The Pursuit of Reality: Narrative History of the Quantum and the Great Minds That Made it 9789819910304, 9789819910311

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Selçuk Ş. Bayın

The Pursuit of Reality Narrative History of the Quantum and the Great Minds That Made it

The Pursuit of Reality

Selçuk S. ¸ Bayın

The Pursuit of Reality Narrative History of the Quantum and the Great Minds That Made it

Selçuk S. ¸ Bayın Institute of Applied Mathematics Middle East Technical University Ankara, Turkey

ISBN 978-981-99-1030-4 ISBN 978-981-99-1031-1 (eBook) https://doi.org/10.1007/978-981-99-1031-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The twentieth century started with a deterministic universe that runs according to the laws of nature with certainty and independent of observers. In this universe, matter was continuous, and physical reality was directly accessible to observers. Uncertainties were only due to the limitations of our instruments, and the probabilities were either due to our lack of interest in the microscopic details of a system or due to the impracticality of obtaining such detailed descriptions−even though the microscopic properties existed and were well defined at all times. Space and time were absolute, and the geometry of space was Euclidean. In this universe, there was no limit to how fast objects could move. During the first thirty years, atoms were discovered, and then with Einstein’s special theory of relativity, we found that space and time are relative, and that there is an upper limit to how fast we could travel. Einstein’s iconic formula, E = mc2 showed that mass is concentrated form of energy, and that energy also has inertia. With the discovery of the general theory of relativity, we learned that gravitation is just a manifestation of the curvature of spacetime. While these revolutionary discoveries were shaking the foundations of our basic understanding of the universe, quantum theory was in the making by a handful of brilliant minds like Heisenberg, Bohr, Pauli, Schrödinger, and friends. Information era and most of the twenty-first-century technologies are based on the concepts of this remarkable theory that turned out to be one of the most successful theories ever devised by physicists. However, all this success came with a heavy price. With its new type of determinism, where probabilities are an essential part of the theory, and with its new concept of physical reality, quantum physics has completely undermined the way we make sense of this world at the level of its fundamental building blocks. The quantum world is counter-intuitive and sometimes downright weird. Einstein challenged this theory by saying “God does not play dice.” Nobel Laureate physicists like Richard Feynman admitted that they do not understand it. Yet, there is not a shred of evidence against quantum mechanics. As the successes of this theory kept accumulating, most physicists gave up philosophical concerns and decided to move on with its practical applications. Today, along with the drama and the conflicts of v

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the bright minds that helped build it, applications of quantum mechanics to quantum computers, spy-proof communication, and teleportation, etc. keep the public interest in the quantum theory and the microcosmos alive. In this book, we present a narrative history of this captivating theory with the new developments that will intrigue all inquisitive minds. This book is going to take not just physicists but all scientists, and science enthusiasts on an exhilarating journey through the intellectual history of the quantum that is turning out to be more alluring every day. In developing my approach to the material presented in this book, I have benefited from many discussions with numerous colleagues and from the books of my fellow authors. I also thank Prof. Önder Arslan and the Hematology Department of the Ankara University Faculty of Medicine for running a world-class hematology service. I thank my chairperson Prof. Sevtap Kestel in the Institute of Applied Mathematics at the Middle East Technical University, where I have always enjoyed the rich vibrant intellectual and congenial environment. IAM has been my home since 2010. I owe a great debt of gratitude to my editor Loyola D’Silva and the publication team at Springer Nature. Finally, last but not least, I thank my beloved wife Adalet and darling daughter Sumru, who is now a mature scientist at the University of Cambridge in the Gurdon Stem Cell Institute as a group leader, for making this universe the best universe for me. Ankara, Turkey December 2022

Selçuk S. ¸ Bayın

Contents

1

The Quest for Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Newton’s Dynamic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Who Needs the First Law? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Newton’s Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Who Invented Calculus? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 First Test of Gravitation in Laboratory . . . . . . . . . . . . . . . . . . . . . . 1.6 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Fictitious Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Mach’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Newton’s Corpuscular Theory of Light . . . . . . . . . . . . . . . . . . . . . 1.11 Young’s Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Fresnel and the Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Maxwell’s Theory and Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4 5 6 8 8 9 10 11 11 12 14 14

2

Perpetual Motion Dream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Entropy and the Laws of Thermodynamics . . . . . . . . . . . . . . . . . . 2.3 Nothing New Left to Discover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Atoms and the Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Entropy in Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Dark Clouds over the Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Planck’s Black Body Radiation Formula . . . . . . . . . . . . . . . . . . . . 2.8 Reluctant Revolutionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 New Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Boltzmann Distribution for Solids . . . . . . . . . . . . . . . . . . . 2.9.2 Boltzmann Distribution for Gases . . . . . . . . . . . . . . . . . . . 2.9.3 Bose-Einstein Distribution . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 19 21 21 22 24 25 26 29 29 30 31

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3

Born Rebellious . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Difficult Times for Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Patent Office Clerk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Miracle Year of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Brownian Motion and the Reality of Atoms . . . . . . . . . . . . . . . . . 3.6 Universe with a Limit to Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Moving Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Covariance Versus Invariance . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Problems with Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Nature of Physical Theories . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 A Stroke of Genius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Speed of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Minkowski Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Lorentz-FitzGerald Contraction . . . . . . . . . . . . . . . . . . . . . 3.8.2 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Regions of Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.4 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 An Iconic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 36 37 38 40 40 41 42 42 42 43 44 44 45 46 49 50 50 51 52

4

Atoms and Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Bohr at Cambridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bohr Meets Rutherford . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bohr at Manchester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Germ of an Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Master Stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 First Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Demon in Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Shell Model and Bohr Festspiele . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 De Broglie and Particle-Wave Duality . . . . . . . . . . . . . . . . . . . . . . 4.10 Einstein Was Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Electron Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 57 58 58 59 60 64 66 68 69 71 72

5

Revolution Within Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Footsteps of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Pauli and the Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 White Dwarfs and Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Spin Without Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Emergence of the New Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Heisenberg the Magician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Dirac Enters the Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 80 84 87 90 94

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6

Schrödiger’s Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Wave Equation and the Hydrogen Atom . . . . . . . . . . . . . . . . . . . . 6.2 First Reactions to Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Heisenberg, Schrödinger, and Bohr Encounter . . . . . . . . . . . . . . . 6.4 Born and the Meaning of  (Psi) . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Does God Play Dice? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 98 100 101 103 107

7

Complementarity Embraces Particles and Waves . . . . . . . . . . . . . . . . 7.1 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Heisenberg’s Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Bohr Returns from the Ski Trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Measurement Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Fifth Solvay Conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Double-Slit Experiment and Einstein—Bohr Debate . . . . . . . . . . 7.9 After the Fifth Solvay Conference . . . . . . . . . . . . . . . . . . . . . . . . .

109 111 112 114 116 117 118 119 121 124

8

Sixth Solvay Conference and Titans Meet Again . . . . . . . . . . . . . . . . . 8.1 Einstein’s Light Box Stuns Bohr . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bohr Could Not Believe His Eyes . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Bohr Turns the Tables Completely . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Tumultuous Years in Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 129 129 131

9

Einstein at Princeton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Einstein-Podolsky-Rosen Argument . . . . . . . . . . . . . . . . . . . . . . . 9.2 Reactions to EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Schrödinger and Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Schrödinger’s Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Einstein, Bohr Meetings at Princeton . . . . . . . . . . . . . . . . . . . . . . .

133 133 136 138 139 140

10 Bohm’s Hidden Variables and Bell’s Inequality . . . . . . . . . . . . . . . . . . 10.1 Quantum Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Reactions to Bohm’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Bell’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Bell’s Challenge and Clauser’s Acceptance . . . . . . . . . . . . . . . . . 10.5 Aspect’s Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Third Generation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Aftermath of Bell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Everett and Many-Worlds Interpretation . . . . . . . . . . . . . . . . . . . .

143 145 147 148 151 152 153 154 157

11 The Gist of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Entanglement and No Signaling Theorem . . . . . . . . . . . . . . . . . . . 11.2 Single-Particle Systems and Quantum Information . . . . . . . . . . . 11.3 Mach-Zehnder Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Mach-Zehnder Interferometer with a Channel-Blocker . . . . . . . . 11.5 Feynman’s Double Slit Thought Experiment . . . . . . . . . . . . . . . .

161 162 164 166 167 170

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11.6 11.7 11.8

Realization of Feynman’s Thought Experiment . . . . . . . . . . . . . . Scully-Drühl Version of the Double Slit Experiment . . . . . . . . . . Quantum Eraser and the Delayed Choice Thought Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Complementarity and the Quantum Eraser Experiment . . . . . . . 11.10 Finally the Delayed Choice Experiment Done . . . . . . . . . . . . . . . 11.11 Quantum Tunneling and More Quantum Weirdness . . . . . . . . . .

171 172

12 Can We Ever Hope to Understand Quantum Mechanics? . . . . . . . . . 12.1 How to Define the State of a System . . . . . . . . . . . . . . . . . . . . . . . 12.2 Weird or Just Counterintuitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Wigner’s Thought Experiment Realized in Laboratory . . . . . . . . 12.4 Non-locality and the EPR Thought Experiment . . . . . . . . . . . . . . 12.5 Does Quantum Mechanics Need Imaginary Numbers? . . . . . . . .

181 181 182 184 185 186

13 Navigating Between the Classical and the Quantum Worlds . . . . . . . 13.1 Coherence Is What Determines “Quantumness” . . . . . . . . . . . . . 13.2 Do All Objects Have a Wavefunction? . . . . . . . . . . . . . . . . . . . . . . 13.3 Where Does the Weirdness Begin or End? . . . . . . . . . . . . . . . . . . 13.4 The Essence of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 13.5 Defining Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Decoherence as a Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Pointer States and Einselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Quantum Darwinism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 190 190 191 191 193 194 195 198

14 Mathematics, Physics, and Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Laws of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Could the New Physics Be Hiding in Living Matter? . . . . . . . . . 14.3 Physics Versus Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Consciousness and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 14.5 Mathematics and Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Cracking the Brain’s Memory Code . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Is Mathematics the only Language for Nature? . . . . . . . . . . . . . . 14.8 Nature and Mankind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 202 205 205 206 207 209 210 211

174 176 178 179

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Chapter 1

The Quest for Knowledge

Since the beginning of time, as soon as humans open their eyes to this universe, they find themselves immersed in a boundless diversity of events. First, they shiver under the daunting task of surviving in such a complex and hostile environment. But they soon realize that this universe is not as belligerent as it looks and that there is some law and order to its ever-changing diversity. As they explore their environment, they notice that some of the stones they kick aimlessly on their path roll away without hurting them. They also recognize that the smaller ones go further than the bigger ones. They quickly learn not to kick large objects which hurt them. Eventually, the sun, to which at first they did not pay too much attention, slowly began to disappear—leaving them cold and in darkness. At first, this scares them a lot, but what a joy it must have been to survive the night and to see the sun appearing on the other side of the horizon. Even for modern humans, these feelings are not impossible to experience when nature is faced unprepared and alone. In his book: Until the End of Time, Brian Green shares his solo camping experience when he was challenged for a brief period in the woods with limited resources. After wasting his three matches in his unsuccessful attempts to build a fire, he says “... I found myself more deeply alone than I had ever been … . As the sun began to set and terror started to rise, I rolled out the sleeping bag, scurried in, and stared at the tarp hovering close above my face. I was just this side of panic.”1 Of course, Brian knows that the sun will rise again, but our subconscious mind quickly recalls the primeval fears and joys experienced by our forefathers. As humans wondered about their daily interactions with their environment, they realized that the order in this universe is not only reliable but also comprehensible. Small stones that did not hurt them, do not hurt them another day, another place. Even though the sun eventually disappears, leaving them cold and in darkness, they are now confident that it will reappear. In time, they learn to live in communities and 1

Green (2021), p. 186.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_1

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1 The Quest for Knowledge

Fig. 1.1 Sir Isaac Newton (1642–1727) working with light and equipment (Bausch and Lomb Inc., courtesy of AIP Emilio Segrè Visual Archives, Physics Today Collection)

develop languages to communicate with each other. As the quality and the number of observations they make increase, they begin to undertake projects that require careful recording and interpretation of data that spans several generations. As in Stonehenge, they even build agricultural computers to keep track of the crop times. For humans to search their place and role in this universe that make them feel weak in every way is an instinctive derive. This drive also gives them a tremendous survival advantage in evolution that helps them make peace with their vulnerabilities and apparent insignificance. Along this journey, they learn to develop primitive technologies that to a certain extent help them control their environment to make it a safer and a more predictable place to live. Eventually, they realize that they can go only so far with the everyday language they have developed to communicate with each other. For a deeper understanding of the universe, a new language, much richer, much more efficient, and much more in tune with the inner logic of the universe, was needed.

