. 47–52, November 2002. Cited on page 150.
L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger & M. Arndt,
Decoherence of matter waves by thermal emission of radiation, Nature 427, pp. 711–714, 2004.
Cited on page 150.

Motion Mountain – The Adventure of Physics

101

bibliography

bibliography

257

107 K. Baumann, Quantenmechanik und Objektivierbarkeit, Zeitschrift für Naturforschung
108
109
110
111
112
113

115

117

118

119

120

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

116

Motion Mountain – The Adventure of Physics

114

25a, pp. 1954–1956, 1970. Cited on page 151.
See for example D. Styer, Physics Today p. 11, September 2000. Cited on page 153.
David B ohm, Quantum Theory, Prentice-Hall, 1951, pp. 614–622. Cited on page 154.
A. Einstein, B. Podolsky & N. Rosen, Can quantum-mechanical description of reality be considered complete?, Physical Review 48, pp. 696–702, 1935. Cited on page 154.
A. Aspect, J. Dalibard & G. Roger, Experimental tests of Bell’s inequalities using
time-varying analyzers, Physical Review Letters 49, pp. 1804–1807, 1982, Cited on page 155.
G. C. Hergerfeldt, Causality problems for Fermi’s two-atom system, Physical Review
Letters 72, pp. 596–599, 1994. Cited on page 155.
An experimental measurement of superpositions of left and right flowing currents with 1010 electrons was J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian,
C. H. van der Wal & S. Lloyd, Josephson persistent-current qubit, Science 285,
pp. 1036–1039, 1999. In the year 2000, superpositions of 1 μA clockwise and anticlockwise
have been detected; for more details, see J.R. Friedman & al., Quantum superposition
of distinct macroscopic states, Nature 406, p. 43, 2000. Cited on page 156.
On the superposition of magnetization in up and down directions there are numerous
papers. Recent experiments on the subject of quantum tunnelling in magnetic systems
are described in D. D. Awschalom, J. F. Smith, G. Grinstein, D. P. DiVicenzo
& D. Loss, Macroscopic quantum tunnelling in magnetic proteins, Physical Review Letters
88, pp. 3092–3095, 1992, and in C. Paulsen & al., Macroscopic quantum tunnelling effects
of Bloch walls in small ferromagnetic particles, Europhysics Letters 19, pp. 643–648, 1992.
Cited on page 156.
For example, superpositions were observed in Josephson junctions by R. F. Voss &
R. A. Webb, Macroscopic quantum tunnelling in 1 mm Nb Josephson junctions, Physical
Review Letters 47, pp. 265–268, 1981, Cited on page 156.
S. Haroche, Entanglement, decoherence and the quantum-classical transition, Physics
Today 51, pp. 36–42, July 1998. An experiment putting atom at two places at once,
distant about 80 nm, was published by C. Monroe, C. Monroe, D. M. Meekhof,
B. E. King & D. J. Wineland, A ‘Schroedinger Cat’ Superposition of an Atom, Science
272, pp. 1131–1136, 1996. Cited on page 156.
M. R. Andrews, C. G. Townsend, H. -J. Miesner, D. S. Durfee, D. M. Kurn &
W. Ketterle, Observations of interference between two Bose condensates, Science 275,
pp. 637–641, 31 January 1997. See also the www.aip.org/physnews/special.htm website.
Cited on page 156.
A clear discussion can be found in S. Haroche & J. -M. Raimond, Quantum computing: dream or nightmare?, Physics Today 49, pp. 51–52, 1996, as well as the comments in
Physics Today 49, pp. 107–108, 1996. Cited on page 157.
The most famous reference on the wave function collapse is chapter IV of the book by
Kurt Gottfried, Quantum Mechanics, Benjamin, New York, 1966. It was the favourite reference by Victor Weisskopf, and cited by him on every occasion he talked about the
topic. Cited on page 158.
The prediction that quantum tunnelling could be observable when the dissipative interaction with the rest of the world is small enough was made by Leggett; the topic is reviewed
in A. J. Leggett, S. Chahravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg &
W. Zwerger, Dynamics of dissipative 2-state systems, Review of Modern Physics 59, pp. 1–
85, 1987. Cited on page 160.

258

bibliography

121 S. Kochen & E. P. Specker, The problem of hidden variables in quantum mechanics,
122

123

124

126
127

129

130

131

132

133

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

128

Motion Mountain – The Adventure of Physics

125

Journal of Mathematics and Mechanics 17, pp. 59–87, 1967. Cited on page 163.
J. F. Clauser, M. A. Horne, A. Shimony & R. A. Holt, Proposed experiment to test
local hidden-variable theories, Physical Review Letters 23, pp. 880–884, 1969. The more general and original result is found in J. S. Bell, On the Einstein Podolsky Rosen Paradox,
Physics 1, p. 195, 1964. Cited on page 164.
D. M. Greenberger, M. A. Horne & A. Zeilinger, Going beyond Bell’s theorem, postprint of the 1989 paper at arxiv.org/abs/0712.0912. The first observation was
D. B ouwmeester, J. -W. Pan, M. Daniell, H. Weinfurter & A. Zeilinger,
Observation of three-photon Greenberger-Horne–Zeilinger entanglement, preprint at arxiv.
org/abs/quant-ph/9810035. Cited on page 164.
Bryce de Witt & Neill Graham, eds., The Many–Worlds Interpretation of Quantum
Mechanics, Princeton University Press, 1973. This interpretation talks about entities which
cannot be observed, namely the many worlds, and often assumes that the wave function of
the universe exists. Both habits are beliefs and in contrast with facts. Cited on page 167.
‘On the other had I think I can safely say that nobody understands quantum mechanics.’ From Richard P. Feynman, The Character of Physical Law, MIT Press, Cambridge,
1965, p. 129. He repeatedly made this statement, e.g. in the introduction of his otherwise excellent QED – The Strange Theory of Light and Matter, Penguin Books, 1990. Cited on page
167.
M. Tegmark, The importance of quantum decoherence in brain processes, Physical Review
D 61, pp. 4194–4206, 2000, or also arxiv.org/abs/quant-ph/9907009. Cited on page 168.
Connections between quantum theory and information theory can be followed in the International Journal of Quantum Information. Cited on page 169.
J. A. Wheeler, pp. 242–307, in Batelle Recontres: 1967 Lectures in Mathematics and Physics, C. DeWitt & J. A. Wheeler, editors, W.A. Benjamin, 1968. For a pedagogical explanation, see John W. Norbury, From Newton’s laws to the Wheeler-DeWitt equation,
arxiv.org/abs/physics/980604 or European Journal of Physics 19, pp. 143–150, 1998. Cited
on page 170.
The most fascinating book on the topic is by Kurt Nassau, The Physics and Chemistry of
Color – the Fifteen Causes of Color, 1983, and the excellent webexhibits.org/causesofcolour
website. Cited on page 171.
Y. Ruiz-Morales & O. C. Mullins, Measured and Simulated Electronic Absorption and Emission Spectra of Asphaltenes, Energy & Fuels 23, pp. 1169–1177,
2009. U. Bergmann, H. Groenzin, O. C. Mullins, P. Glatzel, J. Fetzer &
S. P. Cramer, Carbon K-edge X-ray Raman spectroscopy supports simple, yet powerful description of aromatic hydrocarbons and asphaltenes, Chemical Physics Letters 369, pp. 184–
191, 2003. Cited on page 171.
Two excellent reviews with numerous photographs are E. Grotewohl, The genetics and
biochemistry of floral pigments, Annual Reviews of Plant Biology 57, pp. 761–780, 2006,
and Y. Tanaka, N. Sasaki & A. Ohmiya, Biosynthesis of plant pigments: anthocyanins,
betalains and carotenoids, The Plant Journal 54, pp. 733–749, 2008. Cited on page 179.
L. Pérez-Rodriguez & J. Viñuda, Carotenoid-based bill and eye coloration as honest
signals of condition: an experimental test in the red-legged partridge (Alectoris rufa), Naturwissenschaften 95, pp. 821–830, 2008, Cited on page 179.
R. Pello, D. Schaerer, J. Richard, J. -F. Le B orgne & J. -P. Kneib, ISAAC/VLT
observations of a lensed galaxy at z=10.0, Astronomy and Astrophysics 416, p. L35, 2004.

bibliography

134

135
136

137
138

Cited on page 182.
A pedagogical introduction is given by L. J. Curtis & D. G. Ellis, Use of the Einstein–
Brillouin–Keller action quantization, American Journal of Physics 72, pp. 1521–1523, 2004.
See also the introduction of A. Klein, WKB approximation for bound states by Heisenberg
matrix mechanics, Journal of Mathematical Physics 19, pp. 292–297, 1978. Cited on pages
183 and 188.
J. Neukammer & al., Spectroscopy of Rydberg atoms at ? ∼ 500, Physical Review Letters
59, pp. 2947–2950, 1987. Cited on page 186.
Mark P. Silverman, And Yet It Moves: Strange Systems and Subtle Questions in Physics,
Cambridge University Press 1993. A beautiful book by an expert on motion. Cited on pages
187, 194, and 195.
This is explained by J. D. Hey, Mystery error in Gamow’s Tompkins reappears, Physics
Today pp. 88–89, May 2001. Cited on page 187.
The beautiful experiment was first published in A. S. Stodolna, A. Rouzée,
F. Lépine, S. Cohen, F. Robicheaux, A. Gijsbertsen, J. H. Jungmann,
C. B ordas & M. J. J. Vrakking, Hydrogen atoms under magnification: direct observation of the nodal structure of Stark states, Physical Review Letters 110, p. 213001, 2013.
Cited on pages 185 and 186.
L. L. Foldy, The electromagnetic properties of Dirac particles, Physical Review 83, pp. 688–
693, 1951. L. L. Foldy, The electron–neutron interaction, Physical Review 83, pp. 693–696,
1951. L. L. Foldy, Electron–neutron interaction, Review of Modern Physics 30, pp. 471–
481, 1952. Cited on page 190.

140 H. Euler & B. Kockel, Über die Streuung von Licht an Licht nach der Diracschen The-

141 See the simple explanation by L. J. F. Hermans, Blue skies, blue seas, Europhysics News

37, p. 16, 2006, and the detailed explanation by C. L. Braun & S. N. Smirnov, Why is
water blue?, Journal of Chemical Education 70, pp. 612–614, 1993. Cited on page 194.
142 The discovery is published in T. Friedmann & C. R. Hagen, Quantum Mechanical De-

rivation of the Wallis Formula for π , Journal of Mathematical Physics 56, p. 112101, 2015,
preprint at arxiv.org/1510.07813. See also I Chashchina & Z. K. Silagadze, On the
quantum mechanical derivation of the Wallis formula for π, preprint at arxiv.org/1704.06153.
Cited on page 194.
143 For the atomic case, see P. L. Gould, G. A. Ruff & D. E. Pritchard, Diffraction of

atoms by light: the near resonant Kapitza–Dirac effect, Physical Review Letters 56, pp. 827–
830, 1986. Many early experimental attempts to observe the diffraction of electrons by light,
in particular those performed in the 1980s, were controversial; most showed only the deflection of electrons, as explained by H. Batelaan, Contemporary Physics 41, p. 369, 2000.
Later on, he and his group performed the newest and most spectacular experiment, demonstrating real diffraction, including interference effects; it is described in D. L. Freimund,
K. Aflatooni & H. Batelaan, Observation of the Kapitza–Dirac effect, Nature 413,
pp. 142–143, 2001. Cited on page 195.
144 A single–atom laser was built in 1994 by K. An, J. J. Childs, R. R. Dasari &
M. S. Feld, Microlaser: a laser with one atom in an optical resonator, Physical Review
Letters 73, p. 3375, 1994. Cited on page 195.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

orie, Naturwissenschaften 23, pp. 246–247, 1935, H. Euler, Über die Streuung von Licht an
Licht nach der Diracschen Theorie, Annalen der Physik 26, p. 398, 1936, W. Heisenberg
& H. Euler, Folgerung aus der Diracschen Theorie des Electrons, Zeitschrift für Physik 98,
pp. 714–722, 1936. Cited on page 193.

