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"Christoph Schiller MOTION MOUNTAIN the adventure of physics quantum theory: smallest change www.motionmountain.eu Christoph Schiller Motion Mountain The Adventure of Physics Quantum Theory: The Smallest Change available free of charge at www.motionmountain.eu Editio vicesima secunda. Proprietas scriptoris © Christophori Schiller secundo anno Olympiadis vicesimae nonae. Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliquo, quod partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Twenty-second edition, second printing, ISBN 978-300-021946-7. Copyright © 2009 by Christoph Schiller, the second year of the 29th 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 at no charge. To Esther and Britta τῷ ἐμοὶ δαὶμονι Die Menschen stärken, die Sachen klären. Preface The present introduction to quantum physics is the result of a threefold aim I have pursued since 1990: to present the basics of motion in a way that is simple, up to date and vivid. In the structure of modern physics, shown in Figure 1, this introduction to quantum physics covers the lower left point. 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. Quantum theory is the result of a smallest action, Planck’s quantum of action. In other words, quantum theory follows from the observation that there is a minimum change in nature. 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 vivid, 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 – maybe – we understand the trick. When we look at nature, we often have the same experience. The text tries to intensify this by following a simple rule: on each page, there should be at least one surprise or provocation for the reader to think about. Numerous interesting challenges are proposed. Hints or answers to these are given in an appendix. Giving full rein to one’s curiosity and thought leads to the development of a strong and dependable character. The motto of the text, die Menschen stärken, die Sachen klären, a famous statement by Hartmut von Hentig on pedagogy, translates as: ‘To clarify things, to fortify people.’ Exploring any limit requires courage; and courage is also needed to abandon space and time as tools for the description of the world. Changing habits of thought produces fear, often hidden by anger; but we grow by overcoming our fears. The great adventures in life allow this: exploring love is one, exploring physics is another. Eindhoven and other places, 8 January 2009 “ Primum movere, deinde docere.* Antiquity ” Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Advice for learners In my experience as a teacher, there was one learning method that never failed to transform unsuccessful pupils into successful ones: if you read a book for study, summarize every section you read, in your own words, 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 * ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is often called motivating; both terms go back to the same Latin root. 8 preface PHYSICS: Describing motion with action. (Unified) theory of motion Adventures: understanding everything, intense fun with thinking, catching a glimpse of bliss What are space, time and quantum particles? General relativity Adventures: the night sky, measuring curved space, exploring black holes and the universe, space and time How do everyday, fast and large things move? Quantum theory with gravity Adventures: bouncing neutrons, understanding tree growth Quantum field theory Adventures: building accelerators, understanding quarks, stars, bombs and the basis of life, matter, radiation How do small things move? What are things? Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Classical gravity Adventures: climbing, skiing, space travel, the wonders of astronomy and geology Special relativity Adventures: understanding time dilation, length contraction and E=mc2 c h, e, k G Quantum theory Adventures: death, sexuality, biology, enjoying art, colours in nature, all high-tech business, medicine, chemistry, evolution Galilean physics, electricity and heat Adventures: sport, music, sailing, cooking, using electricity and computers, understanding the brain and people F I G U R E 1 A complete map of physics: the connections are deﬁned by the speed of light c, the gravitational constant G, the Planck constant h, the Boltzmann constant k and the elementary charge e. Copyright © Christoph Schiller November 1997–January 2009 words, aloud. You can do this alone in a room, or with friends, or while walking. If you do this with everything you read, you will reduce your learning and reading time significantly. In addition, 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. Using this book Text in green, as found in many marginal notes, is a link that can be clicked in a pdf reader. Green links can be bibliographic references, footnotes, cross references to other pages, challenge solutions or URLs of other websites. preface 9 Solutions and hints for challenges are given at the end of each part. Challenges are classified as research level (r), difficult (d), standard student level (s) and easy (e). Challenges of type r, d or s for which no solution has yet been included in the book are marked (ny). A request The text is and will