1.1 Newton’s Dynamic Theory

3

At this point, physics and mathematics begin to get acquainted. With the introduction of a coordinate system, which is one of the greatest constructions of the free human mind, the foundations of the connection between physics and mathematics become established. Once a coordinate system is defined, it becomes possible to reduce all the events and their processes in the universe to numbers. Now, the law and order that exists among these events and their processes can be searched among the numbers that represent them and can be expressed efficiently and economically in terms of mathematical constructs.

1.1 Newton’s Dynamic Theory From the tumbling motion of a small stone that they kicked around joyfully to the motions of planets, stars, and galaxies, it is now possible to understand and to describe in terms of Newton’s dynamical theory and his theory of gravitation (Fig. 1.1). Newtonian dynamics is stated in terms of his three laws, which have a rich lineage of great philosophers and scientists ranging from Aristotles to Ptolemy, from Copernicus to Galileo, Kepler, Hook, and to Leibniz, and others. Newton’s theory, also known as the Newton’s dynamical theory, is given in terms of his three laws. Newton’s first law, is basically a definition or a statement of how a free particle should move. It is shaped by our everyday experiences and expectations. However, it is also an idealization. Today, using air tracks and air tables, we can create low friction environments in laboratory for limited distances. But even in space, there is some friction. Satellites and the spacelab have to be boosted periodically to maintain their orbits. Newton’s first law is commonly stated as: If there is no net force acting on a body, then there is no change in its state of motion: If the body is initially at rest, it will remain at rest, and if it is in motion, it will continue its motion along a straight line with a constant velocity. Newton’s second law, explains any deviation from the expected path defined in the first law, by the presence of forces acting upon the body. The second law is generally stated as: Whenever there is a net force F, acting upon a body, it produces an acceleration a, proportional to the net force F acting upon the body. The proportionality constant m, is called the mass of the object, hence F = ma. Mass m is also called the inertia, which is the ability of an object to resist acceleration. Two different masses subjected to the same force will accelerate at different amounts; the one with the larger mass will accelerate less. For example, you can easily push a bicycle, but the same push will hardly get a car moving. Newton’s third law, is a consequence of the second law, which states that every action on a body produces an equal and opposite reaction. It is commonly stated as: For every force that acts on a body, there is a second force equal in magnitude and opposite in direction that acts upon another body. In other words, the mutual actions of two bodies on each other are always equal in magnitude and opposite in direction. Newton published his laws of motion and his law of universal gravitation in three books written in Latin; Mathematical Principles of Natural Philosophy (1687), usually referred as Principia.

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Fig. 1.2 Two boxes, m 1 and m 2 , are moving together under the influence of the force F acting on m 1 with the acceleration a. F12 is the force that m 2 exerts on m 1 and vice versa

As simple as they look, Newton’s equations are the culmination of centuries of human thought based on careful observation and experimentation. During the last couple of decades, quantum mechanics has started to enter our lives through computer chips and electronic devices; however, Newton’s theory is still a dominant force that shapes our intuition and science and engineering applications. Let us now scrutinize these laws more carefully. The second law says that when there is a net force acting on a body, its acceleration, i.e., the rate of change of velocity, is proportional to the force acting on it, where the proportionality constant is its mass or inertia. The third law is very important in identifying the forces acting on a body. However, the third law is also a direct consequence of the second law. Consider two boxes with masses m 1 and m 2 , moving under the influence of the force F acting on the first box (Fig. 1.2). The boxes are moving together along a straight line on a frictionless surface with the acceleration a. Since the boxes are also exerting forces on each other, let F12 be the force exerted by the second box onto the first box and F21 be the force that the first box exerts onto the second box. Newton’s second law for the entire system implies F = (m 1 + m 2 )a. We now consider only the first box and write the second law for m 1 as F + F12 = m 1 a. Since the only force acting on m 2 is F21 , we write the second law for m 2 as F21 = m 2 a. Adding the last two equations: F + F12 + F21 = (m 1 + m 2 )a, and using the first equation, we obtain F12 + F21 = 0. As the third law says, the two boxes exert forces on each other that are equal in magnitude but opposite in direction, that is, F12 = −F21 . Newton used his third law to derive the law of conservation of momentum.

1.2 Who Needs the First Law? When there is no net force acting on an object and since the mass is always different from zero, the second law implies that the acceleration is also zero. This means that the velocity of the object does not change. In other words, if the object is initially at rest, it will remain at rest, and if it is moving, it will continue to move along a straight line with constant velocity. We may be inclined to think that the first law is a special case of the second law, hence redundant. Actually, the first law is the most important of all the three laws. It is the foundation on which the second law is built. It defines how a free object that has no net force acting on it should move or behave. When we observe an object deviating from this behavior, the second law explains the difference in terms of forces acting on

1.3 Newton’s Gravitation

5

the object. For a different definition of the first law, the force needed to explain the difference would be different. Is it than possible to modify or even to get rid of the force concept all together with an appropriate definition of the first law? Newton’s first law is so intuitive and natural that one may wonder what other choices one may have. At the time of Aristotle, all known forces were contact forces. For objects to exert force on each other, they had to be touching. For motion, objects had to be pulled or pushed by an agent. Otherwise, everything eventually came to rest. Gravitation was not known yet. Stars and planets were thought as heavenly bodies that did not interact with matter. For Aristotle, an intuitive definition of the first law was “when there is no net force acting on a body, its natural state is to be at rest.” In principle, it is possible to build a dynamical theory based on Aristotle’s first law. However, such a theory would have an immediate problem; one would need a force to explain the motion of an object moving freely in space with uniform velocity. Such a force would not have any physical agent that could be attributed to it. In other words, it would be a “fictitious” force—a term that will be encountered in Newton’s theory with the accelerating reference frames. In 1916, Einstein discovered that it is possible to eliminate gravitational forces by redefining the first law as “free particles follow the geodesics of the spacetime geometry,” where the geodesics are the shortest paths between two points. For a given mass/energy distribution, spacetime geometry is found from Einstein’s general theory of relativity by using his geometric field equations.

1.3 Newton’s Gravitation In addition to his dynamical theory, Newton also discovered that the observations of great scientists like Copernicus, Galileo, and Kepler can be summarized in terms er . This simple expression of his famous law of gravitation: F = −G(m 1 m 2 /r 2 ) encompasses all earthly phenomena, as well as the motion of planets, structure of stars, and the behavior of galaxies. It displays the elegance and the efficiency of mathematical description. In words, Newton’s law of gravitation says that two-point masses, m 1 and m 2 , attract each other along the line joining them with a force directly proportional to their masses and inversely proportional to the square of their separation. The proportionality constant is the gravitational constant G = 6.67 × 10−8 dyn cm2 gm−2 , where 10−8 is 0.00000001 or 7 zeros to the right-hand side of the decimal point before 1. Such forces are called central forces. For extended objects, we sum over the gravitational forces that the mass elements of the two objects exert on each other. The mass in Newton’s gravitation is the same as the mass, or the inertia, in Newton’s second law. Historically, Galileo is credited for demonstrating this point by dropping two objects with different masses from the leaning tower of Pisa and thereby demonstrating that they both fall at the same time. Equivalence of the gravitational and the inertial mass is known as the “equivalence principle” and played a key role

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Fig. 1.3 Etched portrait of Gottfried Wilhelm von Leibniz, 1646–1716 (AIP Emilio Segrè Visual Archives, E. Scott Barr Collection)

in the discovery of Einstein’s general theory of relativity, which is essentially a geometric theory of gravitation. Equivalence principle was verified to one part in 108 (1 with eight zeros) by the Hungarian physicist Eötvos in 1910.2 In 2022, the French mission put the equality of the gravitational and the inertial mass to test via the MICROSCOPE Satellite3 and found no violation to a few parts in a thousand trillion (1015 ).

1.4 Who Invented Calculus? Starting with the ancient Greeks, calculus was developed over centuries of human thought, experimentation, and observation. But when it comes to the question; who discovered calculus? two names, Isaac Newton and Gottfried Leibniz, tower over the others (Fig. 1.3). Even though Isaac Newton was the first mathematician and scientist credited for inventing calculus during the middle of 1660s, it is also true that Gottfried Leibniz has independently invented calculus somewhere in the middle of 1670s. It is only fair to say that what we take as fundamental to calculus today was quite vague to both Newton and Leibniz and to their contemporaries. For example, 2 3

Eötvos, R.v. (1910), p. 319. Touboul (2022).

1.4 Who Invented Calculus?

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Fig. 1.4 The derivative of f (x) at x1 can be approximated by the slope of the chord between x1 and x2 in the limit as x2 approaches x1

Fig. 1.5 The area under the curve f (x) between a and b can be approximated by the sum of the areas of the trapezoids. In the limit as the base of the trapezoids shrink to zero, x → 0, area becomes equal to the integral b a f (x)d x

the idea of function was formulated only after 1690s. Newton and Leibniz were mostly concerned with infinitesimal durations of time and how far objects will travel in an infinitesimal amount of time. In developing his theories, Newton formulated calculus in terms of geometric arguments about infinitesimally small shapes.4 Among the most rudimentary concepts of calculus, we can name the function, limit, derivative, and integral. Derivative and integral can be introduced most easily by their geometric representations. The derivative of a function, d f (x)/d x, at x1 is the slope of the tangent to f (x) at x1 , which can be approximated by the slope, tan θ = ( f 2 − f 1 )/(x2 − x1 ), of the chord between two points, x1 and x2 (Fig. 1.4). In the limit as x2 approaches x1 , the chord approaches the tangent to f (x) at x1 ; hence, the limit: lim x2 →x1 ( f 2 − f 1 )/(x2 − x1 ), becomes the derivative, d f /d x, of f (x) at x1 . b On the other hand, the Integral: I = a f (x)d x, is the area under f (x) between two points, a and b, which can be approximated by the sum of the areas of the trapezoids shown in Fig. 1.5. In the limit as the base, x, of the trapezoids go to zero, the sum becomes the area under f (x).

4

Hall (1980).

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1.5 First Test of Gravitation in Laboratory The first test of Newton’s law of gravitation in laboratory took place 111 years after the publication of Principia by Henry Cavendish in 1798, who used a torsion balance.5 In Newton’s theory, two masses attract each other without actually touching. This action at a distance bothered none other than Newton himself. In the second edition of Principia in 1713, Newton said “I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses… . It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies.”6 A partial solution came later with Maxwell’s theory in terms of the field concept. Like charged particles in Maxwell’s theory, massive particles also have fields around them. Therefore, a mass attracts another mass by a kind of touching it with its gravitational field. Gravity is 1040 times weaker than the electromagnetic forces that hold atoms together. Since bulk matter contains an equal number of positive and negative charges, at macroscales gravity is the dominant force. Newton’s dynamical theory coupled with his law of gravitation is full of success stories that very few theories will ever have for many years to come. Among the most dramatic is the discovery of Neptune. At that time, small deviations from the calculated orbit of Uranus were observed. At first, the neighboring planets, Saturn and Jupiter, were thought to be the culprit. However, even after the effects of these planets were subtracted, a small unexplained difference remained. Some scientists questioned even the validity of Newton’s theory. However, astronomers putting their trust in Newton’s theory postulated the existence of another planet as the source of these deviations. From the amount of the deviations, they calculated the orbit and the mass of this proposed planet. They even gave a name to it: Neptune. Now the time had come to observe this planet. When the telescopes were turned into the calculated coordinates: Hello! Neptune was there.

1.6 Conservation Laws One of the most fundamental laws of nature that govern all natural phenomena is the conservation of energy. It says that there is a quantity that does not change during natural processes. It is an abstract mathematical principle and as far as we know, it is exact. Newton’s theory respects energy conservation and allows us to calculate its value. A spring with a mass attached to it is called the simple harmonic oscillator. When the mass is slightly displaced from its equilibrium position, it executes a to and fro motion about the equilibrium position. During these oscillations, the total energy of the system, the kinetic energy of the mass plus the potential energy stored in the spring, remains constant. When systems in equilibrium are disturbed slightly, they usually behave like a simple harmonic oscillator. In this regard, this simple 5 6

Cavendish (1798). Westfall (1978).