Motion Mountain – The Adventure of Physics

139

259

260

bibliography

145 An introduction is given by P. Pinkse & G. Rempe, Wie fängt man ein Atom mit einem

Photon?, Physikalische Blätter 56, pp. 49–51, 2000. Cited on page 195.
146 J.P. Briand & al., Production of hollow atoms by the excitation of highly charged ions

in interaction with a metallic surface, Physical Review Letters 65, pp. 159–162, 1990. See
also G. Marowsky & C. Rhodes, Hohle Atome und die Kompression von Licht in Plasmakanälen, Physikalische Blätter 52, pp. 991–994, Oktober 1996. Cited on page 195.
147 G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio & B. Odom, New determina-

tion of the fine structure constant from the electron g value and QED, Physical Review Letters
97, p. 030802, 2006. Cited on page 196.
148 A. Sommerfeld, Zur Quantentheorie der Spektrallinien, Annalen der Physik 51, pp. 1–

94, 1916, and its continuation with the same title on pp. 125–167 in the same volume. The
fine structure constant is introduced in the first paper, but Sommerfeld explains that the
paper is a transcript of talks that he gave in 1915. Cited on page 196.
149 Wolf gang Pauli, Exclusion principle and quantum mechanics, Nobel lecture, 13 Decem150 An informative account of the world of psychokinesis and the paranormal is given by

the famous professional magician James Randi, Flim-flam!, Prometheus Books, Buffalo
1987, as well as in several of his other books. See also the www.randi.org website. Cited on
page 201.
151 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de

152 The bible in the field of time measurement is the two-volume work by J. Vanier &

C. Audoin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A
popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000.
The site opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field.
For precision length measurements, the tools of choice are special lasers, such as modelocked lasers and frequency combs. There is a huge literature on these topics. Equally large
is the literature on precision electric current measurements; there is a race going on for the
best way to do this: counting charges or measuring magnetic forces. The issue is still open.
On mass and atomic mass measurements, see Volume II, on page 71. On high-precision
temperature measurements, see Volume I, on page 544. Cited on page 206.
153 The unofficial SI prefixes were first proposed in the 1990s by Jeff K. Aronson of the Uni-

versity of Oxford, and might come into general usage in the future. See New Scientist 144,
p. 81, 3 December 1994. Other, less serious proposals also exist. Cited on page 207.
154 For more details on electromagnetic unit systems, see the standard text by

John David Jackson, Classical Electrodynamics, 3rd edition, Wiley, 1998. Cited on
page 210.
155 D.J. Bird & al., Evidence for correlated changes in the spectrum and composition of cosmic

rays at extremely high energies, Physical Review Letters 71, pp. 3401–3404, 1993. Cited on
page 211.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI
units are published in the journal Metrologia, edited by the same body. Showing the slow
pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at
www.bipm.fr. See also the www.utc.fr/~tthomass/Themes/Unites/index.html website; this
includes the biographies of people who gave their names to various units. The site of its
British equivalent, www.npl.co.uk/npl/reference, is much better; it provides many details
as well as the English-language version of the SI unit definitions. Cited on page 205.

Motion Mountain – The Adventure of Physics

ber 1946, in Nobel Lectures, Physics, Volume 3, 1942–1962, Elsevier, 1964. Cited on page 197.

bibliography

261

156 P. J. Hakonen, R. T. Vuorinen & J. E. Martikainen, Nuclear antiferromagnetism

in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Letters
70, pp. 2818–2821, 1993. See also his article in Scientific American, January 1994. Cited on
page 211.
157 A. Zeilinger, The Planck stroll, American Journal of Physics 58, p. 103, 1990. Can you
Challenge 204 e

find another similar example? Cited on page 211.
158 An overview of this fascinating work is given by J. H. Taylor, Pulsar timing and relativ-

istic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–134,
1992. Cited on page 211.
159 The most precise clock built in 2004, a caesium fountain clock, had a precision of one

part in 1015 . Higher precision has been predicted to be possible soon, among others
by M. Takamoto, F. -L. Hong, R. Higashi & H. Katori, An optical lattice clock,
Nature 435, pp. 321–324, 2005. Cited on page 212.
160 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met161 See the information on D±? mesons from the particle data group at pdg.web.cern.ch/pdg.

Cited on page 212.
180

Tam and
its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical
Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 212.

162 About the long life of tantalum 180, see D. Belic & al., Photoactivation of

163 See the review by L. Ju, D. G. Blair & C. Zhao, The detection of gravitational waves,

Reports on Progress in Physics 63, pp. 1317–1427, 2000. Cited on page 212.
164 See the clear and extensive paper by G. E. Stedman, Ring laser tests of fundamental physics
165 The various concepts are even the topic of a separate international standard, ISO 5725, with

the title Accuracy and precision of measurement methods and results. A good introduction is
John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 213.
166 P. J. Mohr, B. N. Taylor & D. B. Newell, CODATA recommended values of the fun-

damental physical constants: 2010, preprint at arxiv.org/abs/1203.5425. This is the set of
constants resulting from an international adjustment and recommended for international
use by the Committee on Data for Science and Technology (CODATA), a body in the International Council of Scientific Unions, which brings together the International Union of
Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry
(IUPAC) and other organizations. The website of IUPAC is www.iupac.org. Cited on page 214.
167 Some of the stories can be found in the text by N. W. Wise, The Values of Precision,

Princeton University Press, 1994. The field of high-precision measurements, from which
the results on these pages stem, is a world on its own. A beautiful introduction to it
is J. D. Fairbanks, B. S. Deaver, C. W. Everitt & P. F. Michaelson, eds., Near
Zero: Frontiers of Physics, Freeman, 1988. Cited on page 214.
168 For details see the well-known astronomical reference, P. Kenneth Seidelmann, Ex-

planatory Supplement to the Astronomical Almanac, 1992. Cited on page 220.
169 See the corresponding reference in the first volume. Cited on page 222.
170 A good reference is the Encyclopedia of Mathematics, in 10 volumes, Kluwer Academic Pub-

lishers, 1988−1993. It explains most concepts used in mathematics. Spending an hour with
it looking up related keywords is an efficient way to get an introduction into any part of

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited on page 212.

Motion Mountain – The Adventure of Physics

rology, World Scientific, 1997. Cited on page 212.

262

bibliography

mathematics, especially into the vocabulary and the main connections.
The opposite approach, to make things as complicated as possible, is taken in the delightful text by Carl E. Linderholm, Mathematics Made Difficult, 1971. Cited on page
223.
171 An excellent introduction into number systems in mathematics, including hyperreal
(or nonstandard) numbers, quaternions, octonions, ?-adic and surreal numbers, is the
book by Heinz-Dieter Ebbinghaus, Hans Hermes, Friedrich Hirzebruch,
Max Koecher, Klaus Mainzer, Jürgen Neukirch, Alexander Prestel &
Reinhold Remmert, Zahlen, 3rd edition, Springer Verlag, 1993. It is also available in
English, under the title Numbers, Springer Verlag, 1990. Cited on pages 225, 234, and 235.
172 For a book on how to use hyperreals in secondary school, see Helmut Wunderling,

Analysis als Infinitesimalrechnung, Duden Paetec Schulbuchverlag, 2007. Cited on page 235.
173 A. Waser, Quaternions in Electrodynamics, 2001. The text can be downloaded from vari-

176 Gaussian integers are explored by G. H. Hardy & E. M. Wright, An Introduction to

177 About transfinite numbers, see the delightful paperback by Rudy Rucker, Infinity and

the Mind – the Science and Philosophy of the Infinite, Bantam, 1983. Cited on page 235.
178 E. I. Butikov, The rigid pendulum – an antique but evergreen physical model, European
Journal of Physics 20, pp. 429–441, 1999. D. Easton, The quantum mechanical tipping pencil – a caution for physics teachers, European Journal of Physics 28, pp. 1097–1104, 2007,
Cited on page 242.

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

the Theory of Numbers, 5th edition, Clarendon Press, Oxford, 1979, in the sections 12.2 ‘The
Rational Integers, the Gaussian Integers, and the Integers’, pp. 178–180, and 12.6 ‘Properties
of the Gaussian Integers’ pp. 182–183. For challenges relating to Gaussian integers, look at
www.mathpuzzle.com/Gaussians.html. Cited on page 235.

Motion Mountain – The Adventure of Physics

ous websites. Cited on pages 227 and 232.
174 S. L. Altman, Rotations, Quaternions and Double Groups, Clarendon Press, 1986, and also
S. L. Altman, Hamilton, Rodriguez and the quaternion scandal, Mathematical Magazine
62, pp. 291–308, 1988. See also J. C. Hart, G. K. Francis & L. H. Kauffman, Visualzing quaternion rotation, ACM Transactions on Graphics 13, pp. 256–276, 1994. The latter
can be downloaded in several places via the internet. Cited on page 230.
175 See the fine book by Louis H. Kauffman, Knots and Physics, World Scientific, 2nd edition, 1994, which gives a clear and visual introduction to the mathematics of knots and their
main applications to physics. Cited on page 231.

C R E DI T S

Acknowled gements

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Many people who have kept their gift of curiosity alive have helped to make this project come
true. Most of all, Peter Rudolph and Saverio Pascazio have been – present or not – a constant
reference for this project. Fernand Mayné, Anna Koolen, Ata Masafumi, Roberto Crespi, Serge
Pahaut, Luca Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark,
Kim Jalink, my parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar,
Barbara and Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan
Schiller, Franz Aichinger and, most of all, my wife Britta have all provided valuable advice and
encouragement.
Many people have helped with the project and the collection of material. Most useful was the
help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by
Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Sheldon, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu,
the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious
help of Adrian Kubala.
Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr,
John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas
Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der
Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr,
Allan Hayes, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom Bruce, Ryan
Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Martin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli, Lukas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finkenzeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo
Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vitaliy Solomatin, Carl Offner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre,
Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Martos, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi,
Thomas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Nadia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort,
Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni,
Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jürgen Janek,
Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Damoon Saghian, Wladimir Egorov, Markus Zecherle, Miles Mutka, plus a number of people who wanted to remain
unnamed.
The software tools were refined with extensive help on fonts and typesetting by Michael Zedler
and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help
came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Up-

264

credits

right, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story,
Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez,
Uwe Siart, Will Robertson, Joseph Wright, Enrico Gregorio, Rolf Niepraschk and Alexander
Grahn.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other
countries in the world, allow residents to use the local university libraries.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation.