remain free to download from the internet. In exchange, please send me a short email on the following issues: — What was unclear? — What story, topic, riddle, picture or movie did you miss? — What should be improved or corrected? Feedback on the specific points listed on the www.motionmountain.eu/help.html web page is most welcome of all. You can also add feedback directly to www.motionmountain. eu/wiki. On behalf of all other 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. But above all, enjoy the reading. Challenge 1 s Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Quantum Theory: The Smallest Change In our quest to understand how things move, we discover that there is a minimal change in nature, implying that motion is fuzzy, that matter is not permanent, that boxes are never tight, that matter is composed of elementary units, and that light and interactions are streams of particles. The minimal change explains why antimatter exists, why particles are unlike gloves, why copying machines do not exist, and that probabilities are reasonable. Contents 14 1 Minimum action – quantum theory for poets The effects of the quantum of action on rest 17 • The consequences of the quantum of action for objects 17 • Why ‘quantum’? 19 • The effect of the quantum of action on motion 21 • Quantum limits and surprises 23 • Transformation, growth and Democritus 24 • Randomness 27 • Waves 28 • Quantum flows 28 • Quantum information 29 • Curiosities and fun challenges about the quantum of action 30 L ight – the strange consequences of the quantum of action What is the faintest lamp? 32 • What is light? – Again 37 • The size of photons 38 • Are photons countable? – Squeezed light 38 • The positions of photons 41 • Are photons necessary? 42 • How can a wave be made up of particles? 44 • Can light move faster than light? – Virtual photons 50 • Indeterminacy of electric fields 51 • Curiosities and fun challenges about photons 51 Motion of mat ter – beyond cl assical physics Wine glasses and pencils – no rest 54 • Cool gas 55 • No infinite precision 55 • Flows and the quantization of matter 55 • Knocking tables and quantized conductivity 56 • Quantized liquid flow 57 • Matter quantons and their motion – matter waves 57 • Mass and acceleration of quantons 59 • Velocities of quantons 60 • Why are atoms not flat? Why do shapes exist? 60 • Rotation of quantons – the lack of north poles 61 • Silver, Stern and Gerlach – polarization of quantons 63 • The quantum description of physical systems 65 • The quantum description of motion 66 • The state evolution – dispersion of quantons 68 • Tunnelling and limits on memory – damping of quantons 70 • Phase and the Aharonov–Bohm effect 72 • The motion of quantons with spin 73 • Relativistic wave equations 74 • Virtual particles 75 • Compositeness 75 • Curiosities and fun challenges about quantum matter 77 Colours and other interactions bet ween light and mat ter What are stars made of? 79 • What determines the colours of atoms? 80 • Relativistic hydrogen 84 • Relativistic wave equations – again 85 • Antimatter 87 • Curiosities and fun challenges about colour 88 • The strength of electromagnetism 88 Permu tation of particles – are particles like gloves? Why does indistinguishability appear in nature? 92 • Can particles be counted? 93 • What is permutation symmetry? 93 • Indistinguishability and symmetry 94 • The behaviour of photons 95 • Bunching and antibunching 96 • The energy dependence of permutation symmetry 96 • Indistinguishability in quantum field theory 97 • How accurately is permutation symmetry verified? 98 • Copies, clones and gloves 99 Rotations and statistics – visualizing spin The belt trick 104 • Angels, Pauli’s exclusion principle and the hardness of matter 106 • Spin, statistics and composition 107 • Is spin a rotation about an axis? 108 • Why is fencing with laser beams impossible? 109 • Rotation requires antiparticles 109 • A summary on spin and indistinguishability 110 • Limits and open questions of quantum statistics 111 Superpositions and probabilities – quantum theory withou t ideolo gy Why are people either dead or alive? 32 2 54 3 79 4 90 5 101 6 112 112 7 contents 13 119 124 Summary on decoherence, life and death 118 What is a system? What is an object? Is quantum theory non-local? – A bit about the Einstein-Podolsky-Rosen paradox 120 • Curiosities and fun challenges about superpositions 122 What is all the fuss about measurements in quantum theory? Hidden variables 129 • Summary on probabilities and determinism 131 • What is the difference between space and time? 133 • Are we good observers? 134 • What connects information theory, cryptology and quantum theory? 134 • Is the universe a computer? 135 • Does the universe have a wave function? And initial conditions? 135 Biblio graphy Challenge hints and solu tions Credits Film credits 152 • Image credits 152 137 148 152 Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Chapter 1 MINIMUM ACTION – QUANTUM THEORY FOR POET S C Challenge 2 s Page ?? Page ?? 