1.7 Fictitious Forces

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generic case has found many uses in science and engineering. The simple harmonic oscillator has also played an important role in the discovery of quantum mechanics with its applications to radiation problems. In chemistry and solid-state physics, bonds between atoms can be represented by springs. Energy comes in many forms: Kinetic, electromagnetic, gravitational, elastic, thermal, chemical, etc., which can transmute into each other. The energy conservation law says that in a closed system, the total energy is fixed, and it can neither be created nor destroyed. This law applies both in the macro and the microcosmos. In the late 1920s, scientists were puzzled by the energy balance in nuclear beta decay experiments. No matter how careful the energy of the system before and after the reaction was measured, the conservation of energy equation was not satisfied. In 1930, Pauli was obliged to postulate a new particle that carried the missing energy. This particle had to be light, electrically neutral with spin 1/2, and which virtually interacted with nothing. In 1934, Enrico Fermi called this particle neutrino. Later that year, in a Nature article, Hans Bethe and Rudolf Peierls said “... there is no practically possible way of observing the neutrino.” In 1955, neutrinos were first detected by Clyde Cowan and Frederick Reines. They discovered that in nuclear reactions, a nucleus undergoing beta decay emits a neutrino along with an electron. This discovery was awarded the Nobel Prize in physics after 40 years of its discovery in 1995.7 Another important consequence of Newton’s second law is that if we define the momentum, p, of a particle as its mass times velocity, p =mv, then the second law says that in the absence of a net force acting on a particle, its momentum does not change. Similarly, for a system of particles, the total momentum is conserved if the net force acting on the system is zero.

1.7 Fictitious Forces Once all the forces due to material sources acting on a body are identified and given the initial conditions, Newton’s dynamical theory determines the path of an object precisely. This way, astronomers were able to predict solar eclipses centuries ahead and successfully predicted a new planet Neptune. However, Newton’s dynamical theory fails in accelerating reference frames. In such frames, in addition to the forces with material sources, certain terms have to be added to the Newton’s second law to make it work. These additional terms are called the inertial forces or the fictitious forces. Unlike the other forces, they do not have material sources that can be identified. These additional terms correspond to what you feel when the bus driver puts on the brakes or when the bus is turning a corner. The reference frames in which the Newton’s equations are valid in their original form are called the inertial frames. All reference frames that move with constant velocity with respect to an inertial frame are also inertial. 7

Close (2012).

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1.8 Newton’s Bucket Experiment Newton’s vision of reality was that material particles, or objects, interact through forces that have physical sources. Newton was bothered by the problem that noninertial frames posed, and he tried to explain it by introducing the absolute space concept. Using the rotating bucket thought experiment, he explained this in terms of the shape of the surface of the water in reference to absolute space. But it was never clear what he meant exactly by absolute space. In Principia (1689), Newton describes a simple experiment with a bucket half-filled with water and suspended with a rope from a fixed point in space. In this experiment, first, the rope is twisted tightly, and after the water has settled with a flat surface, the rope is released. Initially, the bucket spins rapidly with the water remaining at rest with its surface flat. Eventually, the friction between the water and the bucket communicates the motion of the bucket to the water and the water begins to rotate. As the water rotates, it also rises along the sides of the bucket. Slowly, the relative motion between the bucket and the water ceases, and the surface of the water assumes a concave shape. Finally, the rope unwinds completely and begins to twist in the other direction, thus slowing and eventually stopping the bucket. Shortly after the bucket has stopped, the water continues its rotation with its surface still concave. The question is; what causes this concave shape of the surface of the water? At first, the bucket is spinning, but the water is at rest and its surface is flat. Eventually, when there is no relative motion between the bucket and the water, the surface is concave. Finally, when the water is spinning but the bucket is at rest, the surface is still concave. From these, it is clear that the relative rotation of the water and the bucket is not what determines the shape of the surface. The crucial question is, what is spinning and with respect to what? Let us try to understand the shape of the surface in terms of interactions. Since the bucket and the water, and the rest of the universe is on average neutral, electromagnetic forces cannot be the reason. The gravitational interaction between the bucket and the water is surely negligible; hence, it cannot be the reason either. Besides, in Newton’s theory, gravity is a scalar interaction, that is, the force between two masses depends only on their separation and not on their relative motion. In this regard, Newton could not have used the gravitational interaction of water with other matter. This led Newton reluctantly to explain the concave shape as due to rotation with respect to absolute space. In other words, the surface of the water is flat when the water is not rotating with respect to absolute space, and when there is rotation with respect to absolute space, the surface is concave. Leibniz objected to Newton’s absolute space and thought that rotational motion is meaningful only with respect to its relation to other matter. Here, what is meant by relation should be understood as interaction with other matter. Today, following Newton, we sidestep this problem and say that Newton’s equations are valid in inertial reference frames that move uniformly with respect to one another. For non-inertial reference frames, certain non-inertial (force) terms have to be added to the second Law.

1.10 Newton’s Corpuscular Theory of Light

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A satisfactory solution to this problem could come only with Einstein’s general theory of relativity (1916), where the gravitational force between two masses depends not just on their separation but also on their relative velocity. This is analogous to Maxwell’s theory, where the electromagnetic interactions are described by a vector potential. Hence, the force between two charged particles has a velocity dependent part aside from the usual Coulomb force. In general theory of relativity, gravity is described by a tensor potential, which is the metric tensor; hence, the velocity dependence in Einstein’s theory is even more complicated. In this regard, in Einstein’s theory, not just the shape of the surface of the water in Newton’s bucket experiment but also all fictitious forces in Newton’s dynamical theory could, in principle, be explained as the gravitational interaction of matter with other matter, that is, with the mean matter distribution of the universe.

1.9 Mach’s Principle In principle, inertial forces have to be explained in terms of (gravitational) interaction with the average matter distribution of the universe. This effect is also referred to as the Mach’s principle. In constructing his theory of gravitation, Einstein was deeply influenced by Ernst Mach. In Newton’s theory, there is no way of explaining the inertial forces in terms of gravitational interaction of local matter with the average matter distribution of the universe. This follows from the fact that in Newton’s theory, gravity is what is called a scalar interaction that depends only on the distance between objects. For example, the gravitational force that the sun exerts on earth is the same regardless of the rotational velocity of the sun. No matter how fast the sun spins, the force that earth feels will be the same. Even though Mach has never given a proper formulation of his principle, Einstein’s theory incorporates Machian effects as demonstrated by Lense-Thirring.8 In Einstein’s theory, rotation of the sun causes an affect called the dragging of the inertial frames. It is an affect that has to be included to make the GPS system work properly.

1.10 Newton’s Corpuscular Theory of Light Newton thought that certain properties of light like reflection and refraction can only be explained if the light was made up of particles called corpuscles that travel in straight lines, as opposed to waves, which do not. Unlike the photons in quantum mechanics, corpuscles were classical particles that are supposed to be single, infinitesimally small with shape, size, and color. What separated corpuscles from atoms was that atoms are indivisible but corpuscles can be divided. Newton’s corpuscular theory was an extension of his vision of reality to light, where material 8

Pfister (2006).

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Fig. 1.6 ThomasYoung, 1773–1829 (AIP Emilio Segrè Visual Archives)

particles interact through forces. Newton’s corpuscular theory of light was the subject of the centuries old debate between Newton, Huygens, and Young about whether light is corpuscular or a wave. Newton’s corpuscular hypothesis was able to explain reflection, refraction, and the bending of light as light passes from a less dense to a dense medium. However, the Dutch physicist Christien Huygens argued that light is a wave traveling through ether like the ripples on a lake when a stone is dropped. Huygens asked if light is corpuscular, then where is the evidence of colliding light particles when two light beams cross. Even though both theories explained reflection and refraction, at the time sufficiently accurate experiments were not possible to make the difference appear. Given Newton’s preeminence and also the Huygen’s early death, 32 years before Newton in 1695, Newton’s corpuscular theory was unquestioned until the British scientist Thomas Young came along (Fig. 1.6).

1.11 Young’s Double-Slit Experiment Thomas Young shone monochromatic light, which is light with a single wavelength, to a screen with a single slit, which then passed through a second screen with a double slit and then fell onto a screen some distance away (Fig. 1.7). The first screen

1.11 Young’s Double-Slit Experiment

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Fig. 1.7 Thomas Young discovered that monochromatic light passing through two closely spaced holes produces alternating bright and dark fringes. Young argued that this could only be explained by light waves emanating from the two holes interacting constructively for the light and destructively for the dark fringes

with the single slit formed a point source, while the second screen with the double slit formed two-point sources. According to the corpuscular theory, one expects to see two bright spots with a dark region in between, corresponding to the images of the two slits. Instead, Young observed a central bright region followed by alternating bright and dark fringes. To explain this, Young argued; imagine two pebbles dropped onto a lake simultaneously. Each pebble will produce spherical waves emanating from the point where the pebbles have hit the water. At each point on the lake surface, where the two waves meet, if the two troughs or the two crests coincide, we have constructive interference; hence, the combined wave will have a new trough or a crest. Where the troughs and the crests of the individual waves coincide, then we have destructive interference; hence, the two waves cancel each other leaving the water surface undisturbed at that point. In the double-slit experiment, Young realized that the only way to get this interference pattern is to have light as a wave with the dark and the bright fringes corresponding to the points where the light waves coming from the individual slits interfere destructively or constructively, respectively. When Young first presented his conclusion that light is a wave in 1801, he was ruthlessly attacked for challenging Newton. Lord Brougham declared his attempt as “destitute of every species of merit.”9 To defend himself, Young had to put out a pamphlet to let people know his feelings about Newton, which he expressed as “But as much as I venerate the name of Newton, I am not therefore obliged to believe that he was infallible. I see not with exultation, but with regret, that he was liable to err, and that his authority has, perhaps, sometimes even retarded the progress of science.”10 Only a single copy of the pamphlet was sold.

9

Baggott (2016), p. 2. Robinson (2006), p. 96.

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Fig. 1.8 James Clerk Maxwell, 1831–1879 (Engraved by G. J. Stodart, courtesy of AIP Emilio Segrè Visual Archives)

1.12 Fresnel and the Wave Theory Only after the 1820s and after Augustine Fresnel repeated Young’s double slit experiment and did much more, the wave theory begin to gain some approval among the reputable scientists of that time. Fresnel’s experiments were much more precise than Young’s and were also backed with careful and detailed analysis. One of the outstanding objections raised against the wave theory was that sound can turn corners but light cannot. Fresnel also showed that light also turns corners. Since light waves have wavelengths millions of times shorter than the sound waves, light bends only around objects whose size is comparable to its wavelength, which was not possible to detect before Fresnel.

1.13 Maxwell’s Theory and Light As the number of converts to the wave nature of light increased, its detailed nature remained unknown until Maxwell’s theory of electromagnetism came in 1864 in terms of Maxwell’s equations. About 150 years ago, guided by the experiments of Faraday, Oerstead, and Ampere, James Clerk Maxwell introduced his theory of

1.13 Maxwell’s Theory and Light

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electromagnetism (Fig. 1.8). Up to that time, electricity and magnetism were thought to be separate phenomena. With Maxwell’s theory, electric and magnetic fields were unified elegantly under one roof that also set an example for future attempts of unified field theories. In Maxwell’s theory, time dependent electric fields generate a magnetic field and time dependent magnetic fields produce an electric field. The four Maxwell’s equations not only explained a whole wealth of experimental data but also showed that electromagnetic waves are possible. Given an oscillating electric field in the z-direction, it generates a magnetic field oscillating in the perpendicular x-direction and vice versa. When one begins to decrease, it generates the other, thus moving together in the y-direction with the speed of light: 3 × 1010 cm/s. Unlike water or sound waves, an electromagnetic wave can travel in vacuum unattenuated for billions of light years.

Chapter 2

Perpetual Motion Dream

Two important consequences of the Newton’s theory are the conservation of energy and the conservation of momentum. Even though these laws are consequences of the Newton’s theory, their universality extends them to all branches of science at all scales. Let us now consider the following set of events that we are all familiar. You get into your car in the morning and start your engine. You drive around the neighborhood to visit friends and maybe do some shopping. For simplicity, we assume that there are no hills in your district. Therefore, we can ignore the gravitational potential energy. After driving for a while, you return home and park your car. The following morning when you come back, you notice that the engine, which was hot when you left, has now cooled down to the same temperature that it was the previous morning. The location of your car is the same as before. Everything looks the same except your fuel gauge, which shows less fuel than before. In other words, your car is in the same state as you started your trip, except the gas in your tank is less. We have seen that energy comes in different forms and that it could change from one form into another. The chemical energy generated by burning the fuel in the cylinders has not only gone into the kinetic energy of the car, but also some of it has showed up as heat on the engine. Also, the friction in different parts of the car, like the brakes, bearings, as well as the friction between the tires and the road, etc. has caused additional heating in the bulk of the car. Noise that the engine has produced is lost into the atmosphere as sound energy, which eventually dissipated as heat. Some electromagnetic energy is also lost as heat through the wiring of the car. All in all, during all these changes taking place in your car, a substantial amount of energy is lost into the atmosphere as heat. But the atmosphere is so large that its temperature has not changed. It is still at the same temperature. Actually, not just you, but all the other drivers in the city have lost a lot of energy to the atmosphere, and yet, the atmosphere is still at the same temperature. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_2

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Some people thought that it would be nice to build an engine that could absorb some of that energy lost into the atmosphere and do useful work; like running a pump. Since the atmosphere is so large that the amount of energy taken to run the pump would not change its temperature, the pump would be running perpetually. The sole action of such an engine would be to absorb some heat from the atmosphere without changing its temperature and to convert it into useful work. As far as Newton’s theory is concerned, during this process, both the energy and the momentum would be conserved. Energy would neither be created nor destroyed. Whatever we have taken from the atmosphere would be converted into work. No more no less. However, there is a law of thermodynamics, the second law, that prohibits us from building such an engine. Not just in nineteenth century, but even today, a lot of people chase this illusion and apply for patents for their designs. Some of these ideas may even look very ingenious and sophisticated, but when analyzed in detail, they wind up being nothing but perpetual motion machines. When Einstein was working at the Swiss Patent Office, a lot of the patent applications he examined were perpetual motion machines.