Film credits

The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The photograph of a glow worm on page 14 is copyright and courtesy of John Tyler, and
found on his beautiful website at www.johntyler.co.uk/gwfacts.htm. The photograph of a glass
butterfly on page 16 is copyright and courtesy of Linda de Volder and found on her site at
www.flickr.com/photos/lindadevolder. The photograph of a train window on page 32 is copyright and courtesy of Greta Mansour and found at her website www.flickr.com/photos/wireful/.
The graphics of the colour spectrum on page 41 is copyright and courtesy of Andrew Young
and explained on his website mintaka.sdsu.edu/GF/explain/optics/rendering.html. The images
of photographic film on page 42 are copyright and courtesy of Rich Evans. The images of photomultipliers on page 42 are copyright and courtesy of Hamamatsu Photonics. The pictures of
the low-intensity photon interference experiment of page 43 are copyright of the Delft University of Technology, courtesy of Silvania Pereira, and found on the website www.optica.tn.tudelft.
nl/education/photons.asp. The photograph of the Compton effect apparatus on page 46 was
taken by Helene Hoffmann and is courtesy of Arne Gerdes from the University of Göttingen;
it is found at the physics teaching website lp.uni-goettingen.de. The graph on page 50 is courtesy and copyright of Rüdiger Paschotta and found in his free and wonderful laser encyclopedia
at www.rp-photonics.com. The photograph of the Mach–Zehnder interferometer on page 51 is
copyright and courtesy of Félix Dieu and Gaël Osowiecki and found on their websites www.
flickr.com/photos/felixdieu/sets/72157622768433934/ and www.flickr.com/photos/gaeloso/sets/

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Image credits

Motion Mountain – The Adventure of Physics

The hydrogen orbital image and animation of page 80 were produced with a sponsored copy of
Dean Dauger’s software package Atom in a Box, available at daugerresearch.com. The coloured
animations of wave functions on page 90, page 94, page 95, page 99, page 110, page 190 and
page 192 are copyright and courtesy by Bernd Thaller; they can be found on his splendid website
vqm.uni-graz.at and in the CDs that come with his two beautiful books, Bernd Thaller,
Visual Quantum Mechanics Springer, 2000, and Bernd Thaller, Advanced Visual Quantum
Mechanics Springer, 2004. These books are the best one can read to get an intuitive understanding
for wave functions and their evolution. The animation of the belt trick on page 131 is copyright
and courtesy by Greg Egan; it can be found on his website www.gregegan.net/APPLETS/21/21.
html. The beautiful animation of the belt trick on page 131 and the wonderful and so far unique
animation of the fermion exchange on page 134 are copyright and courtesy of Antonio Martos.
They can be found at vimeo.com/62228139 and vimeo.com/62143283.

credits

265

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

72157623165826538/. The photograph on page page 53 is copyright of John Davis and courtesy
of . The telescope mirror interference image on page page 57 is copyright and courtesy of Mel
Bartels and found on his site www.bbastrodesigns.com. The speckle pattern image is copyright
and courtesy of Epzcaw and found on Wikimedia Commons. On page page 58, the double slit interference patterns are copyright and courtesy of Dietrich Zawischa and found on his website on
beauty and science at www.itp.uni-hannover.de/~zawischa. The interference figure of Gaussian
beams is copyright and courtesy of Rüdiger Paschotta and found on his free laser encyclopedia at www.rp-photonics.com. The blue sky photograph on page 69 is courtesy and copyright of
Giorgio di Iorio, and found on his website www.flickr.com/photos/gioischia/. The images about
the wire contact experiment on page 69 is courtesy and copyright of José Costa-Krämer and
AAPT. The famous photograph of electron diffraction on page 76 is copyright and courtesy of
Claus Jönsson. The almost equally famous image that shows the build-up of electron diffraction
on page 77 is courtesy and copyright of Tonomura Akira/Hitachi: it is found on the www.hqrd.
hitachi.co.jp/em/doubleslit.cfm website. The hydrogen graph on page 85 is courtesy and copyright of Peter Eyland. The photographs of the Aharonov–Bohm effect on page 99 are copyright
and courtesy of Doru Cuturela. The images of DNA molecules on page 101 are copyright and
courtesy by Hans-Werner Fink and used with permission of Wiley VCH. The experiment pictures
of the bunching and antibunching of 3 He and 4 He on page 119 are from the website atomoptic.
iota.u-psud.fr/research/helium/helium.html and courtesy and copyright of Denis Boiron and
Jerome Chatin. The molten metal photograph on page 172 is courtesy and copyright of Graela and
found at flickr.com/photos/alaig. The sparkler photograph on page 172 is courtesy and copyright
of Sarah Domingos and found at her flickr.com website. The reactor core photograph on page 172
is courtesy NASA and found on the grin.hq.nasa.gov website. The discharge lamp photographs
on page 172 are courtesy and copyright of Pslawinski and found at www.wikimedia.org. The aurora photograph on page 172 is courtesy and copyright of Jan Curtis and found at his climate.gi.
alaska.edu/Curtis/curtis.html website. The coloured flames photograph on page 172 is courtesy
and copyright of Philip Evans and found at his community.webshots.com/user/hydrogen01 website. The iceberg photograph on page 173 is courtesy and copyright of Marc Shandro and found
at his flickr.com/photos/mshandro website. The malachite photograph on page 173 is copyright
and courtesy of Stephan Wolfsried and found on the www.mindat.org website. The shadow mask
photograph on page 173 is courtesy and copyright of Planemad and found at the www.wikimedia.
org website. The mineral photographs on page 173 and later are copyright and courtesy of Rob
Lavinsky at irocks.com, and taken from his large and beautiful collection there and at www.
mindat.org/photo-49529.html. The narcissus photograph on page 174 is courtesy and copyright
of Thomas Lüthi and found at his website www.tiptom.ch/album/blumen/. The photograph with
a finger with blood on page 174 is courtesy and copyright of Ian Humes and found at his website
www.flickr.com/photos/ianhumes. The berries photograph on page 174 is courtesy and copyright
of Nathan Wall and found at his website www.flickr.com/photos/ozboi-z. The photograph of a
red-haired woman on page 174 is by dusdin and courtesy of Wikimedia. The rare photograph
of a living angler fish on page 174 is courtesy and copyright of Steve Haddock and found at his
website www.lifesci.uscb.edu/~biolum/. The magnetite photograph on page 175 is copyright and
courtesy of Stephan Wolfsried and found on the www.mindat.org website. The desert photograph on page 175 is copyright of Evelien Willemsen, courtesy Raf Verbeelen and found at www.
flickr.com/photos/verbeelen. The tenor saxophone photograph on page 175 is courtesy and copyright of Selmer at www.selmer.fr. The photograph of zinc oxide on page 175 is by Walkerma and
courtesy of Wikimedia. The fluorescing quantum dot photograph on page 175 is courtesy and
copyright of Andrey Rogach, Center for Nanoscience, München. The zirconia photograph on
page 176 is courtesy and copyright of Gregory Phillips and found at the commons.wikimedia.
org website. The Tokyo sunset on page 176 is courtesy and copyright of Altus Plunkett and found

266

credits

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

at his www.flickr.com/photos/altus website. The blue quartz photograph on page 176 is courtesy
and copyright 2008 of David K. Lynch and found at his www.thulescientific.com website. The
snowman photograph on page 177 is courtesy and copyright of Andreas Kostner and found at his
www.flickr.com/photos/bytesinmotion website. The endangered blue poison frog photograph on
page 177 is courtesy and copyright of Lee Hancock and found at the www.treewalkers.org website.
The ruby glass photograph on page 177 is courtesy and copyright of the Murano Glass Shop and
is found at their murano-glass-shop.it website. The photograph of a ring laser with second harmonic generation on page 177 is courtesy and copyright of Jeff Sherman and found at his flickr.
com/photos/fatllama website. The abalone photograph on page 177 is courtesy and copyright of
Anne Elliot and found at her flickr.com/photos/annkelliot website. The photograph of polarization colours on page 177 is copyright of Nevit Dilmen and courtesy of Wikimedia. The mallard
duck photograph on page 178 is courtesy and copyright of Simon Griffith and found at his www.
pbase.com/simon2005 website. The opal photograph on page 178 is courtesy and copyright of
Opalsnopals and found at his www.flickr.com website. The aeroplane condensation photograph
on page 178 is courtesy and copyright of Franz Kerschbaum and found at the epod.usra.edu website. The CD photograph on page 178 is courtesy and copyright of Alfons Reichert and found at
his www.chemiephysikskripte.de/artikel/cd.htm website. The liquid crystal pattern on page 178
is courtesy and copyright of Ingo Dierking and Wiley/VCH; it is found in his wonderful book
Ingo Dierking, Textures of Liquid Crystals, Wiley-VCH, 2003. See also his website reynolds.
ph.man.ac.uk/people/staff/dierking/gallery. The measured colour spectrum on page 180 is copyright and courtesy of Nigel Sharp, NOAO, FTS, NSO, KPNO, AURA and NSF. The photograph of
a hydrogen discharge on page 181 is copyright and courtesy of Jürgen Bauer and found at the
beautiful website www.smart-elements.com. The illustrations of hydrogen orbitals on page 187
are courtesy of Wikimedia. The images of the nodal atomic structures on page 185 are courtesy
of Aneta Stodolna and copyright and courtesy of the American Physical Society; they are found
at journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.213001. The graphs of the squeezed light
states on page 241 are courtesy of G. Breitenbach and S. Schiller and copyright of Macmillan.
The historical portraits of physicists in the text are in the public domain, except where mentioned. The photograph on the back cover, of a basilisk running over water, is courtesy and copyright by the Belgian group TERRA vzw and found on their website www.terravzw.org. All drawings
are copyright by Christoph Schiller. If you suspect that your copyright is not correctly given or
obtained, this has not been done on purpose; please contact me in this case.