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, such as the one of Figure 2, we enjoy its bright colours, its wild smell, its soft and delicate shape or the fine details of its symmetry. None of the three classical descriptions of nature can explain any of these properties; neither do they explain the impression that the flower makes on our senses. Classical physics can describe certain aspects of the impression, but it cannot explain their origins. For such an explanation, we need quantum theory. In fact, we will discover that in life, every type of pleasure is an example of quantum motion.** Take any example of a pleasant situation: for example, a beautiful evening sky, a waterfall, a caress, or a happy child. Classical physics is not able to explain it: the colours, shapes and sizes involved remain mysterious. In the early days of physics, this limitation was not seen as a shortcoming, because neither senses nor material properties were thought to be related to motion – and pleasure was not considered a serious subject of investigation for a respectable researcher. However, we have since learned that our senses of 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 behaves unclassically. Therefore the inability of classical physics to describe matter and the senses is indeed due to its intrinsic limitations. * ‘Nature [in its workings] makes no jumps.’ ** The photograph on page 11 shows a female glow worm, Lampyris noctiluca, as found in the United Kingdom (© John Tyler, www.johntyler.co.uk/gwfacts.htm). “ Natura [in operationibus suis] non facit saltus.* 15th century ” Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Ref. 1 minimum action – quantum theory for poets 15 F I G U R E 2 An example of a quantum machine (© Ata Masafumi) Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu F I G U R E 3 Max Planck (1858–1947) Ref. 2 In fact, every failure of classical physics can be traced back to a single, fundamental discovery made in 1899 by Max Planck:* ⊳ In nature, actions smaller than ħ/2 = 0.53 ⋅ 10−34 Js are not observed. All attempts to observe actions smaller than this fail.** In other words, in nature – as in a good film – there is always some action. The existence of a minimal action – the so-called quantum principle – is in complete contrast with classical physics. (Why?) However, it has * For more about Max Planck and his accomplishments, see the footnote on page ??. ** In fact, the quantum principle cited here is a slight simplification: the constant originally introduced by Planck was the (unreduced) h = 2πħ. The factors 2π and 1/2 leading to the final quantum principle were found 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; this may be the first time in many decades that it has been used in a teaching text. Niels Bohr (b. 1885 Copenhagen, d. 1962 Copenhagen) made Copenhagen University into one of the centres 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. Copyright © Christoph Schiller November 1997–January 2009 Challenge 3 s Ref. 3 16 1 minimum action – quantum theory for poets F I G U R E 4 Niels Bohr (1885–1962) TA B L E 1 How to convince yourself and others that there is a minimum action, or minimum change ħ in nature Issue Local change or action values < ħ are not observed Change values < ħ are non-local Local change or action values < ħ cannot be produced Local change or action values < ħ cannot be imagined A smallest local change or action value ħ is consistent Method check all observations check all observations check all attempts solve all paradoxes 1 – show that all consequences, however weird, are confirmed by observation 2 – deduce quantum theory from it and check it Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Page ?? passed an enormous number of experimental tests, many of which we will encounter in this part of our mountain ascent. Planck discovered the principle when studying the properties of incandescent light, i.e., 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 space-time. Incidentally, the factor 1/2 results from historical accidents in the definition of the constant ħ (which is pronounced ‘aitch-bar’). Despite this missing factor, ħ is called the quantum of action, or alternatively Planck’s (reduced) constant. The quantum principle states that no experiment can measure an action smaller than ħ/2. 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. In physics – as in the theatre – action is a measure of the change occurring in a system. Therefore, a minimum action implies that there is a minimum change in nature. Thus the quantum of action would perhaps be better named the quantum of change. If we compare two observations, there will always be change between them. (What is called ‘change’ in everyday life is often called ‘change of state’ by physicists; the meaning is the same.) Can a minimum change exist in nature? Table 1 shows that we need to explore three Copyright © Christoph Schiller November 1997–January 2009 minimum action – quantum theory for poets 17 points to accept the idea. We need to show that no higher speed is observed in nature, that no higher speed can ever be observed, and show that all consequences of invariance, however weird they may be, apply to nature. In fact, this is all of quantum theory. 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 present some of its more surprising consequences. The effects of the quantum of action on r..."

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