2.1 Thermodynamics During nineteenth century, Newton’s mechanics and Maxwell’s electrodynamics were joined by thermodynamics to start the industrial revolution. Thermodynamics is the branch of science that deals with the relationships between heat, work, energy, and temperature. Its key concept is that heat is a form of energy that can be used to do mechanical work. In particular, the Watt steam engine, developed by James Watt during the years 1763–1775 started the industrial revolution both in England and the rest of the world.1 Modern thermodynamics was historically developed to increase the efficiency of heat engines. In particular, the French physicist Sadi Carnot with his book, Reflections on the Motive Power of Fire, published in 1824 (Paris) is often referred as the father of thermodynamics. The concept of motive power is defined by Carnot as the useful effect that a motor is capable of producing. The modern definition of work was introduced by Carnot as the work done by lifting a weight over a certain height. In 1843, James Joule experimentally found the mechanical equivalent of heat.2,3 This lead to the first law of thermodynamics, which states that in thermodynamic processes, a change in the internal energy of the system is equal to the heat added minus the work done on the environment; such as lifting a weight by pushing a piston. This explained how heat can do work. We have seen that energy comes in different forms and that it could transform from one form into another. Now, the first

1

Cardwell (1971). Cardwell (1991). 3 Joule (1850). 2

2.2 Entropy and the Laws of Thermodynamics

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law of thermodynamics says that in natural processes, we also have to include the heat energy into the conservation of energy equation. In 1850, Rudolf Clausius introduced the term entropy as the heat lost or turned into waste. In 1854, he published his famous statement in German and in English in 1856,4 which is known as the second law of thermodynamics: “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.” Rephrased slightly; heat cannot flow spontaneously from a cold object to a hot object. In 1848, Lord Kelvin (William Thomson) introduced the idea of absolute zero. The name thermodynamics was introduced by Lord Kelvin in 1854 in his paper On the Dynamical Theory of Heat.5 In 1906, Walther Nernst introduced the third Law of thermodynamics.6

2.2 Entropy and the Laws of Thermodynamics Mainly developed during the nineteenth century, thermodynamics can be summarized in terms of four laws: Zeroth law of thermodynamics: When two systems are in thermodynamic equilibrium with a third system, then the first two systems are also in thermal equilibrium. This allows the third system to be used as a thermometer. First law of thermodynamics: This is basically an expression of the energy conservation principle with the inclusion of the heat added to the system. It is important to note that heat was not accepted as a form of energy until about 1798, when Sir Benjamin Thomson, noticed that the heat generated during the boring of cannons is proportional to the work done by turning the boring tool. Second law of Thermodynamics: There are several equivalent ways of expressing the Second law: (i) Heat does not flow spontaneously from a cold object to a hot object. Invention of the refrigerator, where the cold inside of the refrigerator becomes colder and the hot outside, the room, becomes hotter does not violate the second law since the refrigerator has to be plugged in, thus explaining the word spontaneous. (ii) An equivalent way to express the second law was given by Lord Kelvin7 as that heat at a given temperature cannot be converted entirely into work. In other words, you cannot build a heat engine whose entire action is to absorb heat from a reservoir, say atmosphere, and then to convert it entirely into useful work. Reservoir is a heat bath that can exchange heat at constant temperature. According to the second law, a heat engine whose sole action is to absorb some 4

Clausius (1867), Ch. V. Thomson (1854). 6 Coffey (2008), pp. 78–81. 7 Kumar (2014), p. 388. 5

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heat from a reservoir, and then to convert it all into useful work is forbidden. What can be done is that you can absorb some heat from a reservoir and do some useful work as long as you pay a price, that is, dump some of the energy into another reservoir at a different temperature. The efficiency of such an engine is related to the temperature difference of the two reservoirs. The second law says that you cannot have an heat engine that is 100 percent efficient, which is true only when the two reservoirs are at the same temperature. (iii) In a closed system, heat energy per unit temperature called entropy increases toward a maximum value, which is reached at thermodynamic equilibrium, where no energy is available to do useful work. A closed system is a system that does not exchange energy with the outside. When a hot object and a cold object are brought into contact, as far as the conservation of energy is concerned, the direction of heat flow could be either way. Heat could equally flow from the cold object to the hot object making the cold colder and the hot hotter. However, in reality, heat flows from the hot object toward the cold. The second law of thermodynamics says that there is a quantity called entropy that increases in this process, eventually reaching a maximum when the two objects reach thermal equilibrium at an intermediate common temperature. At this point, there will be no more energy left to do useful work. Energy conservation requires just the constancy of the combined energy of the two objects. The second law says in which direction the process should go. In this regard, the second law is also considered as determining the direction of time. How sure are we about the second law of thermodynamics? Very sure. So far no body has seen a cold object brought into contact with a hot object getting colder and the hot object getting hotter. No body has been able to build a perpetual motion machine that keeps running on its own. Thermodynamics was initially developed in response to improving the efficiency of the steam engines; however, its sweeping generality makes it applicable to both physical and biological systems and even to the entire universe. In his seminal paper in 1864, Clausius ended with the following statements: “The energy of the universe is a constant.” and “The entropy of the universe tends to a maximum.”8 Do not expect a mathematical proof of the second law. The proof is all around us in our experiences. Living things get old and eventually die. Cars and objects age and deteriorate. Stars, galaxies, and even the universe age and die. The direction of many processes in physics, chemistry, and biology is determined by the second law of thermodynamics. Third law of Thermodynamics: Entropy of a system approaches zero as temperature approaches absolute zero (−273 ◦ C). This helps define an absolute scale for entropy.

8

Clausius (1867), Ch. X.

2.4 Atoms and the Kinetic Theory

21

2.3 Nothing New Left to Discover At the beginning of the twentieth century, after Isaac Newton and James Clerk Maxwell, and finally the discovery of the laws of thermodynamics, there were many reasons to believe that physics may have come to an end. In reflecting the sentiments of that era, in 1900, Lord Kelvin declared to the British Association for the Advancement of Physics that “there is nothing new to be discovered in physics now. All that remains is more and more precise measurements.”9 In fact, young Max Planck was advised by his professor at the University of Munich not to consider a carrier in theoretical physics. The same Max Planck, however reluctantly, will soon kindle the discovery of quantum mechanics. So far we have said nothing about atoms. At the turn of the nineteenth century, there were a lot of prominent scientists like Ernst Mach and the physical chemist Wilhelm Ostwald who did not believe in the existence of atoms. Thermodynamics was developed in terms of the macroscopic properties of matter like pressure, temperature, entropy, etc., where changes among these variables took place continuously.

2.4 Atoms and the Kinetic Theory During the second half of the nineteenth century, along with thermodynamics, a parallel development led by Ludwig Boltzmann and James Clerk Maxwell was taking place in the kinetic theory of gases. Boltzmann believed that matter is made up of atoms from which the macroscopic properties of gases like pressure, density, etc. can be derived. In 1860, Maxwell discovered the mean distribution of the velocity of simple molecules without needing the velocity of single molecules. Using Maxwell’s statistical and probabilistic approach, Boltzmann went further to develop the kinetic theory of gases, where the average kinetic energy of the atoms is proportional to the temperature of the gas. The most rudimentary kinetic model was that gases are composed of large number of identical atoms or molecules. The average separation of these atoms is large compared to their sizes and they undergo perfectly elastic collisions with each other, and with the walls of the container. In between collisions, atoms are assumed to move freely. With these simplifying assumptions, using kinetic theory, it was possible to drive the ideal gas law—one of the successful applications of the kinetic theory. Based on the kinetic theory, pressure on the walls is due to the random collisions of the atoms with the walls of the container. At first, the kinetic theory was not well received. Boltzmann was heavily criticized by contemporaries like Planck. Their main objection was that when you mix two gases at different temperatures in an isolated container, entropy increases, eventually reaching a maximum at equilibrium. According to atomists, behavior of gases is due to the mechanical properties of atoms, and the equilibrium is just the most probable state among the many possible microstates that the gas could assume. According 9

Baggott (2016), p. 4.

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Fig. 2.1 Ludwig Boltzmann, 1844–1906 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

atomists, all microstates that yield the same macroscopic variables are equiprobable. The critiques argued that according to Newton’s laws, collisions between the atoms are reversible. That is, when you reverse all the velocities and consider individual collisions between the atoms backwards, nothing peculiar about them would appear. But nobody has witnessed a case where the two gases spontaneously separate and return to their initial temperatures, thus reducing the entropy in the process. This was in complete contrast with the second law of thermodynamics.

2.5 Entropy in Kinetic Theory Using Maxwell’s discovery, in 1870s, Boltzmann went ahead to give a statistical interpretation of entropy and the second law of thermodynamics, which remained somewhat elusive and abstract in thermodynamics (Fig. 2.1). Boltzmann’s approach to entropy was that the total energy of a gas can be organized into a series of boxes with energies ε0 , ε1 , ε2 , . . .. For most practical applications, it is sufficient to consider gases as collection of independent atoms or molecules, which move freely except for the brief moments during collisions. We then distribute the gas molecules into these boxes, while conserving the total energy and the total number of particles. At a given time and temperature, the state of a gas can be described by giving the number of atoms in each box.

2.5 Entropy in Kinetic Theory

23

Let us now consider a simple 3-atom gas. Assume that the internal energy of the system is 3ε. Among the three indistinguishable atoms (a, b, c) this energy could be distributed in a number of different ways: a ε 3ε 0 0 2ε 2ε ε ε 0 0 b ε 0 3ε 0 ε 0 2ε 0 2ε ε . c ε 0 0 3ε 0 ε 0 2ε ε 2ε We see that all together there are 10 possible configurations or microstates, in which the 3ε amount of energy can be distributed among the three atoms. Note that the sum in each column gives the total internal energy of our 3-atom gas. Since atoms interact, no matter how briefly, through collisions, the system fluctuates between these possible microstates. Boltzmann postulated a priori that all possible microstates are equally probable. If we look at the microstates carefully, we see that they can be grouped into three states (S1 , S2 , S3 ) with respect to the number of particles (n 0 , n ε , n 2ε , n 3ε ) in each box as S1 S2 S3 n0 0 2 1 nε 3 0 1 . n 2ε 0 0 1 n 3ε 0 1 0 Note that only 1 microstate corresponds to state S1 , where all the atoms are in the box with the energy ε. Similarly, 3 microstates to S2 and 6 microstates to state S3 . Since all the microstates are equiprobable, probabilities of finding the system in the states S1 , S2 , and S3 , respectively, are 1/10, 3/10, and 6/10. This means that if sufficiently many observations are made, 6 out of 10 times the system will be seen in state S3 , 3 out of 10 times it will be in state S2 , and only 1 out of 10 times it will be in state S1 . In other words, if we wait long enough, the system can be seen in all three states. But this simple 3-atom model will spend most of its time in state S3 , which can be considered as its equilibrium state. For the case where we mix two gases at different temperatures in an isolated container, among the microstates, there will be states that correspond to atoms separating on their own, thus reducing the entropy. However, compared to the equilibrium state, one may have to wait longer than the age of the universe to see one. In his kinetic theory Boltzmann defined entropy as proportional to the logarithm of the number of microstates, W, that yields the same macrostate as S = k log W.

(2.1)

In conformity with the third law of thermodynamics, at absolute zero, all the atoms will be in the lowest energy state with the energy ε,, hence W will be 1, thus giving zero for the entropy S. The proportionality constant k is known as the Boltzmann constant.