NA M E I N DE X

A
AAPT

Bohr, Niels 17, 37, 65, 78, 79,
143, 144, 166, 182, 256
life 17
Boiron, D. 254
Boiron, Denis 119, 265
Bombelli, Luca 263
Boone, Roggie 263
Bordas, C. 259
Borgne, J.-F. Le 258
Born, Max 25, 87, 89, 248, 252
life 22
Bose, S.N. 253
Bose, Satyenra Nath
life 118
Bouwmeester, D. 254, 258
Bradley, C.C. 250
Brahmagupta 223
Brandes, John 263
Braun, C.L. 259
Brebner, Douglas 263
Breitenbach, G. 241, 249, 266
Brendel, J. 250
Brezger, B. 256
Briand, J.P. 195, 260
Brillouin, Léon 182
Broglie, L. de 251
Broglie, Louis de 76
life 34
Bronshtein, Matvei 8
Brown, R. Hanbury 249
Bruce, Tom 263
Brumberg, E.M. 40, 248
Brune, M. 256
Bub, J. 248
Buchmann, Alfons 263
Budney, Ryan 263
Bunsen, Robert 180

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

B
Babinet, Jacques
life 206
Bader, Richard F.W. 247
Baez, John 263
Bagnoli, Franco 263
Balibar, Françoise 248, 255
Balmer, Johann 182
Barberi Gnecco, Bruno 263

Bardeen, John 256
Barnett, S.M. 251
Bartel, Elmar 263
Bartels, Mel 57, 265
Batelaan, H. 259
Bauer, Jürgen 181, 266
Baumann, K. 257
Bautista, Ferdinand 263
Baylor, D.A. 249
Beaty, William 263
Beeksma, Herman 263
Belfort, François 263
Belic, D. 261
Belinfante, F.J. 255
Bell, J. 163
Bell, John 258
life 164
Belsley, M. 254
Bergmann, U. 258
Bergquist, J. 261
Bernu, J. 256
Berroir, J.-M. 254
Berry, M.V. 255
Berry, Michael 255
Bessel 180
Beutelspacher, Albrecht 169
Beyer, Lothar 263
Biggar, Mark 263
Bird, D.J. 260
Björk, G. 249
Blair, D.G. 261
Blumensath, Achim 263
Boas, P. Van Emde 253
Bocquillon, E. 254
Bohm, D. 252
Bohm, David 257
Bohr, N. 247, 250, 251

Motion Mountain – The Adventure of Physics

A
AAPT 75, 265
Aedini, J. 250
Aflatooni, K. 259
Aharonov, Y. 252
Aichinger, Franz 263
Alber, G. 256
Allison, S.W. 252
Altman, S.L. 262
An, K. 259
Anderson, Carl 192
Anderson, M.H. 250
Andrews, M.R. 250, 257
Anton, A. 250
APS 185
Aquinas, Thomas 136
Arndt, M. 251, 256
Aronson, Jeff K. 260, 263
Arseneau, Donald 263
Aspect, A. 254, 257
Aspect, Alain 155
Aspelmeyer, M. 250
Ata Masafumi 263
Audoin, C. 260
Augel, Barbara 263
Augel, Edgar 263
Awschalom, D.D. 257

268
Busch, Paul 248
Butikov, E.I. 262
Buzek, V. 254
Böhncke, Klaus 263

B
Busch

Curtis, L.J. 259
Cuturela, Doru 99, 265

E
Easton, D. 262
Ebbinghaus, Heinz-Dieter 262
Egan, Greg 131, 264
Egorov, Wladimir 263
Einstein, A. 249, 253, 257
Einstein, Albert 30, 55, 56, 105,
118, 182, 252
Ekstrom, C.R. 252
Elitzur, Avshalom 69
Elliot, Anne 177, 266
Ellis, D.G. 259
Els, Danie 264
Elswijk, Herman B. 263
Emelin, Sergei 263
Engels, F. 248
Engels, Friedrich 39, 74
Englert, Berthold-Georg 248
Epicurus 40, 44
Epzcaw 57, 265
Erdős, Paul
life 223
Erschow, A. 255
Euler, H. 259
Euler, Hans 193
Euler, Leonhard 229
Evans, Philip 172, 265
Evans, Rich 42, 264
Everitt, C.W. 261
Eyland, Peter 85, 265
F
Facchi, P. 251
Fairbanks, J.D. 261
Faragó, B. 252
Farinati, Claudio 263
Feld, M.S. 259
Fermani, Antonio 263
Fermi, Enrico
life 118
Fetzer, J. 258
Feynman, R. P. 255
Feynman, Richard 167
Feynman, Richard P. 250, 258
Feynman,
Richard (‘Dick’) Phillips
life 60

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

D
Daffertshofer, A. 254
Dahlman, John 263
Dalibard, J. 257
Danecek, Petr 263
Daniell, M. 258
Darley, Vincent 264
Darre, Daniel 263
Dasari, R.R. 259
Dauger, Dean 80, 264
Davis, John 53, 265
Davis, K.B. 250
Deaver, B.S. 261
Degen, Carl Ferdinand 234
Degiovanni, P. 254
Dehmelt, H. 253, 254
Dehmelt, Hans 254
Delft University of
Technology 43, 264
Deléglise, S. 256
DeWitt, C. 258
Dicke, Robert H. 251
Dieks, D. 254
Dieks, Dennis 122
Dierking, Ingo 178, 266
Dieu, Félix 51, 264
Diez, Ulrich 264
DiFilippo, Frank 263
Dilmen, Nevit 177, 266
Diophantus of Alexandria 226
Dirac 195
Dirac, P.A.M. 250
Dirac, Paul 59, 66, 189
life 189
Dirr, Ulrich 264
Diu, B. 248
DiVicenzo, D.P. 257
Dobra, Ciprian 263
Domingos, Sarah 172, 265
Dorsey, A.T. 257
Dreyer, J. 256
Druten, N.J. van 250
Duck, I. 255
Duran, E. 255
Durfee, D.S. 250, 257
dusdin 174, 265

Dyson, Freeman 263

Motion Mountain – The Adventure of Physics

C
Caldeira, A.O. 256
Candela, D. 251
Carlip, Steve 263
Carr, Jim 263
Carruthers, P. 251
Cato, Marcus Porcius 171
Cavanna, A. 254
Cayley, Arthur 233, 234
Center for Nanoscience,
München 265
Chahravarty, S. 257
Chang, H. 254
Chang, I. 249
Chapman, M.S. 252
Chashchina, O.I 259
Chatin, Jerome 119, 265
Childs, J.J. 259
Chu, Steven 195
Cicero, Marcus Tullius 72
Cirone, M.A. 253
Clauser, J.F. 258
Clifton, R. 248
Cohen, S. 259
Cohen-Tannoudji, C. 248
Cohen-Tannoudji, Claude 195
Colazingari, Elena 263
Colella, R. 252
Collins, D. 254
Compton, A.H. 249
Compton, Arthur 45
Conti, Andrea 263
Corovic, Dejan 263
Correia, A. 251
Costa-Krämer, J.L. 250, 251
Costa-Krämer, José 74, 75, 265
Costella, J.P. 253
Courtial, J. 251
Cramer, S.P. 258
Crane, H.R. 253
Crespi, Roberto 263
Crommle, M.F. 242
Cser, L. 252
Curtis, Jan 172, 265

name index

name index

F
Fink

H
Haake, F. 256
Haas, Arthur Erich 21, 186
Haber, John 263
Hackermüller, L. 251
Hackermüller, L. 256
Haddock, Steve 174, 265
Hagen, C. R. 259
Hagley, E. 256
Hajnal, A. 253
Hakonen, P.J. 260
Haley, Stephen 263
Halvorson, H. 248
Hamamatsu Photonics 42
Hamilton 229
Hamilton, William Rowan
life 227
Hammond, T.D. 252
Hanbury Brown, Robert 52
Hancock, Lee 177, 266
Hanneke, D. 260

Hannout, M. 252
Hardcastle, Martin 263
Hardy, G.H. 262
Haroche, S. 256, 257
Haroche, Serge 150, 256
Hart, J.C. 262
Hartung, R.W. 255
Hausherr, Tilman 263
Hayes, Allan 263
Heaviside 232
Hegerfeldt, Gerhard 155
Heisenberg, W. 259
Heisenberg, Werner 22, 25,
78, 193
life 25
Helmond, Tom 263
Henderson, Paula 263
Hergerfeldt, G.C. 257
Hermann, Armin 248
Hermans, L.J.F. 259
Hermes, Hans 262
Herneck, Friedrich 250
Hertz 232
Hertz, Heinrich 54, 189
Hertzlinger, Joseph 264
Hess, Victor 92
Heumann, John 263
Hewitt, Susan 248
Hey, J.D. 259
Higashi, R. 261
Hilbert, David 223
Hilborn, R. 255
Hilgevoord, Jan 248
Hillery, M. 254
Hillman, Chris 263
Hirzebruch, Friedrich 262
Hitachi 77, 265
Hite, Kevin 264
Hitler, Adolf 17
Hoffman, Richard 263
Hoffmann, Helene 46, 264
Hogervorst, W. 254
Holt, R.A. 258
Hong, C.K. 253
Hong, F.-L. 261
Hoppeler, R. 254
Hornberger, K. 256
Horne, M.A. 258
Howell, J.C. 254

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

G
Gabrielse, G. 254, 260
Galilei, Galileo 21, 24
Galle, Johann Gottfried 180
Garcia, N. 250, 251
García-Mochales, P. 250, 251
Gardner, Martin 253
Garg, A. 257
Gasiorowicz, Stephen 251
Gasparoni, S. 250
Gauß, Carl-Friedrich 235
Gelbaum, Bernard R. 238
Gell-Mann, M. 255
Gell-Mann, Murray 143, 167
Georgi, Renate 263
Gerdes, Arne 264
Gerlach, W. 251
Gerlach, Walther
life 83
Gibbs, Josiah Willard
life 114
Gijsbertsen, A. 259
Gillies, G.T. 252
Gisin, N. 250, 254
Glanz, J. 249
Glassey, Olivier 263
Glatzel, P. 258

Glauber, R.J. 250
Glauber, Roy 63
Gleyzes, S. 256
González, Antonio 263
Gottfried, Kurt 257
Goudsmit, S. 252
Goudsmit, Samuel 104
Gould, P.L. 259
Gould, R. 255
Graela 172, 265
Graham, Neill 258
Grahn, Alexander 264
Graves, John 234
Gray, C.G. 252
Greenberg, O.W. 254, 255
Greenberg, Oscar 142
Greenberger, D.M. 255, 258
Greenbergest, T.V. 255
Gregorio, Enrico 264
Greiner, Jochen 263
Griffith, Simon 178, 266
Grinstein, G. 257
Grit, C.O. 242
Groenzin, H. 258
Grotewohl, E. 258
Guerlin, C. 256
Gácsi, Zoltán 263

Motion Mountain – The Adventure of Physics

Fink, Hans-Werner 101, 252,
265
Finkenzeller, Klaus 263
Firestone, R.B. 254
Fischbach, E. 249
Fischer, Ulrike 263
Fisher, M.P.A. 257
Foldy, L.L. 106, 252, 259
Foley, E.L. 251
Fonseca, E.J.S. 249
Francis, G.K. 262
Franke-Arnold, S. 251
Fraunhofer, Joseph
life 180
Freimund, D.L. 259
Freulon, V. 254
Friedman, J.R. 257
Friedmann, T. 259
Fumagalli, Giuseppe 247
Furrie, Pat 263
Fève, G. 254

269

270
Hoyt, S. 252
Huber, Daniel 263
Huff, D. 255
Hulet, R.G. 250
Humes, Ian 174, 265
Hurwitz, Adolf 234
Hänsch, Theodor 194

H
Hoy t

I
Icke, Vincent 248
Iorio, Giorgio di 69, 265
Itano, W.M. 256
Ivanov, Igor 263

L
Lagrange, Joseph 229
Laloë, F. 248
Lamas-Linares, A. 254
Langel, R.A. 249
Lavinsky, Rob 173, 176, 265
Leach, J. 251
Leggett, A.J. 256, 257
Lenstra, J.K. 253
Leonardo da Vinci 81
Levitov, L. 257
Lindblad, G. 256