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Fig. 2.2 Comparison of Wien’s and Rayleigh-Jeans limit formulas and the Planck’s formula

2.6 Dark Clouds over the Horizon At the turn of the nineteenth century Newton’s theory, Maxwell’s electrodynamics, and the thermodynamics, what is collectively called the classical physics was considered by many physicists as a complete description of nature. However, there were already dark clouds looming on the horizon for classical physics. Principles of thermodynamics reaffirmed the classical view of the universe, where energy flowed continuously between matter and radiation. Atomists on the other hand offered another view, where matter is not continuous but made up of discrete particles as atoms or molecules. These atoms were not visible themselves, but their existence can be proven by obtaining the observable macroscopic properties of matter via the mechanical properties of the atoms using the kinetic theory and statistics. As a young student, Max Planck was advised not to go into theoretical physics, since there was nothing interesting left to discover.10 In 1897, as a leading expert in thermodynamics and the second law, Planck focused his attention on the black body radiation problem; a problem which tantalized the physicists of that era. A part of Planck’s motivation was his dislike of the atomistic approach, which he expressed his feelings as opening the door for some discomforting consequences.11 Considering the problems that the Boltzmann’s kinetic theory is having with the second law, Planck thought that he could use this problem to refute the atomic view once in for all. After all, the physics of the black body radiation appeared not to have any link to atoms or molecules.

10 11

Cline (1987), p. 34. Baggott (2016), p. 8.

2.7 Planck’s Black Body Radiation Formula

25

All objects when heated radiate with an intensity and color depending on their temperature. When a piece of iron is heated, at first the radiation is not visible because it is in the infrared region of the electromagnetic spectrum. Today, we can see it with an infrared camera. As the temperature rises, it first becomes red and then yellowish-orange and eventually becomes bluish-white. In 1666, Newton passed sunlight through a prism and showed that it is composed of the rainbow colors: red, orange, yellow, green, blue, indigo, and violet. The fact that red to violet represents the range of sensitivity of the human eye was shown only in 1800. To investigate the radiation problem, Gustav Kirchhoff in 1859 introduced the black body concept, which is a perfect absorber and an emitter. The name black body was appropriate since a perfect absorber would absorb all radiation, thus reflect nothing, hence would appear black. Kirchhoff considered his black body as a hollow sphere with perfectly absorbing walls and a small hole through which radiation could enter and leave. Since all radiation that hits the hole, regardless of its frequency, would go in, the hole would mimic a perfect absorber. Once the radiation is in, bouncing back and forth from the walls, it would eventually thermalize. Now the hole would act as a perfect emitter for the trapped radiation. This problem was not just for academic curiosity, it was also of interest to the German Bureau of Standards as reference for rating electric lamps.12 In 1860, Kirchhoff theoretically proved that the energy density of the radiation inside a cavity depends only on the frequency and the temperature of the black body, that is, it is independent of the shape, size, and the type of material that the black body is made up of. This implied that something fundamental is going on. However, the exact form of the black body radiation formula was left to be discovered. In 1893, Wilhelm Wien proposed a simple formula that seemed to agree with the existing data (Fig. 2.2). Later, as photometric measurement techniques improved, Wien’s formula was shown to fail at higher wavelengths. In 1900, Rayleigh-Jeans gave another limit law for the black body that agreed with experiments for the higher wavelengths, but this time failed at the lower wavelengths.13

2.7 Planck’s Black Body Radiation Formula An experimental physicist Heinreich Rubens visited Planck in his home in October 7, 1900, and told him that with his coworker Ferdinand Kurlbaum, they obtained some new data about the black body radiation at even higher wavelengths that showed the breakdown of the Wien’s formula. After Rubens left, with the fresh data in his hand and with the Wien’s limit formula, Planck immediately got to work on a new formula (Fig. 2.3). After a few unsuccessful attempts, he eventually managed to obtain the black body radiation formula that fit the new data (Fig. 2.2). The formula depended on two constants; the temperature of the black body and a constant that will later be called the Planck constant. He immediately mailed his result to Rubens. After a 12 13

Baggott (2016), p. 10. Baggott (2016), p. 11.

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Fig. 2.3 Max Planck, 1858–1947 (AIP Emilio Segrè Visual Archives, W. F. Meggers Gallery of Nobel Laureates Collection)

couple of days, Rubens returned by saying that he checked the formula and found complete agreement with all cases. Now, the black body radiation formula for all temperatures and for the entire range of the spectrum was at hand, but a physical derivation was still to come.

2.8 Reluctant Revolutionary Planck had an excellent background in classical physics; in particular, he was a leading expert in the second law of thermodynamics.14 Planck was a conservative person that preferred a slow peaceful life. This was also reflected in his character as a scientist and in his opposition to the atomists and Boltzmann. Despite the successful applications of the kinetic theory, Planck thought that eventually it had to be abandoned in favor of continuous matter.15 In fact, when he started working on the 14 15

Heilborn (2000), p. 10. Heilborn (2000), p. 14.

2.8 Reluctant Revolutionary

27

black body problem, his aim was to disprove the atomic view, which he considered as opening the door for some disturbing consequences. Planck knew that the real jewel in his formula was in the physics behind it. He now devoted all his time to understand what this formula actually means and at whatever cost it bears. He recalls, the next six weeks of his life as “the most strenuous work of my life,” after which “the darkness lifted, and an unexpected vista began to appear.”16 As a leading authority in classical physics, from Maxwell’s theory, Planck knew that oscillating charges radiate and absorb energy only at a specific frequency, where the frequency ν and the wavelength λ are related as ν = c/λ. He modeled the interior walls of the black body as an enormous collection of oscillators. Since each oscillator emits or absorbs radiation at a fixed frequency, together they would emit all the frequencies found in the black body spectrum. Planck envisioned his oscillators as massless (frictionless) springs with an electric charge attached to one end. Since the frequency of the oscillations of a spring is proportional to its stiffness, Planck assumed the collection of springs to be composed of varying stiffness. Whether a spring is oscillating or not depended only on the temperature of the walls. It could emit or absorb energy from the radiation in the cavity. When the temperature of the radiation inside the cavity becomes equal to the temperature of the walls, thermal equilibrium would be reached and the number of springs absorbing energy would be balanced by the number of springs that emit radiation. Planck was a firm believer of the second law of thermodynamics, which he took as absolutely true. For him, entropy always increases. In Boltzmann’s kinetic theory, entropy had a probabilistic interpretation, which involved counting number of microstates that correspond to a given macrostate. Among these states, there would be cases where entropy decreases, but such states would be highly unlikely. On the other hand, thermal equilibrium corresponds to a case, where one has the maximum number of microstates leading to a given macrostate, hence with the maximum probability or entropy. In finding the physical basis of his black body formula, Planck had no choice but to resort to Boltzmann’s probabilistic approach. He expressed his view of the second law as “Until then I had paid no attention to the relationship between entropy and probability, in which I had little interest since every probability law permits exceptions; and at that time I assumed that the second law of thermodynamics was valid without exception.”17 In 1884, Boltzmann had already verified that the total energy radiated by a black body is proportional to the fourth power of temperature, which is known as the Stephan-Boltzmann law. Thermal equilibrium is the state of maximum entropy, which is the most probable state with the highest probability. Planck now faced the problem of distributing the total energy of the black body among his oscillators. Here, he ran into another problem. His oscillators can only oscillate at a fixed frequency, where the energy of an oscillator is proportional to the square of its amplitude. A spring can store more energy when it is stretched more. When you stretch a spring to a distance x0 and then let go, the spring will execute simple harmonic 16 17

Planck (1993), p. 106. Hermann (1971), p. 16.

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oscillations with the total energy proportional to the square of the amplitude x0 . Applying the Boltzmann’s entropy definition, his formula left him no choice but to accept that these oscillators exchange energy only in packets of energy proportional to the frequency of the oscillators. Planck called this the “most essential point of the whole calculation.”18 Energy was no longer exchanged continuously, but only in discrete units, which were later called quanta. Now Planck had to divide the energy into hν amount of chunks, where h is a constant, later called the Planck constant in his honor, and ν is the frequency of the oscillator. Each oscillator can now exchange energy in integer multiples of hν as nhν, where n = 1, 2, 3, . . . . Planck’s dilemma was that his atomic sized oscillators absorbed and emitted radiation only in chunks of hν, while the macroscopic oscillators absorbed energy continuously. Sometimes in physics to facilitate the evaluation of a continuous quantity, we consider it in discrete units or steps and then at the end take the continuum limit as the step size goes to zero. When Planck considered these chunks of energy in the continuum limit as h → 0, his quantum oscillators began to behave like classical oscillators, but this time, to his dismay, the black body radiation formula also disappeared, which Planck was not willing to give up. In other words, there was no escape from the quantum concept. Planck’s son Ervin, who was seven-year-old at that time, remembers that as they were strolling along with his father he said to him “Today I have made a discovery as important as that of Newton.”19 The exact date of this conversation is not known, but probably it was before December 14, 1900, the day when Planck presented the physics behind his formula to the members of the German Physical Society. Apparently, Planck was aware of the revolutionary implications of his chopping energy into chunks of hν. Like him most of the physicists at the conference thought that it is a neat trick that will somehow be ironed out in time in favor of continuous energy. Planck acknowledged his appreciation of Boltzmann by naming the constant k in his formula as the Boltzmann constant. On the other hand, Boltzmann suffered from ill health, not to mention the severe manic depression he suffered. Even though he was among the most appreciated and respected physicists of his time, he felt increasingly deserted and unappreciated. At the age of 62 in September 1906, the tragic news that he hanged himself shocked his friends and the scientific community. Even though Planck’s black body radiation formula and his chopping energy into chunks of hν ushered in the quantum era, his deep conviction to classical physics, where energy is exchanged continuously made him spend most of his remaining life trying to look for a way to avoid the quantum. After his death in 1947, at the age of 89, his student James Frank recalled him as “a revolutionary against his own will.”20 In

18

Planck (1900), p. 84. Born (1948), p. 170. 20 Kumar (2014), p. 29. 19

2.9 New Statistics

29

a letter to Nature in December 18, 1926, a renounced physicist and chemist, Gilbert Newton Lewis, coined the term “photon” as the quantum of the electromagnetic radiation.

2.9 New Statistics Planck’s black body radiation formula involved one more surprise, which was a different statistics then the one given by Boltzmann. This point was noticed by the Indian mathematician and physicist Satyendra Nath Bose (1924–25) and later the idea was adopted and extended by Albert Einstein in collaboration with Bose. This new statistics is known as the Bose-Einstein statistics. To elaborate this point, we first start with the Boltzmann distribution for solids and gases.

2.9.1 Boltzmann Distribution for Solids Consider a simple model for a solid with the energy levels ε1 , ε2 , . . . . A particular state can be specified by giving the occupancy numbers n 1 , n 2 , . . . of these energy levels. The problem is now how to distribute N distinguishable atoms into boxes labeled ε1 , ε2 , . . . . For the first atom, we have N choices, this leaves (N − 1) choices for the second atom. Similarly, for the third atom we will have (N − 2) choices. All together, N atoms can be arranged in N (N − 1)(N − 2) . . . 2.1 ways. This is nothing but N factorial, defined as N ! = N (N − 1)(N − 2) . . . 2.1. There are now n 1 atoms in box 1, n 2 atoms in box 2, and in general, n i atoms in the ith box. Since atoms are distinguishable in the sense that they can be identified in terms of their positions within the lattice, and since how atoms are arranged in each box is irrelevant, there are n i ! number of arrangements of n i atoms in the ith box that does not lead to a new microstate; hence, we can write the number of microstates corresponding to a particular state as N! W = . (2.2) n 1 !n 2 ! . . . The most probable state is the one with the maximum entropy S, where S = k log W,

(2.3)

hence we look for the maximum number of microstates subject to the conditions that the total number of particles and the total energy are fixed. Mathematically, this problem is solved by finding the occupancy numbers that make W a maximum, which yields the occupancy numbers as n i = Aeβεi .

(2.4)

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Here, A is a constant to be determined from the total number of particles, N = n 1 + n 2 + · · · , and β is related to the temperature as β = −1/kT, where k is the Boltzmann constant.

2.9.2 Boltzmann Distribution for Gases Compared to solids, gases have basically two differences: Due to the fact that atoms are free to roam around the entire volume of the system, they are not localized. On the other hand, the distribution of the energy levels is practically continuous, and there are many more energy levels than the number of atoms. Therefore, the occupancy number of each level is usually either 0 or 1. Mostly 0 and almost never greater than 1. In 1 cc of helium gas at 1 atm and 290 K there are approximately 106 times more levels than atoms. Since atoms are no longer localized, we treat them as indistinguishable. We now group neighboring energy levels into bundles so that a microstate is defined by giving the occupation numbers n k , k = 1, 2 . . . , which is the number of particles in the kth bundle of gk levels with the energy εk . As long as n k is sufficiently large, the value of gk is quite arbitrary. Also, gk must be sufficiently large so that each bundle can be approximated by the average energy εk . For the maximum number of microstates, we again need the most probable values of n k . However, W is now somewhat more complicated. We first consider on the kth bundle, where there are gk levels available for n k atoms. For the first atom, all gk levels are available, for the second atom, there will be (gk − 1) levels left, and so on. If we keep going on going like this, we find the number of different possibilities as the product (2.5) gk (gk − 1) · · · (gk − n k + 1). There are n k terms in this product, and since gk  n k , to a high level of accuracy, we can approximate this product as gkn k . Within the kth bundle, since it does not matter how n k particles are ordered, we also divide gkn k with n k ! to obtain the number of distinct microstates for the kth bundle as gkn k . nk !