Linde, Johan 264
Linderholm, Carl E. 262
Lintel, Harald van 263
Lloyd, S. 257
Lockyer, Joseph 181
Lombardi, Luciano 263
Loss, D. 257
Loudon, Rodney 248
Louisell, W.H. 251
Lovász, P. 253
Lui, A.T.Y. 249
Lundeen, J.S. 249
Lynch, David 176
Lépine, F. 259
Lévy-Leblond, Jean-Marc
248, 255
Lüthi, Thomas 174, 265
M
Maali, A. 256
Macmillan 241, 266
Magyar G. 67
Magyar, G. 250
Mahoney, Alan 263
Mainzer, Klaus 262
Maitre, X. 256
Malik, J. 248
Mandel, L. 67, 250, 253
Mansour, Greta 32, 264
Mariano, A. 251
Mark, Martin van der 263
Marowsky, G. 260
Marqués, M. 251
Martikainen, J.E. 261
Martini, K.M. 251
Martos, Antonio 131, 134, 263,
264
Massar, S. 254
Mattheck, Claus 248
Maxwell 232
Mayné, Fernand 263
Mayr, Peter 263
McKellar, B.H.J. 253
McNamara, J.M. 254
McQuarry, George 263
Meekhof, D. M. 257
Meekhof, D.M. 256
Mensky, M.B. 247
Merrit, John 263

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

K
K. Lynch, David 266
Köppe, Thomas 263
Kan, A.H.G. Rinnooy 253
Kapitza 195
Karl, G. 252
Katori, H. 261
Kauffman, L.H. 262
Kauffman, Louis H. 262

Keller, C. 251
Keller, Joseph 182
Kells, W. 254
Kelu, Jonatan 263
Kerschbaum, Franz 178, 266
Ketterle, W. 250, 257
Kidd, R. 250
Kielpinski, D. 256
King, B. E. 257
King, B.E. 256
Kinoshita, T. 260
Kirchhoff, Gustav 180
Kiss, Joseph 263
Klauder, J.R. 250
Klaus Tschira Foundation 264
Klein, A. 259
Klein, Oskar 192
Kloor, H. 249
Kneib, J.-P. 258
Kochen, S. 163, 258
Kockel, B. 259
Koecher, Max 262
Koolen, Anna 263
Kostner, Andreas 177, 266
Krachmalnicoff, V. 254
Krexner, G. 252
Krijn, Marcel 263
Kronig, Ralph 104
Kryowonos, A. 252
Królikowski, Jarosław 263
Kubala, Adrian 263
Kuhr, S. 256
Kurn, D.M. 250, 257
Kuzin, Pavel 263
Kuščer, I. 252
Küster, Johannes 264

Motion Mountain – The Adventure of Physics

J
Jackson, John David 260
Jacobson, J. 249
Jalink, Kim 263
Jamil, M. 263
Jammer, Max 247
Janek, Jürgen 263
Janssen, Jules 181
Jeltes, T. 254
Jin, Y. 254
Johansson, Mikael 263
Johnson, Samuel 247
Jones, Quentin David 263
Jones, Tony 260
Jong, Marc de 263
Joos, E. 256
Joos, Erich 256
Jordan, Pascual 22, 25
Joseph Bohm, David
life 154
Joyce, James 152
Ju, L. 261
Jungmann, J.H. 259
Jönsson, C. 251
Jönsson, Claus 76, 77, 265

name index

name index

M
Metikas

P
Pádua, de 55
Padgett, M. 251
Page, Don 263
Pahaut, Serge 85, 263
Pan, J.-W. 249, 258
Papini, G. 251
Parks, David 263
Pascal, Blaise
life 46
Pascazio, S. 251
Pascazio, Saverio 263
Paschotta, Rüdiger 50, 58,
264, 265
Pasi, Enrico 263
Paul, H. 250
Pauli, W. 255
Pauli, Wolfgang 25, 61, 133,
196, 260
life 105
Paulsen, C. 257
Payne, Cecilia
life 181
Peeters, Bert 263
Pello, R. 258
Penrose, R. 255
Peredo, M. 249
Pereira, Silvania 264
Peres, Asher 248
Perini, Romano 263
Perrin, A. 254
Perrin, J. 251
Perrin, Jean 79

Philips, William 195
Phillips, Gregory 176, 265
Photonics, Hamamatsu 264
Pinkse, P. 260
Pittman, T.B. 253
Planck, Erwin 248
Planck, Max 20, 47, 55, 105,
247
life 17
Planemad 173, 265
Plastino, A. 254
Plastino, A.R. 254
Plaçais, B. 254
Plunkett, Altus 176, 265
Podolsky, B. 257
Poincaré, Henri 55, 249
Popescu, S. 254
Prentis, J.J. 249
Prestel, Alexander 262
Prewitt, O. 255
Pritchard, Carol 263
Pritchard, D.E. 252, 259
Pritchard, David 85
Proença, Nuno 263
Pslawinski 172, 265
Purves, William 263
Pádua, S. de 249
Pérez-Rodriguez, L. 258
R
Rahtz, Sebastian 264
Raimond, J.-M. 256, 257
Raimond, J.M. 256
Ramberg, E. 122, 254
Randi, James 260
Rankl, Wolfgang 263
Redondi, Pietro 263
Reichert, Alfons 178, 266
Reichl, Linda 256
Remmert, Reinhold 262
Rempe, G. 260
Renselle, Doug 263
Reppisch, Michael 263
Rhodes, C. 260
Richard, J. 258
Rieke, F. 248, 249
Rindler, W. 255
Rivas, Martin 263
Robbins, J.M. 255

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

N
Nagaoka Hantaro 79
Nagaoka, H. 251
Nairz, O. 251
Namouni, Fathi 263
NASA 172, 265
Nassau, Kurt 258
Neukammer, J. 259
Neukirch, Jürgen 262
Neumaier, Heinrich 263
Neumann, John von 144
life 163
Neumann, János
life 163
Newell, D.B. 261
Newton 70
Newton, T.D. 106, 252
Nienhuis, G. 252
Niepraschk, Rolf 264
Nieto, M.M. 251
Nieuwpoort, Frans van 263
Nio, M. 260
Norbury, John W. 258
Novikov, V.A. 252

O
Oberdiek, Heiko 264
Oberquell, Brian 263
Odom, B. 260
Offner, Carl 263
Ohanian, Hans C. 255
Ohmiya, A. 258
Olmsted, John M.H. 238
Oostrum, Piet van 263
Opalsnopals 178, 266
Orlando, T.P. 257
Orlitzky, A. 253
Osowiecki, Gaël 51, 264
Ou, Z.Y. 253
Overhauser, A.W. 252

Motion Mountain – The Adventure of Physics

Metikas, G. 253
Mewes, M.-O. 250
Meyenn, K. von 252
Meyer, J.C. 242
Michaelson, P.F. 261
Miesner, H.-J. 257
Migdall, A. 253
Milonni, P.W. 249
Misner, C.W. 255
Mitchell, M.W. 249
Mlynek, J. 249
Mohr, P.J. 261
Monken, C.H. 249
Monroe, C. 256, 257
Mooij, J.E. 257
Moortel, Dirk Van de 263
Moser, Lukas Fabian 263
Mullins, O.C. 258
Murdock, Ron 263
Murillo, Nadia 263
Mutka, Miles 263
Muynck, Wim de 263
Myatt, C.J. 256

271

272

R
Robertson

T
Tacitus 199

Takamoto, M. 261
Tanaka, Y. 258
Tarko, Vlad 263
Taylor, B.N. 261
Taylor, J.H. 261
Taylor, John R. 261
Tegelaar, Paul 263
Tegmark, M. 256, 258
Tetrode, Hugo 114
Thaler, Jon 263
Thaller, Bernd 90, 94, 95, 99,
110, 190, 192, 264
Thies, Ingo 263
Thomas, L. 252
Thomas, Llewellyn 105
Thorne, K.S. 255
Tian, L. 257
Tiberius 199
Tino, G. 255
Tittel, Wolfgang 250
Tollett, J.J. 250
Tonomura Akira 77, 265
Townsend, C.G. 257
Townsend, Paul 263
Trevorrow, Andrew 264
Tschira, Klaus 264
Tuominen, M.T. 251
Tuppen, Lawrence 263
Turchette, Q.A. 256
Twamley, J. 255
Twiss, R.Q. 249
Twiss, Richard 52
Tyler, John 16, 264
Török, Gy. 252
U
Uguzzoni, Arnaldo 263
Uhlenbeck, G.E. 252
Uhlenbeck, George 104
Upright, Craig 263
Ursin, R. 250
V
Vaidman, Lev 69
Vanier, J. 260
Vannoni, Paul 263
Vardi, Ilan 253
Vassen, W. 254
Vavilov, S.I. 40, 248

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

S
S.R. Madhu Rao 263
Sabbata, V. de 252
Sackett, C.A. 250, 256
Sackur, Otto 114
Sagan, Hans 238
Saghian, Damoon 263
Salam, Abdus 248
Sasaki, N. 258
Sayrin, C. 256
Schaerer, D. 258
Schellekens, M. 254
Schiller, Britta 263, 264
Schiller, Christoph 266
Schiller, Friedrich
life 40
Schiller, Isabella 263
Schiller, Peter 263
Schiller, S. 249, 266
Schiller, Stephan 263
Schleich, W.P. 253, 256
Schmiedmayer, J. 252
Schrödinger, E. 252
Schrödinger, Erwin 35, 182
life 92
Schubert, Max 248
Schucking, E. 252
Schwenk, Jörg 169
Schwinger, Julian 102, 103, 248
Schönenberger, C. 254
Scott, Jonathan 263
Seidelmann, P. Kenneth 261

Selmer 175, 265
Serena, P.A. 250, 251
Sergienko, A.V. 253
Shandro, Marc 173, 265
Sharkov, I. 252
Sharp, Nigel 180, 266
Shaw, George Bernard 72
Sheldon, Eric 263
Shepp, L. 253
Sherman, Jeff 177, 266
Shih, Y.H. 253
Shimony, A. 258
Siart, Uwe 264
Sierra, Bert 263
Silagadze, Z.K. 259
Silverman, M.P. 252
Silverman, Mark 194
Silverman, Mark P. 259
Simon, C. 254
Simon, Julia 263
Singleton, Douglas 263
Sivaram, C. 252
Slabber, André 263
Smirnov, S.N. 259
Smith, J.F. 257
Snow, G.A. 122, 254
Solomatin, Vitaliy 263
Sommerfeld, A. 260
Sommerfeld, Arnold 186, 188,
196
Specker, E.P. 163, 258
Stedman, G.E. 261
Steinberg, A.M. 249
Stern, O. 251
Stern, Otto
life 83
Stodolna, A.S. 259
Stodolna, Aneta 185, 186, 266
Story, Don 264
Strekalov, D.V. 253
Strunz, W.T. 256
Styer, D. 257
Subitzky, Edward 248
Sudarshan, E. C. G. 255
Sudarshan, E.C.G. 250
Surdin, Vladimir 263