(2.6)

Similar expressions for all the other bundles can be written. Hence, the total number of microstates is now expressed as their product: W =

g1n 1 g2n 2 . ··· , n1! n2!

(2.7)

Again, W has to be maximized subject to the condition that the total number of particles and the total energy of the system are constant. This yields the most probable number of particles, n k , in the kth bundle as

2.9 New Statistics

31

n k = Agk eβεk ,

(2.8)

where A is obtained by using the condition that the total number of atoms is fixed as N , while the condition that the total energy of the atoms is equal to the internal energy of the gas gives β as −1/kT. Using the Boltzmann distribution, we can drive the ideal gas law—one of the successful predictions of statistical mechanics.

2.9.3 Bose-Einstein Distribution For the Bose-Einstein distribution, there is no restriction on the number of particles that we can put in each level, hence we remove the restriction on n k . We first consider the number of different ways that we can distribute n k particles over the gk levels of the kth bundle. This is equivalent to finding the number of different ways that one can arrange N indistinguishable particles in gk boxes. As a simple case, consider 2 balls and three boxes, gk = 3, where n k could take the values 0, 1, and 2. The possible arrangements of two balls in three boxes are given as  − −   −  −  −  − −   − −  where there are six distinct possibilities. To see how this number emerges, notice that for three boxes, there are two partitions, shown by the double lines, which is one less than the number of boxes. The number of arrangements of the number of balls plus the number of partitions is given by [2 + (3 − 1)]!. However, the balls and the number of partitions can also be arranged among themselves as 2! and (3 − 1)! times, respectively, which do not lead to any new configurations; hence, this gives a total number of 6 distinct possibilities as 6=

[2 + (3 − 1)]! . 2!(3 − 1)!

(2.9)

Since the number of partitions is always 1 less than the number of boxes, we can generalize this formula for n k balls in gk boxes as (n k + gk − 1)! . n k !(gk − 1)!

(2.10)

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For the whole system, we can write W as the product W =

(n 1 + g1 − 1)! (n 2 + g2 − 1)! . ··· . n 1 !(g1 − 1)! n 2 !(g2 − 1)!

(2.11)

As in the previous cases, we maximize W subject to two conditions: N = n 1 + n 2 + · · · and U = n 1 ε1 + n 2 ε2 + · · · , where N is the total number of particles and U is the internal energy of the gas, which gives the Bose-Einstein distribution as nk =

gk −α−βε k e

−1

,

(2.12)

where β = −1/kT . Using the Bose-Einstein distribution, one can derive the Planck’s black body radiation formula. However, one of its limits turned out to be very interesting that lead to a new form of matter. For high temperatures, the − 1 in the denominator is negligible; hence, the Bose-Einstein distribution reduces to the Boltzmann distribution Eq. (2.8). However, for low temperatures, all the particles can populate the lowest energy level, thus giving rise to what is known as the Bose-Einstein condensation. This form of matter was observed only decades later in 1995 by Eric Cornell and Carl Wieman by cooling a gas of rubidium atoms to 1.7 × 10−7 K. Along with Wolfgang Ketterle, who created a Bose-Einstein condensate with the sodium atoms, they received the 2001 Nobel Prize for physics. Bose-Einstein condensation has not only helped discover new phenomena like superfluidity, where fluids flow with no friction, and superconductivity, where electrons flow through a material with no resistance, but also extended our understanding of quantum mechanics.

Chapter 3

Born Rebellious

After graduating from the Gymnasium, Einstein wanted to enter the Eidgenössische Technische Hochschule (ETH). After failing the first time, Einstein took the ETH exam in the Summer of 1896 and passed at the age of seventeen. ETH did not require graduation from a gymnasium as a precondition. Among his classmates were Marcell Grossmann and Mileva Maric, whom later they got married. Lectures were usually an intrusion to him. He was already clear that his interest is in physics rather than mathematics. During his four-year education at ETH, he had to pass two major exams. Thanks to his classmate Grossmann’s meticulous lecture notes and who shared them generously with Einstein, he graduated in 1900. Grossmann was a fast learner and a brilliant mathematician. The two became close friends quickly. Grossmann’s lecture notes allowed Einstein the time to follow his own path and master Maxwell’s theory on his own, which to his disappointment was not covered much in lectures.

3.1 Difficult Times for Einstein With graduation came the difficult times for Einstein (Fig. 3.1). With his distrust to authority, he had already alienated his professors, among whom was none other than Heinrich Weber, a leading authority of that time. Weber expressed his irritation with Einstein as “You’re a clever fellow! But you have one fault. You won’t let anyone tell you a thing.”1 He was now 21 and looking for a university position (assistantship). He kept getting rejections on grounds that he was not in good terms with his former professors at ETH. During these turbulent times, while trying to make ends meet with temporary jobs like teaching and private tutoring, he still found some comfort in science when new ideas started to dance in his mind. He wrote a research article on capillarity that was published in Annalen der Physik in 1901, a leading journal at 1

Hoffman and Ducas (1972), p. 32.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_3

33

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Fig. 3.1 Albert Einstein, 1879–1955 (AIP Emilio Segrè Visual Archives)

that time. Young Einstein had high hopes on this paper. He immediately sent a copy along with an application for an assistantship to Prof. Wilhelm Ostwald, who was a leading chemist at that time and who would later win the Nobel Prize. Unfortunately, Ostwald never returned Eintein’s inquiry for a position. Maybe he never got his letter, but Ostwald would be the first to recommend Einstein for the Nobel Prize.2

3.2 Patent Office Clerk At this point, his close friend Marcell Grossmann comes to rescue. Grossmann himself was an assistant; hence, he was in no position to offer Einstein a job, but he mentioned Einstein’s situation to his father and requested help. Grossmann was not only a close friend but also somebody who appreciated Einstein’s potential and talent. With Grossmann’s father’s help, Einstein gets a provisional job as a clerk in the Swiss Patent Office in Bern. While waiting for the job at the Patent Office to realize, Einstein completes another paper on thermodynamics, which he sends to the University of Zurich to get his doctoral degree. However, it gets rejected by Prof. Kleiner as a PhD thesis.3 Shortly after this rejection, the news that his paper got accepted

2 3

Pais (1982), p. 45. Pais (1982), p. 46.

3.2 Patent Office Clerk

35

for publication in Annalen der Physik came.4 While his PhD was still in limbo, on June 23, 1902, Einstein officially started to work as a probationary Technical Expert, Third Class, with a modest salary of 3500 francs a year.5 Finally, Einstein had a steady job and an income. While working at the Patent Office, Einstein was also following the developments in physics and publishes his third paper in Annalen der Physik. It is remarkable how Einstein maintained his productivity with whatever little time is left from his duties at the Patent Office, which also demanded excellence. His fourth paper was followed in 1904 by his fifth, both published in Annalen der Physik and both were on thermodynamics. These papers will later turn out to be the precursors, if not the prerequisites, of what is to come in 1905.6 Einstein was primarily self-taught. His expertise on Maxwell’s theory and Boltzmann’s kinetic theory, which were still yet to gain general acceptance and which were not covered in lectures properly during his student years at ETH. Maxwell’s theory was confirmed only in 1888 when Heinrich Hertz electromagnetically generated and detected radio waves. Einstein’s impressive knowledge of Maxwell’s electrodynamics was instrumental in getting the job at the Patent Office. Einstein’s mastery of thermodynamics and the second law of thermodynamics and his knowledge of the kinetic theory was also primarily self-taught. With his limited access to a scientific library, Einstein has probably covered some of the ground already covered by Boltzmann and Williard Gibbs,7 but this helped him to deepen his insight and also helped him understand and internalize these new and complex concepts, and then go beyond the works of these great scientists. At the beginning, even though Einstein would have preferred an academic job, the Patent Office was a blessing in disguise to him that helped sharpen his mastery of Maxwell’s theory and thermodynamics by looking for fallacies in other’s ideas. During his job at the Patent Office, he probably examined quite a few patent applications involving perpetual motion machines, which are prohibited by the second law. In an academic job, he would be forced to work on problems that are more in line with the mainstream, and hence, his work on his revolutionary ideas would have been hampered. His probationary position at the Patent Office changed to a permanent position in September 1904. At this time, he urged his old friend Michele Besso to take a job at the Patent Office. Besso was a talented and a passionate Italian engineer. As new ideas were blossoming in Einstein’s head, with his enthusiasm and critical comments, Besso was just the perfect listener that Einstein needed. Stimulating conversations with Besso, both at work and on the way home helped Einstein sharpen his ideas and helped prepare answers for potential objections. The year 1905 is usually referred to as the Annus Mirabilis; the Miracle Year in Latin, when Einstein’s genius went into high gear and produced five papers. Each one of these papers was major contributions that shaped our understanding of the universe and formed the foundations of modern physics. Historians usually compare it with the years 1965–66, when the plague4

Pais (1982), p. 45. Pais (1982), p. 47. 6 Pais (1982), p. 55. 7 Pais (1982), p. 55. 5

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stricken England forced Newton to leave Cambridge and worked at home in secrecy to develop calculus, his theory of light, and laid the groundwork of his theory of gravitation.8

3.3 Miracle Year of Physics 1905 is usually referred as the Miracle Year of Physics. In 1905, Einstein published five papers in Annalen der Physik. The first paper written on March was on the photoelectric effect and was sent to the Annalen der Physik three days after his 26th birthday. The second one was sent almost one month after the first one. It was about a new theoretical method for calculating molecular radii and Avogadro’s number. It was entitled “A New Determination of the Sizes of the Molecules.” It was also submitted to the University of Zurich as a potential PhD thesis. After the rejection of his first attempt in 1901 by Kleiner, this one, only 24 pages, also gets rejected on grounds of being too short. Without any delay, Einstein resubmits it with a single sentence added. This time it gets accepted. He is now Herr Doktor Albert Einstein.9,10,11 Einstein’s paper on the Brownian motion was received by Annalen der Physik on May 11, 1905, only 11 days after his thesis was finished.12 As Abraham Pais says “It is not sufficiently realized that Einstein’s thesis is one of his most fundamental papers.”13 Pais also continues “In my opinion, the thesis is on par with the Brownian motion article. In fact, in some—not all—respects, his results on Brownian motion are by-products of his thesis work.” Historians usually refer to the May 11 paper as one of the Miracle Year papers. Einstein’s papers on Brownian motion are collected in a nice little book by Fürth.14,15 The fourth Miracle Year paper was the one on special relativity entitled “On the Electrodynamics of Moving Bodies” completed in June and the fifth one entitled “Does the Inertia of a Body Depend on its Energy Content?” was published in September.16 What is incredible is that Einstein was juggling all these revolutionary ideas at the same time in his mind, while attending his duties at the Patent Office.

8

Hoffman and Ducas (1972), p. 42. Baggott (2016), p. 20. 10 Hoffman and Ducas (1972), p. 55. 11 Pais (1982), p. 89. 12 Fürth (1956). 13 Pais (1982), p. 89. 14 Fürth (1956). 15 Pais (1982), footnote p. 93. 16 Baggott (2016), p. 20. 9

3.4 Photoelectric Effect

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3.4 Photoelectric Effect The first paper on the photoelectric effect introduced his light-quanta hypothesis using Boltzmann’s statistics. It also gave the first correct application of the equipartition theorem to radiation. From late 1900 to 1905, Planck’s quantum was treated by the physics community with silence. To Planck, the quantum hypothesis was a distasteful concept and like Planck, most other physicists thought it as a mathematical artifact that will go away when the proper derivation of the black body radiation formula is found.17 The photoelectric effect is the emission of conduction electrons when light hits a metal plate. Einstein started his paper by pointing out the conflict between Maxwell’s theory and the experimental results.18 According to the classical theory, light transfers energy to the electrons continuously, which are then released when sufficient amount of energy is accumulated. With increased intensity, energy transfer is expected to increase, and hence, the number of electrons released should also increase. However, experiments showed that electrons are released only when the frequency of the light exceeded a certain value—regardless of the intensity of the impinging radiation or its duration. On the other hand, low-frequency radiation regardless of its intensity and duration did not release any electrons. This led Einstein to postulate that light is not a wave but a swarm of particles, later called photons. This was no lucky guess or blind assumption, Einstein had solid arguments behind it. He started with Wien’s limit formula for the black body radiation, which failed for low frequencies but worked almost perfectly for high frequencies. This helped him avoid a particular model that Planck had used in terms of springs or oscillators. He then used a formula that Wien has given for the entropy of radiation and applied Wien’s limit formula to it. The result was interesting. It looked like the entropy of a gas of particles.19 Next, he turned to Boltzmann’s probabilistic definition of entropy and showed that the energy of these particles of light is proportional to their frequency with the proportionality constant being equal to h, later called the Planck constant, the same constant that Planck had used in his black body radiation formula. The bold conclusion of this paper was that Planck’s black body radiation formula made sense only if radiation is thought as a gas composed of discrete packets (particles) of energy, which Einstein called light-quanta.20 Einstein was taking the bold step of extending Planck’s quanta to light itself.21 With this, Einstein was not scrapping the entire wave mechanics. There was just too much evidence for the wave properties of light like diffraction and interference. Einstein reconciled the wave and the particle properties of light by saying that the wave phenomena are just the time average of the collective effect of the light-quanta.22 17

Hoffman and Ducas (1972), p. 49. Hoffman and Ducas (1972), p. 51. 19 Hoffman and Ducas (1972), p. 51. 20 Baggott (2016), p. 20. 21 Hoffman and Ducas (1972), p. 52. 22 Baggott (2016), p. 22. 18

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Using his light-quanta, Einstein now turned to the photoelectric effect, which was observed by Phillip Lenard in 1902, who also pointed out the fact that his results are in conflict with Maxwell’s theory.23,24 Einstein not only showed that his lightquanta could explain Lenard’s results with utmost ease, but went beyond Lenard’s experiments and predicted that a plot of the voltage of the photo-ejected electrons versus the frequency of the incident light would be a straight line with the slope independent of the type of the metal used. The slope was the Planck constant h.25 Physicists tried to check Einstein’s predictions, but even as late as 1913 the results were inconclusive. That same year, Einstein was nominated for membership to the Prussian Academy by Planck and some other distinguished members.26,27 They acknowledged his remarkable contributions, but also acknowledged that “he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light-quanta, can not be really held against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk.”28 Apparently, Planck and majority of the establishment still thought that light is continuous and Planck’s fixed energy quanta could still be explained in terms of some yet unknown properties of the atoms that formed the inside of the black body. An experimental physicist, Robert Millikan, who accurately measured the charge of the electron, decided to investigate the photoelectric effect, to show once in for all that Einstein was wrong. After struggling for ten years, to his surprise, he found complete agreement of his results with Einstein’s predictions. His final results were published in 1916, but he still could not believe the revolutionary idea of light-quanta.29 In 1921, Einstein won the Nobel Prize in physics, where the photoelectric effect was the only paper cited. Aside from establishing the quantum property of light, today photoelectric effect has found many interesting uses like photomultipliers, image sensors, and night vision devices.

3.5 Brownian Motion and the Reality of Atoms Einstein’s third paper entitled “On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular Kinetic Theory of Heat”30 was received on May 11 and published on July 18. At that time, the kinetic theory, despite its successful results, was still considered controversial. A lot of the prominent

23

Baggott (2016), p. 23. Hoffman and Ducas (1972), p. 52. 25 Baggott (2016), p. 23. 26 Baggott (2016), p. 24. 27 Hoffman and Ducas (1972), p. 54. 28 Baggott (2016), p. 24. 29 Hoffman and Ducas (1972), p. 54. 30 Fürth (1956). 24

3.5 Brownian Motion and the Reality of Atoms

39

scientists of the time like Planck and Mach did not believe in the existence of atoms, which were at most considered as useful constructs but not real entities. In 1827, botanist Robert Brown investigated pollen dust suspended in water under a microscope and discovered that the pollen particles exhibited a random continuous zigzag motion.31 At first, Brown thought that he discovered the “primitive molecule” that is alive. Early research eliminated this as well as other explanations like temperature gradients, mechanical disturbances, capillary actions, and convection.32 For decades, this event lacked satisfactory explanation, until Einstein explained it as due to the random collisions of the water molecules with the pollen particles. The rapid zigzag motion of the pollen particles was difficult to observe. Since the pollen particles are bombarded by the water molecules randomly from all directions, their average displacement is zero, x = 0. Einstein showed that due to fluctuations, these rapid zigzag motions in time cause a slow drift of the pollen particles in a random direction as33    2 RT t, (3.1) x = 3π N aη   where x 2 is the mean square displacement. This was Einstein’s fundamental result for the Brownian motion, where R is the gas constant, T is the temperature, N is the Avogadro number, a is the average size of the pollen particles, and η is the viscosity of the fluid. To arrive at this result, Einstein considered the problem of small hard spheres (Pollen particles) in a fluid, where the fluid is assumed to have molecular structure and also assumed that the spheres are enormous compared to the molecules so that their motion could be observed with a microscope. An individual molecule would not be enough to budge the sphere, but many molecules might. However, the fluid molecules would be hitting the spheres randomly from all directions, and hence their effect on a sphere would cancel each other giving zero for the average displacement x of the sphere. However, Einstein showed that statistical fluctuations cause imbalances that are large enough to give the spheres an erratic zigzag motion that could be observed with a microscope. The erratic motion of the spheres is too fast to be observed, but given enough time, statistical fluctuations would cause  the spheres to drift relatively slowly in a random direction, thus giving a nonzero x 2 large enough to be observed with a microscope. Einstein also noticed that the migratory motion of these spheres is essentially a diffusion process that he investigated in his thesis. Combining the two results, he obtained the formula in Eq. (3.1). All the parameters in Eq. (3.1) are measurable quantities. In 1908, Perrin’s experiments verified this result to the desired accuracy demanded by Einstein’s predictions. After Perrin’s experiments, in 1908, Wilhelm Ostwald, who was one of the leaders of the anti-atom school, was convinced to the existence of atoms.34 However, another 31

Clark (1972), p. 61. Pais (1982), p. 93. 33 Pais (1982), pp. 88–97. 34 Pais (1982), p. 103. 32

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prominent member of the anti-atom school, Ernst Mach, died still unconvinced in 1916.35 Jean Baptiste Perrin was the recipient of the 1926 Nobel Prize in Physics for his work on the discontinuous structure of matter.36 Today, using a method called electron ptychography scientists have actually reached a level of accuracy where they could take detailed images of atoms.37 Brownian motion is also considered today as the prototype of many different phenomena in diffusion, colloid chemistry, polymer physics and even in finance, where small objects interact randomly with their environment.

3.6 Universe with a Limit to Speed Einstein’s fourth Miracle Year paper entitled On the Electrodynamics of Moving Bodies was on the special theory of relativity.

3.6.1 Reference Frames The connection between nature and mathematics starts with the definition of a coordinate system. Since we live in a three-dimensional universe, each point has three codes, called the coordinates. The simplest coordinate system that we can use is the Cartesian coordinate system defined by three orthogonal lines, usually called the x-, y-, z-axes (Fig. 3.2). Think of them as one of the corners in your room. Anything happening in or outside your room, such as the presence of a particle at some point P, can be defined by giving its coordinates as P(x, y, z). If the particle is moving, then its coordinates will depend on time as (x(t), y(t), z(t)). An extended object will be represented by a set of coordinates moving together, which represents the parts of the object. Once the events in the universe are reduced to numbers, then the regularities among them can be represented in terms of mathematical objects defined in the world of numbers. Let us now consider another observer living in a neighboring apartment and using the corners of his/her living room walls as the coordinate axes. Naturally, the coordinates that the second observer will assign to the same event will be different as (x  (t), y  (t), z  (t)). Since in Newton’s theory time is absolute, that is, universal, time t is the same, t = t  , for both observers. In other words, both observers will have the same time. Once they synchronize their clocks, the clocks will always remain synchronized. Naturally, the true laws of nature should not depend on how the events are codified. For example, Newton’s law, F = ma, in one Cartesian coordinate system, should be 35

Pais (1982), p. 103. Pais (1982), p. 103. 37 Blaustein (2021). 36

3.6 Universe with a Limit to Speed

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Fig. 3.2 In Cartesian coordinates, coordinates of a point P are found by dropping perpendiculars to the coordinate axes

F = ma in another Cartesian coordinate system. Of course, the coordinates corresponding to the events will be different for different observers. This is called form invariance, or covariance. In this example, both observers are at rest with respect to each other. The only difference between them is that their Cartesian coordinate systems have origins displaced with respect to each other and the orientation of their axes are different. Cartesian coordinates were originally introduced by Rene Decartes (1596–1650) in the seventeenth century. Decartes’ work provided the groundwork that leads to the discovery of calculus by Newton and Lebniz. Cartesian coordinates are one of the many coordinate systems available in physics. Depending on the symmetries of the physical system, it may be advantageous to use some other coordinate system. For example, Newton’s law of gravitation in spherical polar coordinates (r, θ, φ) depends only on the radial coordinate r . Using a suitable coordinate system in physics simplifies the algebra and makes the interpretation of the solution easier. It is possible to write transformation equations that relate two Cartesian coordinate systems. As a matter of fact, in calculus different coordinate systems and their transformation properties are studied under the title generalized coordinates.

3.6.2 Moving Frames Let us now consider another observer living in a mobile home and traveling on the highway with uniform velocity V . His assigned coordinates for the same events will be (x  (t), y  (t), z  (t)). Since time is universal in Newton’s theory, time is still t and not t  . Since all three observers are inertial frames, observer in the mobile home will also find Newton’s second law as F = ma . In other words, Newton’s law will have the same form for all inertial observers, and thus, they are covariant. In classical physics, if we have an inertial reference frame, we can construct infinitely many others. All reference frames moving with uniform velocity with respect to an inertial frame will also be an inertial frame. Therefore, uniform motion is relative and all the

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inertial reference frames has the right to think that they are at rest and the others are moving. If the mobile home has no windows, then there is no experiment that the observer in the mobile home can do to detect uniform motion. If he can look outside, he will think he is at rest and the other observers are moving in the other direction.

3.6.3 Covariance Versus Invariance Covariance is sometimes confused with invariance. An invariant property is a property whose value does not change when you change your coordinate system. For example, even though the coordinates assigned to a point are different in different coordinate systems, the distance between two points is an invariant. In other words, the same number for all observers or coordinate systems. In this regard, in Newton’s theory space and time are both absolute. That is, intervals in space and time are the same for all inertial observers. Such quantities that preserve their values under coordinate transformations are called scalars. Another important scalar quantity in Newton’s theory is the mass.

3.6.4 Galilean Transformations Naturally, we would like to have a dictionary that allows us to translate the coordinates assigned in one inertial frame to another. In Newton’s theory, Galilean transformations are the dictionary needed. An important property of the Galilean transformations is that clocks run at the same rate for all inertial observers. Once the clocks are synchronized, they remain synchronized. Even though different inertial reference observers assign different coordinates to the same event, distances in space and time are invariant. In other words, space and time are absolute. A key feature of the Galilean transformations is known as the addition of velocities. Consider a car moving with uniform velocity V and a ball is thrown from the car in the direction the car is moving with velocity v. An observer at rest on the ground will see the velocity of the ball as v + V. This simple and very intuitive fact was where one of the first cracks in Newton’s theory appeared as a 16-year-old Einstein started daydreaming in science classes about what happens if he moves along with a light beam.

3.7 Problems with Newton’s Laws In 1887, American physicists Albert Michelson and Edward Morley attempted to detect the existence of ether. Like sound waves traveling in air, ether was thought to be the medium that permeate all space and needed for the propagation of light waves. Michelson and Morley used an interferometer to compare the speed of light in two

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perpendicular directions. They expected to see a result in line with the addition of velocities that Galilean transformations imply. The null result of this experiment has been repeated many times over the past years. It is generally considered as the spark that ignited a series of results that culminated in 1905 with the discovery of special relativity by Albert Einstein in his famous paper “On the Electrodynamics of Moving Bodies.” At first, the ether hypothesis was not easy to give up. Length contraction was postulated by George FitzGerald (1889) and Hendrik Lorentz (1892) to save the stationary ether concept. Lorentz-FitzGerald contraction was also extended by Lorentz (1892–1904) and Joseph Larmor (1897–1900) to show that in moving frames clocks run slower. These two results, unlike what they were intended to do, showed that space and time are not absolute. Distances in space and time intervals are relative and change when viewed from a moving frame. Furthermore, in 1904 Lorentz showed that unlike Newton’s equations, which are covariant under Galilean transformations, Maxwell’s equations are covariant under a new set of transformation equations called the Lorentz transformations. In 1906, Henri Poincare established the group property of the Lorentz transformations, thus establishing its mathematical basis. Physics community was faced with a dilemma. Newton’s theory, which for over two centuries have been tremendously successful in explaining countless earthly and astronomical data and covariant under the Galilean transformations, which are also extremely intuitive and a part of our everyday experiences. On the other hand, Maxwell’s electromagnetic theory, which was also an extremely successful theory that unified electric and magnetic phenomenon, but covariant under a new set of transformation equations called the Lorentz transformations.