Motion Mountain – The Adventure of Physics

Robertson, Will 264
Robicheaux, F. 259
Rogach, Andrey 175, 265
Roger, G. 257
Roos, Hans 248
Rosen, N. 257
Rosenfeld 65
Rosenfeld, L. 250
Rouzée, A. 259
Ruben, Gary 263
Rubin, M.H. 253
Rucker, Rudy 262
Rudolph, Peter 263
Ruff, G.A. 259
Ruiz-Morales, Y. 258
Rydberg, Johannes 182

name index

name index

V
Verbeelen

Verbeelen, Raf 265
Viana Gomes, J. 254
Vico, Giambattista
life 165
Viñuda, J. 258
Volder, Linda de 16, 264
Volin, Leo 263
Vos-Andreae, J. 251
Voss, Herbert 264
Voss, R.F. 257
Vrakking, M.J.J. 259
Vuorinen, R.T. 261

Wright, Joseph 264
Wunderlich, C. 256
Wunderling, Helmut 262
Y
Yamamoto, Y. 249
Yao, E. 251
Yazdani, A. 253
Young, Andrew 41, 263, 264
Z
Zaccone, Rick 264
Zalm, Peer 263
Zawischa, Dietrich 58, 265
Zbinden, H. 250
Zecherle, Markus 263
Zedler, Michael 263
Zeh, H.D. 256
Zeh, Hans Dieter 256
Zeh, Heinz Dieter 144
Zeilinger, A. 250, 251, 256, 258,
261
Zeilinger, Anton 109
Zetti, A. 242
Zhao, C. 261
Zouw, G. van der 251
Zurek, W.H. 122, 253, 254
Zurek, Wojciech H. 255, 256
Zuse, Konrad 163
Zwerger, W. 257

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

Wheeler, John A. 255
Widom, A. 252
Wierda, Gerben 263
Wierzbicka, Anna 263
Wiesner, S.J. 254
Wiesner, Stephen 122
Wigner, E. 254
Wigner, E.P. 106, 252
Wigner, Eugene
life 125
Wigner, Eugene P. 248
Wijk, Mike van 263
Wikimedia 187, 265, 266
Wiley VCH 101, 265
Wiley/VCH 266
Willemsen, Evelien 175, 265
Wineland, D. J. 257
Wineland, D.J. 256
Wise, N.W. 261
Witt, Bryce de 258
Wittke, James P. 251
Woerdman, J.P. 252
Wolfenstätter, Klaus-Dieter
169
Wolfsried, Stephan 173, 175,
265
Wollaston, William 180
Wolsky, A.M. 253
Wootters, W.K. 254
Wootters, W.L. 122
Wouthuysen, S.A. 106, 252
Wright, E.M. 262

Motion Mountain – The Adventure of Physics

W
Wal, C.H. van der 257
Walkerma 175, 265
Wall, Nathan 174, 265
Walther, P. 249
Warkentin, John 264
Waser, A. 262
Webb, R.A. 257
Weber, Gerhard 248
Wehinger, S. 252
Weinberg, Steven 108, 143, 250
Weinfurter, H. 258
Weiss, Martha 263
Weisskopf, Victor 257
Werner, S.A. 252
Westbrook, C.I. 254
Weyl, Hermann 130
Wheeler, J.A. 255, 258
Wheeler, John 239

273

SU B J E C T I N DE X

single 156
size 197
size of 21
atomic mass unit 129, 216
atto 207
aurora 172
average 149
Avogadro’s number 217
axiom
definition 224
axis
of rotation 82
azimuthal quantum number
187
B
Baker–Campbell–Hausdorff
formula 238
Balinese candle dance 130
baryon number density 221
base units 205
basis of vector space 237
bath 151
physical 146
BCH formula 238
beans, dangers of 37
beauty 129
becquerel 207
bell
and exclusion principle 136
Bell’s inequality 164
belt trick 130, 139–142, 230,
264
Benham’s wheel 178
Bennett–Brassard protocol
123
betalains 179

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

in lower left corner 153
annihilation operator 121
anthocyanins 179
anti-bunching 54
anticommutator bracket 121
antimatter see antiparticle, 192
antiparticles 201
anyon 142
aphelion 219
apogee 219
apparatus
classical 166
definition 160
irreversible 166
APS 266
arc lamp 172
argon lamp 172
arm 131, 230
arrow
rotating 89
arrows
rotating 89
artefact
for measurement units 206
asphaltenes 171
astrology 201
astronomical unit 220
atmosphere
pressure 218
atom
and electronium 110
and senses 17
finite size of 136
handling of single 195
hollow 195
rotation 85
shape of 185

Motion Mountain – The Adventure of Physics

A
acausality 153
acceleration
Coriolis 194
maximum 79
Planck 209
quantum limit 79
accuracy 212
limits to 214
action
EBK 188
Planck 209
quantum of 198
action, quantum of, ℏ 18
physics and 8
addition 224
ageing 29
Aharonov–Bohm effect 98
Aharonov–Casher effect 100
Alectoris rufa 258
algebra, alternative 233
ampere
definition 205
amplitude
and complex numbers 227
angelology 255
angels 199
and quantum theory
135–137
and the exclusion principle
136
angular momentum
indeterminacy relation 82
intrinsic 83
of electron 138
smallest measured 202
animation

subject index

B

in quantum theory 80, 85
quantum 79
CODATA 261
coherence 144, 156
definition 101
length 54
of cars 78
of electrons 101–102
time 53
transversal 101
coherence length 52, 60
coherence time 52
coherence volume 60
coherent 151
collapse
of the wave function 93, 153
definition 158
formula 162
of wave function 166
colour 44
charge 129
first summary on 197
origin of 171
colour causes
table of 172–179
colour centres 176
colours 197
Commission Internationale
des Poids et Mesures 205
commutation of Hamiltonian
and momentum operator
109
commutation, lack of 35
commutative 224
complementarity 78
complementarity principle 37,
78
completeness property of sets
224
complex conjugate 225
complex number 225–227
as arrow 226
compositeness 107
criteria for 107–108
Compton (wave)length 108
Compton scattering 70
Compton wavelength 202, 217
computer
universe not a 169

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

C
candela
definition 206
candle colour 172
cans of beans, dangers of 37
car
and garage 96

on highways 78
theft 96
cardinals 235
carotenoids 179
Casimir effect 202
cat
Schrödinger’s 144
causality 162
Cayley algebra 233
Cayley numbers 233
centi 207
centre, quaternion 231
Čerenkov radiation 172
CERN 211
challenge
classification 9
change
measured by action 18
quantum of 18
quantum of, precise value
215
characteristic 224
charge
elementary ?, physics and
8
positron or electron, value
of 215
charge inversion 126
charm quark 128
mass 216
chimaera 123
chlorine 173
classical physics
allows no measurements 15
defines no scales 15
lack of precision 201–203
limits of 15
no length and time scales
15
classification
of concepts 223
cleveite 181
clocks 26
clone
biological 124
physical 122–124
cloud
quantum 91
clouds

Motion Mountain – The Adventure of Physics

bioluminescence 174
biphoton 55
BIPM 205, 206, 208
bits to entropy conversion 218
blood colour 174
blue colour
of the sea 194
of water 194
body
rigid 37
Bohm’s thought experiment
154
Bohr magneton 104, 217
Bohr radius 186, 217
bioluminescenceBoltzmann constant 149
discovery of 17
Boltzmann constant ? 215
physics and 8
bomb
triggered by a
single-photon 69
bond
chemical 80
Bose–Einstein condensate 74,
253
bosons 63, 118, 121
bottom quark 129
mass 216
bottomness 129
Bragg diffraction 70
brain 97
breaking 29
Bremsstrahlung 172
Brillouin scattering 70
bromine 173
Bronshtein cube 8
bulge
as quantum particle 120
Bureau International des
Poids et Mesures 205
butterfly 15, 16

275

276

C
compu ter

Bronshtein 8
physics 8
current
Planck 209
curve
space filling 238
cyclotron frequency 217

E
Earth
age 218
average density 218
equatorial radius 218
flattening 218
gravitational length 218
mass 218
normal gravity 218
radius 218
EBK quantization 182
edge
is never sharp 109
eigenfunction 161
eigenstates 88
eigenvalue 88
and measurement 158
definition 158
of velocity 106
eigenvector 88, 158
definition 158
eigenvectors 88
eight-squares theorem 234
Einstein–Podolsky–Rosen

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

D
daemons 199
damping 148
dance 131
day
sidereal 218
time unit 207
de Broglie wavelength 206
death 29, 150
deca 207
decay 200
deci 207
decoherence 36, 146–167
of light 156
process 145
time 147, 149
decoherence process 157
degree
angle unit 207
degree Celsius 207
Dendrobates azureus 177
density
Planck 209
density functional 145
density matrix 145
detachable 152
detector 160
determinism 167
deviation
standard, illustration 213
devils 199
different 59
diffraction
and scattering 70
as colour cause 178
definition of 60
of gratings 62
of matter by light 195
of quantum states 93
pattern 157
dimension 237

dimensionless 216
dimensions, three spatial 135
disentanglement 145, 152
disentanglement process 157
disinformation 39
dispersion 94, 176
of wave functions 95
distinction
macroscopic 144
distribution
Gaussian 212
normal 212
division 224
division algebra 231
donate
to this book 10
Doppler effect 182
double cover 231
double numbers 235
down quark 128
mass 216
dwarfs
none in nature 21
dyadic product 145

Motion Mountain – The Adventure of Physics

computer science and
quantum theory 35
concepts
classification of 223
condensate 242
condom problem 112
conductance quantum 217
conductivity
quantization of 74–76
cones, in the retina 249
Conférence Générale des
Poids et Mesures 205
configuration space 135
Conférence Générale des
Poids et Mesures 206
consciousness 166
not of importance in
quantum theory 166
constants
table of astronomical 218
table of basic physical 214
table of cosmological 220
table of derived physical
217
Convention du Mètre 205
copy, perfect 122
copying machine 35
definition 122
function 123
lack of 123
Coriolis acceleration in atoms
194
corrected Planck units 210
cosmological constant 220
coulomb 207
Coulomb gauge 153
coupling minimal 190
CPT 105
cream
whipped 22
creation 193
creation operator 121
cross product 234
cryptoanalysis 169
cryptography 169
cryptography, quantum 123
cryptology 169
cryptology, quantum 169
cube

subject index

subject index

E
Ekert

evolution
equation, first order 94
evolution equation 92
Exa 207
excitations in gases 172
exclusion principle 135–137
and angels 136
expansion
periodic decimal 235
explanation 167
eye and the detection of
photons 40

G
g-factor 107
?-factor 104
G-parity 129
Galileo and quanta 24
gas
simple 113
gas constant, universal 217
gas lasers 172
gases 113
gauge, Coulomb 153
Gaussian distribution 212
Gaussian integers 235
Gaussian primes 235
Gaussian unit system 210
Gedanken experiment see
thought experiment
gelatine 198
generators 228
genius 55
ghosts 142, 199
giants
none in nature 21
Gibbs’ paradox 114
Giga 207
Glauber state 48
glove problem 112
gloves 124
glow worm 16
glow-worms 174
gluon 128, 216
goddesses 199
gods 165, 196, 199
gold
yellow colour 195
graphics, three-dimensional
231
grating 62

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

F
fall, free 19
farad 207
Faraday’s constant 217
femto 207
fencing 139
Fermi coupling constant 215
fermion
no coherence 141
fermions 118, 121
field, mathematical 224
field, number 224
film
and action 17
in lower left corner 153
fine structure 188
fine-structure constant 186,
188, 196, 197, 203, 209, 215,
216
fire colour 172
firework colour 172
first property of quantum
measurements 158
flashlamp colour 172
flight simulation 231
floor
why it does not fall 136
flowers 179
flows
are made of particles 74
must fluctuate 74
fluctuations 146
Fock states 49
foundation
of quantum physics 17
four-momentum 126

four-squares theorem 228
fractals 37
do not appear in nature 85
Fraunhofer lines 180
French railroad distance 236
friction 148
full width at half maximum
213
fuzziness
fundamental 74

Motion Mountain – The Adventure of Physics

paradox 154
Ekert protocol 123
electrodynamics 227
electromagnetic coupling
constant
see fine structure constant
electromagnetic unit system
210
electromagnetism, strength of
196
electron 128
classical radius 217
g-factor 217
interference 101
magnetic moment 217
mass 215
radius 138
Trojan 110
electron volt 211
value 218
electronium 110
electrostatic unit system 210
elementary particle
see also particle
emotion
is a quantum process 17
energy
Planck 209
energy levels 184
energy width 129
ensemble 114
entangled systems 35
entanglement 36, 152, 154
entanglement, degree of 156
entropy
Planck 209
environment 146
EPR 123, 154
equilibrium 146
error
in measurements 212
random 212
relative 213
systematic 213
total 213
escape velocity 184
Euclidean vector space 237
eumelanin 174
europium 173

277

278

G

I
ice colour 173
ice, blue 173
images 201
immediate 154
impenetrability
of matter 142, 200
impenetrability of matter 29,
139
incandescence 18, 172
indeterminacy principle
see indeterminacy relation
temperature-time 86
indeterminacy relation
extended 141
for angular momentum 82
for many fermions 141
indeterminacy relations 25, 78
indistinguishable 114

indoctrination 39
information science and
quantum theory 35
inhomogeneous Lorentz
group 125
inner product 236
inner product spaces 236
inorganic charge transfer 175
integers 223
interference 144
and bombs 69
and photons 56–60
as colour cause 177
fringes 57
of electrons 101
of photons 66
quantum 93
interferometer 51
for matter 77
picture of 51
interferometers 212
intermediate bosons 107
International Astronomical
Union 220
International Geodesic Union
220
interpenetration
of atoms and bonds 80
of light vs. matter 140
of matter 136
interpretation
of quantum mechanics 167
interpretation of quantum
mechanics 144
invariant
see also action, quantum of
see also Lorentz invariance
see also Planck units
see also speed of light
iodine 173
ionization energy 184
irreducible representation 125
irreversible 148
isotopes 122
IUPAC 261
IUPAP 261
J
Jarlskog invariant 215

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

H
H2 O 22
half-life 129
Hall effect
fractional quantum 142
Hamilton
function 106
Hamiltonian 92
Hanbury Brown-Twiss
experiment 63
Hanbury Brown–Twiss
effect 56
hand, for quaternion
visualization 230
Heaviside–Lorentz unit
system 210
hecto 207
Heisenberg picture 143, 155
Heisenberg’s indeterminacy
relations 78
helicity 45, 126
helium 107, 150, 181
atom 110
bunching 119
discovery of 181
in Sun 181
hemoglobin 179
henry 207
Hermitean vector space 237
hertz 207

hidden variables 163
Higgs boson 108, 129
Higgs mass 216
Hilbert space 88, 91, 237
Hiroshima 38
Hitachi 253
hologram
electron beam 101
homogeneous 229
horizon
motion and quantum
aspects 198
horseshoe 37
hour 207
Hubble parameter 220
human observer 166
hydrogen
atomic size 79, 110
atoms, existence of 109
colours of 181–184
colours of atomic 211
energy levels 92
heat capacity 85
imaging of 242
in Sun 181
in water 21
orbitals 80
hydrogen atoms 253
hyperreals 235

Motion Mountain – The Adventure of Physics

of light 195
gravitational constant
geocentric 218
heliocentric 219
gravitational constant ? 215
physics and 8
gravitational coupling
constant 215
graviton 35, 127
gray 207
ground state 184
group 224
group velocity 94
growth 31
gravitational Gulliver’s travels 21
gyromagnetic ratio 107
electron 202
Göttingen 24

subject index

subject index
Josephson effect 100, 206
Josephson frequency ratio 217
joule 207
Journal of Irreproducible
Results 248
Jupiter
properties 219

J
Josephson

and quantum physics 15,
204
is a quantum process 17
lifetime 129
lifetime, atomic 202
light 46
see also speed of light
coherent 48, 50
incoherent 156
intensity fluctuations 48
macroscopic 156
made of bosons 139, 140
non-classical 47–51
squeezed 47–51
thermal 48
tunnelling 97
light grating 195
light quanta 40, 46
light quantum 34
light year 218, 220
lightbulb 172
Lilliput 201
limits
to precision 214
linear spaces 236
linear vector spaces 236
linearity of quantum
mechanics 144
link, open 142
litre 207
locality 162
Lorentz group
inhomogeneous 125
Lorentz symmetry
see Lorentz invariance
Loschmidt’s number 217
lumen 207
luminary movement 46
luminous bodies 46
lux 207
Lyman-alpha line 182
M
macroscopic system 151
magic 203
magma colour 172
magnetic flux quantum 217
magnetite 175

magneton 107
magneton, nuclear 217
many worlds interpretation
167
marker
bad for learning 9
Maslov index 183
mass
Planck 209
mass ratio
muon–electron 217
neutron–electron 218
neutron–proton 218
proton–electron 217
material properties 196
first summary on 197
material research 196
materials science 196
materials, dense optically 62
matter
density of 141
motion of 72–111
size of 141
matter wavelength 202
maximum speed
see speed of light ?
measured 161
measurement
comparison 208
definition 205, 208
error definition 212
irreversibility 208
meaning 208
no infinite precision 73
precision see precision
process 208
vs. state 87–89
measurement apparatus 166
measurement results 88
measurements 88, 157
measurements disturb 166
Mega 207
megatonne 38
melanin 179
memory 97, 157, 159
mercury
liquid state of 195
mercury lamp 172
mesoscopic systems 24

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

L
Lagrangian operator 103
lake
blue colour 194
Lamb shift 202
Lampyris noctiluca 16
Laplace operator 92
laser
cavity 47
coherence 54
cooling 244
sword 139
Laue scattering 70
lava colour 172
lawyers 39
learning
best method for 9
without markers 9
without screens 9
Lego 17
length
coherence 52, 60
Planck 209
scale, not in classical
physics 15
length scales 201

life

Motion Mountain – The Adventure of Physics

K
kelvin
definition 205
kilo 207
kilogram
definition 205
kilotonne 38
Klitzing, von – constant 217
knocking
and the fermionic
character of matter 137
on tables 74
Korteweg–de Vries equation
110

279

280

M
metallic

bound quantum 107
is fundamental 206
of matter 72–111
of quantons 111
quantons and 198
motion backwards in time 27
motion detector
senses as 17
motion inversion 126
Motion Mountain
aims of book series 7
helping the project 10
supporting the project 10
mozzarella 23
multiplication 224
muon 128
anomalous magnetic
moment 202
g-factor 217
muon magnetic moment 217
muon mass 215
muon neutrino 128
muonium
hyperfine splitting 202
myoglobin 179

O
oaths
and the quantum of action
39
object 151
made of particles 198
tethered 130–135
observables 88
do not commute 35
observation 159
observations 157
observer
made of radiation 168
octaves 233
octonions 233
ohm 207
operator, adjoint 121
operators 88
orbit
inside atoms 181
order structure 224
order, total 224
ordinals 235
organic compounds 174
P
π, normality of 222
pair creation 202
paradox
EPR 154

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

N
nano 207
nanoscopic systems 24
natural unit 216
see also Planck units
nature 165
nature and computer science
35
neon lamp 172
Neumann, von, equation 145
neutrino 147
masses 215, 216
PMNS mixing matrix 215
neutrino, electron 128
neutron 107
Compton wavelength 218
magnetic moment 218
mass 216
new age 167
newton 207
Newtonian physics
see Galilean physics
no-cloning theorem 122, 123,

254
non-classical light 48, 55
non-local 153
non-unitarity 166
nonstandard analysis 235
norm 225, 228, 236
normality of π 222
North Pole 82
nuclear magneton 217
nuclear warhead 38
nucleus 83
number 223–235
double 235
field 224
hypercomplex 233, 234
theory 235
number states 49
nymphs 199

Motion Mountain – The Adventure of Physics

metallic bands 175
metre
definition 205
metre rules 26
metric space 236
micro 207
microscope 24
magnetic resonance force
105
microscopic system 151
definition 24
microscopic systems 24
microwave background
temperature 221
Mie scattering 70
mile 208
Milky Way
age 219
mass 219
size 219
milli 207
mind 166
minimal coupling 190
minimization of change
see least action
Minion Math font 264
minute 207
definition 220
mirror 60
mirrors 97
mixed state 145
mixing matrix
CKM quark 215
PMNS neutrino 215
molar volume 217
mole 122
definition 205
molecular vibrations and
rotations 173
molecule size 202
momentum
Planck 209
Moon
density 219
properties 219
Moore’s law 37
motion
and measurement units
206

subject index

subject index

P
parit y

constant 17
Planck’s constant 18, 44
Planck’s natural units 208
plankton 194
plants
flowering 179
plate trick 130
pleasure 17
is a quantum process 17
pointer 161
polarization 63, 176
polarization of light 45
police 96
position 168
positron 192
positron charge
specific 217
value of 215
potential
spherical 183
praesodymium 173
precision 212
limits to 214
no infinite measurement
73
of quantum theory
201–203
prefixes 207, 260
SI, table 207
prefixes, SI 207
principle
of complementarity 78
of least action 102
quantum 17
prison 39
probability 158
probability amplitude 165
probability distribution 80
product
vector 234
properties
intrinsic 199
proton 107
Compton wavelength 217
g factor 218
gyromagnetic ratio 218
magnetic moment 217
mass 216
specific charge 217

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

definition 227
of wave function 97–101
phase space cell 60
phase, thermodynamic 114
phasor space 48
pheomelanin 174
Philippine wine dance 130
philosophers 46
phosphorus 196
photochromism 176
photon 34, 128
detection without
absorption 196
faster than light 64–65
interference 66
localisation 51–54
mass 216
number density 221
position of 51–54
radio wave detection 241
virtual 64
photon as elementary particle
47
photon cloning 254
photon-photon scattering 202
photons 40, 43, 46, 63, 200
and interference 59
as arrows 57
to waves 63–64
photons and naked eye 40
photons, entangled 55
photons, eye detection of
single 44
photons, spin of 45
physics
map of 8
physics cube 8
pico 207
Planck action ℏ
see action, quantum of
Planck constant
value of 215
Planck constant ℏ
see action, quantum of
Planck stroll 211
Planck units
as limits 210
corrected 210
Planck’s (unreduced)

Motion Mountain – The Adventure of Physics

parity 129
parsec 218
particle 120
countability 117
elementary 125, 199
elementary, definition of
125
real, definition 193
simple 113
speed 94
virtual 64
virtual, definition 193
see also elementary
particle
see also matter
see also quanton
see also virtual particle
particle counting, limits to 193
pascal 207
passion
hiding 223
path integral formulation 102
paths 33
Paul trap 110
Pauli equation 105
Pauli exclusion principle 122,
133, 139, 140
Pauli pressure 136
Pauli spin matrices 231
Pauli’s exclusion principle see
exclusion principle
penetrability of matter 29
perfect copy 122
perigee 219
perihelion 219
periodic systems of the
elements 136
permanence 27, 168
permanganate 175
permeability
vacuum 215
permittivity
vacuum 215
permutation
of particles 112–124
permutation symmetry 116
Peta 207
phase 33
and complex numbers 225

281

282
proton radius 107
proton volt 211
pure 229
pure state 144
puzzle
glove 112

P
proton

R
radian 206
radiation

observers made of 168
radiative decay 202
radio interference 66
radioactivity 114
rainbow
and Sun’s composition 180
rainbows and the elements in
the Sun 180
RAM 97
Raman scattering 70
inverse 70
random-access memory 97
randomness
and quantum of action
31–33
randomness, experimental 158
rational coordinates 222
rational numbers 224
Rayleigh scattering 70
reactions 31
real numbers 224
real particle
definition 193
recognition 121
record 157
reflection 60
reflection, total
and light amplification 195
refraction 62, 176
refraction and photons 67
refraction of matter waves 85
relaxation 148
representation 226, 231, 233
irreducible 125
reservoir 146
rest 19
does not exist 199
lack of 72–74
rigidity 37
ring 223
unital 224
ring interferometers 212
robotics 231
rods in retina 249
rotation 137, 229
of atoms 85
ruby glass 177
Rydberg atoms 186
Rydberg constant 182, 202, 217

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

summary of motion 111
quantum particles
arrows and 200
clouds and 200
indistinguishability 200
interactions 200
phase of 200
waves and 200
quantum phase 89
quantum physics
see also quantum theory
as magic 203
finite precision and 201
for poets 15
fundamental discovery 17
in a nutshell 198–204
lack of infinitely small 198
life and 15, 204
precision of 201–203
probabilities in 200
quantum principle 17
quantum state 91
quantum states 89
quantum theory 22
see also quantum physics
interpretation 144
summary and main results
198–204
quantum theory and
computer science 35
quark
bottom 129
charm 128
down 128
mixing matrix 215
strange 128
top 129
up 128
quaternion
basic unit 228
conjugate 228
imaginary 229
quaternions 227
quaternions in astronomy 231
qubits 157

Motion Mountain – The Adventure of Physics

Q
q-numbers 235
QED 191
quanta
and Galileo 24
quanti, piccolissimi 24
quantization 44
quantization, EBK 182
quanton
see also particle
elementary 199
motion of 199–201
speed 94
summary of motion 111
quantons 46, 76, 198
quantum
origin of the term 22–24
quantum action 102
quantum action principle 103
quantum computers 123
quantum computing 157, 257
quantum cryptography 123
quantum cryptology 169
quantum electrodynamics 191
quantum field theory 122
quantum interference 93
quantum mechanical system
151
quantum mechanics
origin of the term 22
see also quantum physics
see also quantum theory
quantum mechanics applied
to single events 167
quantum money 243
quantum numbers 127, 129
quantum of action 17, 18, 198
precise value 215
quantum of change 18
quantum of circulation 217
quantum particle
as bulge 120

subject index

subject index

S

squeezed light 48, 55
standard deviation 212
illustration 213
star colours 172
state 143, 170
bound 110
bound, unusual 110
coherent 48
quantum 91
vs. measurement 87–89
state function 165
state sum 243
states 88
are rotating arrows 89
steel, hot 172
Stefan–Boltzmann black body
radiation constant 202, 218
steradian 206
Stern–Gerlach experiment 83,
87
stone 35
stones 30, 62, 76, 135, 198
strange quark 128
mass 216
strength of electromagnetism
196
string trick 130
strong coupling constant 215
Sun’s age 219
Sun’s lower photospheric
pressure 219
Sun’s luminosity 219
Sun’s mass 219
Sun’s surface gravity 219
superconducting quantum
interference devices 100
supernatural phenomena 201
superposition
coherent 144
incoherent 145
support
this book 10
surreals 235
symmetry
of physical system 125
Système International
d’Unités (SI) 205
system 143, 151
bound 110

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

definition 205
prefixes 207
supplementary 206
siemens 207
sievert 207
single events in quantum
mechanics 167
sizes of atoms 197
sizes of tings 197
skew field 224
smartphone
bad for learning 9
Smekal–Raman scattering 70
SO(3) 126
sodium 85
sodium nucleus 83
sodium street lamps 172
soliton 110
soul 199
sources 66
space
metric 236
space, linear 235
sparkler colour 172
sparks 172
spatial parity 126
special orthogonal group 230
spectrum 158
spectrum of hot objects 202
speed
of light ?
physics and 8
sperm 23
spin 83, 104–105, 125, 126, 200
1/2 104
magnitude definition 126
use of value 126
spin 1/2 and quaternions 230
spin and rotation 138
spin myth 137
spin–statistics theorem 140
spinor 135, 191
spinors 230
spirits 199
sponsor
this book 10
spreading
of wave function 94
squark 263

Motion Mountain – The Adventure of Physics

S
Sackur–Tetrode formula 114
samarium 173
sapphire 175
Sargasso Sea 194
scalar 235
scalar multiplication 235
scalar part of a quaternion
228
scalar product 236
scattering 176
definition 69
geometric 70
types of 69
Sackur–TetrodeSchrödinger euqation 91–93
Schrödinger picture 143
Schrödinger’s cat 144, 152
Schrödinger’s equation of
motion 92
Schwarzschild radius
as length unit 210
science fiction 139
scissor trick 130, 255
sea
blue colour 194
sea, bluest 194
second 207
definition 205, 220
second property of quantum
measurements: 158
sedenions 234
semi-ring 223, 224
semiconductor bands 175
sense
as motion detector 17
senses 17
separability 152
sesquilinear 237
sexuality 31
shape 22
of atom 185
shapes 79
shell, atomic 136
SI
new 208
prefixes
table of 207
units 205, 214
SI units

283

284
classical 26
definition in quatum
physics 155
macroscopic 24
microscopic 24
system, cloning of
macroscopic 123

S
system

U
udeko 207
Udekta 207
uncertainty see indeterminacy
relative 213
total 213
uncertainty principle
see indeterminacy relation
uncertainty relation 25, 78
see also indeterminacy
relation
understanding
quantum theory 39
unit 229
astronomical 218
unitarity 162, 166
unitary vector space 237
units 205
natural 208
non-SI 208

Planck’s 208
Planck’s naturalsee Planck
units, natural units
provincial 208
SI, definition 205
true natural 210
universe
initial conditions do not
exist 170
not a computer 169
wave function of 169
up quark 128
mass 216
V
vacuoles 179
vacuum 116, 194
see also space
motion and quantum
aspects 198
permeability 215
permittivity 215
state 121
wave resistance 217
vacuum polarization 193
value, absolute 225
vanishing 145
variable
hidden 163–165
variance 213
Vavilov–Čerenkov radiation
172
vector 229, 235
part of a quaternion 228
product 234
vector space 235
vector space, Euclidean 237
vector space, Hermitean 237
vector space, unitary 237
velocity
Planck 209
vendeko 207
Vendekta 207
video
bad for learning 9
viewpoint changes 88
virtual particle 64, 116
definition 193
virtual photons 64

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

mass 216
topness 129
touch
basis for 140
trace 146
train windows 32
transfinite number 235
transition metal compounds
173
transition metal impurities
173
tree
noise of falling 166
trick
belt 131
plate 131
scissor 131
tropical year 218
truth 129
fundamental 204
tunnelling 95–97, 200
of light 97
tunnelling effect 29, 96
TV tube 97
twin
exchange 121
two-squares theorem 225
Tyndall scattering 70

Motion Mountain – The Adventure of Physics

T
table
of colour causes 172–179
of precision of quantum
theory 201–203
tachyons 27
tau 128
tau mass 215
tau neutrino 128
tax collection 205
teaching
best method for 9
telekinesis 201
teleportation 157, 201
temperature
Planck 209
tensor product 145
Tera 207
terabyte 97
tesla 207
tether 130–135
thermal de Broglie
wavelength 150
thermodynamics, third ‘law’
of 72
third ‘law’ of
thermodynamics 72
Thomson scattering 70
time
coherence 52, 53, 60
Planck 209
scale, not in classical
physics 15
time of collapse 162
time scales 201
time travel 27
TNT 38
TNT energy content 218
Tom Thumb 36
tonne, or ton 207
top quark 129

subject index

subject index
volt 207

V
volt

phase of 97–101
spreading of 94
symmetry of 117
visualization 89–91
wave interference 66
wave–particle duality 46
wavelength
de Broglie 206
weak charge 129
weak isospin 129
weak mixing angle 215
weber 207
weko 207
Wekta 207
Wheeler–DeWitt equation
170
Wien’s displacement constant
202, 218
windows in trains 33
wine

and water 148
glass 72
World Geodetic System 220
X
X-rays 45
scattering 70
xenno 207
Xenta 207
Y
yocto 207
Yotta 207
Z
Z boson 128
mass 216
zepto 207
zero-point fluctuations 74
Zetta 207

Motion Mountain – The Adventure of Physics

W
W boson 128
mass 216
waiting
as quantum effect 20
water
blue colour 173, 194
watt 207
wave
and complex numbers 225
equation 92
evanescent 97
from photons 63–64
wave function 89, 91, 92, 165
as rotating cloud 98
collapse 93, 153
dispersion of 94
is a cloud 109

285

copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net

MOTION MOUNTAIN
The Adventure of Physics – Vol. IV
The Quantum of Change

How can we see single photons?
How do colours appear in nature?
What does ‘quantum’ mean?
What are the dangers of a can of beans?
Why are Gulliver’s travels impossible?
Is the vacuum empty?
What is the origin of decay?
Why is nature random?
Do perfect copying machines exist?
Answering these and other questions on motion,
this series gives an entertaining and mind-twisting
introduction into modern physics – one that is
surprising and challenging on every page.
Starting from everyday life, the adventure provides
an overview of modern results in mechanics,
heat, electromagnetism, relativity,
quantum physics and unification.
Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. He wrote this
book for his children and for all students, teachers and
readers interested in physics, the science of motion.

Pdf file available free of charge at
www.motionmountain.net