3.7.1 Nature of Physical Theories Neither Newton’s equations nor Maxwell’s equations are laws in the strict sense. They are based on some assumptions. It is probably more appropriate to call them theories or models. Assumptions are frequently used in science. Sometimes to concentrate on special but frequently encountered cases, we keep some of the parameters constant and vary the others to avoid unnecessary complications. At other times, because of the complexity of the problem, we restrict our treatment to certain domains like small velocities, high temperatures, and weak fields. However, the most important of all are the assumptions that sneak into our theories without us noticing. Such assumptions are actually manifestations of our prejudices about nature. They look so natural that we usually don’t recognize them as assumptions. In fact, it sometimes takes generations before they are recognized as assumptions. Once they are noticed and theories are reformulated, dramatic changes take place in our understanding of the universe.

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3.7.2 A Stroke of Genius In fact, it took Einstein’s genius to notice that space and time are not absolute and all laws of nature, not just Maxwell’s equations, should be covariant under Lorentz transformations, Einstein gave an insightful derivation of the Lorentz transformations when he introduced the Special Theory of Relativity, which he based on two postulates or principles: (1) The laws of physics are covariant in all inertial reference frames. (2) The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the light source or the observer. Among its many revolutionary consequences, this new theory also answered the 16year-old Einstein’s question; what happens if I move along with a beam of light? In Newton’s theory, there is no upper limit to the speed that one can travel. In special relativity, we can increase the energy of a particle as much as we want but its speed will not increase beyond the speed of light. For massive particles, speed can approach the speed of light but it can never exceed it. This follows from another relativistic effect, which says that mass increases with velocity. As the speed of the particle approaches the speed of light, it becomes infinitely heavy, thus making it impossible to reach the speed of light by adding more energy. In the 27-kilometer ring of the large hadron collider speed of the protons is 0.999999990c. By building a bigger collider, we can increase the energy of the protons, but as far as their speed is concerned, we can only increase the number of nines on the right-hand side of the decimal point. Only the particles like photons and neutrinos with zero rest mass can move with the speed of light. Rest mass is defined as the mass of a particle when it is at rest. In Lorentz transformations, the Galilean velocity addition formula is replaced by a new one. If one shines a light beam from a car moving with the speed of light in the direction the car is moving, the light beam will still be moving with the speed of light for an observer at rest.

3.7.3 Speed of Light The speed of light is very large but it is still finite. It is so large that astronomical distances are measured in terms of the distance that light travels in one year, 9.6 × 1010 km, which is called the light year. The nearest star Alpha-Centauri is 4.2 light years away from us and the nearest galaxy Andromeda is 2.5 million light years away. The sun from earth is only 8 min away. In other words, the light we see now is the light that has left the sun 8 min ago. If something happens to the sun, we will feel it only 8 min later. Light from Mars takes approximately 21 min (as of 15 Nov. 2021). On earth, when you say stop to the Mars rover, it will keep on going for 21 more minutes. That is why we need intelligent guidance systems that can decide on the spot. When we look at the galaxies, we not only see their past but also the past of

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the universe. Photons from the cosmic background radiation (CBR) are the photons that were last scattered when our 13.8-billion-year-old universe was only 380,000 years old and have been traveling freely since then. In this regard, CBR is packed with information about the early universe. The second postulate of Einstein is not just about the invariance of the speed of light, but it is also the fastest speed that anything can interact with anything else in the universe. That is, it is the maximum signal speed that one can use to communicate. Occasionally, we run into situations where the speed appears to exceed the speed of light. For example, in ultra dense matter, sometimes the phase velocity of the speed of sound exceeds the speed of light. However, it is the group velocity of the sound that information travels, which is always less than the speed of light. Consider two entangled electrons that are light years apart with spins pointing in opposite directions. In quantum mechanics, when the spin of one of the electrons is measured, instantaneously the spin of the other electron flips the other way. This is what Einstein called “spooky action at a distance” and worried none other than Bohr for a while, but then it became clear that this cannot be used to send signals faster than the speed of light. We will come back to this point in detail with the quantum mechanics.

3.7.4 Space and Time Space and time are absolute in Newton’s theory. These facts are manifested in Galilean transformations, which are nothing but a dictionary that allows one to translate the coordinates and time that one inertial observer assigns to another. Since in Galilean transformations time runs at the same rate for all inertial observers, all inertial observers can use the same time. In other words, in Newton’s theory time is absolute and thus treated as a parameter. On the other hand, the absolute space concept is somewhat different. Unlike time, different inertial observers assign different coordinates to the same point, but the spacial relations between these points, like their separations, remain the same for all inertial observers. This also implies that rigid objects preserve their shape under Galilean transformations. This is also tied to the Newtonian concept of simultaneity, where infinite signal speed is possible. Simultaneous events like measuring the distance between two points will also be simultaneous in another inertial frame. In Einstein’s special relativity, both space and time are relative as described by the Lorentz contraction and time dilation formulas. The time dilation is conveniently demonstrated by the light clock as shown in Fig. 3.3. For an observer moving with the clock, the light beam follows the straight line path taking the time 2t between the two successive ticks of the clock. For the observer at rest, the clock is moving with the velocity v, and thus, light takes the longer path giving 2t  for the time between the two ticks. The two times are related by (Fig. 3.3) t = t  (1 − v 2 /c2 )1/2 .

(3.2)

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Fig. 3.3 Time between the two tics of a moving clock is longer, which is found from the relation c2 t 2 = v 2 t 2 + c2 t 2

In other words, time runs slower for the moving clock. Of course, this is a light clock, but special relativity says that all clocks regardless of their internal structure will slow down by the same amount. This has been experimentally verified in several ways, but a striking evidence comes from the μ-meson decay, which are produced in the upper atmosphere. According to their lifetime measured in laboratory, they should not be reaching the lower atmosphere where they are observed. The answer comes from the fact that due to time dilation their clock runs slower than the clocks on earth, thus allowing them to reach the lower atmosphere.

3.8 Minkowski Space-Time In Einstein’s special relativity space and time are relative but they are not treated the same way. In some papers, as Hans Reichenbach says “the treatment of time as parallel to that of space has been detrimental.”38 The differences between time and space manifest itself in several ways. For example, space and time are not symmetric; that is, one can be at the same point in space at different times but one cannot be at different places at the same time. Also, as Reichenbach continues “time order is possible, in a realm which has no spatial order, namely the world of the psychic experiences of an individual human being.”39 In relation to this, the second law of thermodynamics states that the total entropy, or disorder, of a system always increases or at most remains constant in any spontaneous process. The second law of entropy can also be stated as that the entropy of isolated systems left to spontaneous evolution cannot decrease. They always arrive at a state of thermal equilibrium, where the entropy is the highest. In cosmology, assuming that a finite universe is an isolated system, the second law states that its total entropy always increases. In this regard, entropy is usually considered as defining the direction of time. Let us now consider a vase fallen off a table and got broken into pieces. This naturally increases the entropy of the universe. However, you can pick up the pieces and then glue them together, which you may think will decrease entropy. Yes, it does decrease the entropy of the broken vase, but in the process, you have crouched to 38 39

Reichenbach (1958), pp. 109–110. Reichenbach (1958), pp. 109–110.

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Fig. 3.4 Hermann Minkowski, 1864–1909 (H.A. Lorentz, A. Einstein, H. Minkowski DAS Relatitatsprlnzip, 1915, courtesy of AIP Emilio Segrè Visual Archives, Born Collection)

pick up the pieces, then went to your desk to search for an appropriate glue, and then struggled to figure out how to bring the jumbled up pieces together in the right order. In this process, you have got tired, frustrated, and worked up a sweat. Yes, when you finally glued the pieces together, entropy of the broken pieces has decreased, but the heat you have generated and eventually dumped into the environment has actually increased the entropy of the universe more than the decreased entropy of the glued pieces. Nobody has seen the pieces of a broken vase coming together on their own and forming the vase. As far as the conservation of energy is concerned, in principle, it is possible, but the broken vases always remain broken unless somebody glues the pieces in the right order and contribute to the disorder in the universe by getting tired. This is why time rather than space has been the center of attention in philosophy and psychology. In neuroscience, biological processes involved in the brain in regard to our perception of time are also being actively investigated. All this being said, the treatment of time as the fourth number, x0 , in a fourdimensional manifold as (x0 , x1 , x2 , x3 ) has some advantages in mathematical physics. The first number, x0 , corresponds to the time of an event, ct, and the next three numbers (x1 , x2 , x3 ) correspond to the codes, that is, the coordinates giving the spacial position (x, y, z) of the event in three dimensions. By all means, this does not allow us to interpret time as the fourth coordinate. This representation merely

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Fig. 3.5 An event P in four-dimensional space-time is represented by four numbers (x0 , x1 , x2 , x3 ) giving the when and where information of that event

points out the fact that one needs four numbers to define a world event; that is, you have to tell where and when an event has happened. In ordinary Euclidean manifold, the distance between two infinitesimally close points is given by the Pythagorean theorem: (3.3) ds 2 = d x02 + d x12 + d x22 + d x32 . In 1908, Hermann Minkowski introduced the space-time representation of the world that is compatible with the postulates of special relativity and one that also displays the singular nature of time (Figs. 3.4 and 3.5). In Minkowski space-time, the interval between two infinitesimally close events is given by ds 2 = c2 dt 2 − d x 2 − dy 2 − dz 2 ,

(3.4)

where ds is invariant for all inertial observers. In other words, even though the spacial distances and the time intervals are no longer preserved under Lorentz transformations, distances in space-time, ds, are invariant. This is basically the first principle of relativity. For two events connected by a light wave ds is zero, hence d x 2 + dy 2 + dz 2 = c2 dt 2 .

(3.5)

This is the second postulate of special relativity, which says that for all inertial observers the speed of light is the same. This is one of the main points that separate the four-dimensional space-time manifold from the four-dimensional Euclidean space Eq. (3.3). The distance between two events, like the emission of a photon from a galaxy and its subsequent absorption in a detector billions of light years away on earth has zero distance between them in Minkowski space-time. Assuming that Eq. 3.4 transforms as ds 2 = c2 dt 2 − d x 2 − dy 2 − dz 2 ,

(3.6)

which implies a linear transformation, we can obtain the Lorentz transformations that relate two inertial observers S  and S, where S  is moving with respect to S with

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constant velocity v along the common x, x  -direction as x = 

1

(x − vt), 1 − v 2 /c2 y  = y, z  = z, 1 t =  (t − vx/c2 ). 1 − v 2 /c2

(3.7) (3.8) (3.9)

In other words, derivation of the Lorentz transformations is possible by imposing the invariance of ds. Reichenbach says “The Lorentz transformations can therefore be exhaustively defined in a purely mathematical fashion, if we specify that it leaves expression Eq. (3.4) invariant… .”40 Minkowski space-time representation of special relativity is neither a new nor a different theory. However, it not only broadens the scope of special relativity, but also paves the way to Einstein’s general theory of relativity. In the limit as the speed of light c goes to infinity, or when v 2 /c2  1, Lorentz transformations reduce to the Galilean transformations: x  = x − vt,

(3.10)

y  = y, z  = z, t  = t.

(3.11) (3.12)

The inverse transformations are simply obtained by changing v to −v.

3.8.1 Lorentz-FitzGerald Contraction We now consider a rod of length L 0 lying along the x-axis in the S frame. Using Lorentz transformations, we can find how it looks in the S  frame moving with respect to the S frame along the common x, x  -axis with the velocity v. Using the first transformation Eq. (3.7), we can compare space intervals in both frames as  x  = x (1 − v 2 /c2 ).   Taking L 0 = x we find L 0 = L  / (1 − v 2 /c2 ), or L 0 (1 − v 2 /c2 ) = L  . In other words, from the moving frame the rod at rest in S appears shorter. Since v appears as v 2 , it does not matter which frame we call S and which frame S  . According to length contraction, a rocket will appear shorter from the ground when launched. Of 40

Reichenbach (1958), p. 177.

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course, for ordinary objects usually v 2 /c2  1, hence this effect is noticeable only as we approach the speed of light.

3.8.2 Time Dilation Another consequence of the Lorentz transformations is the time dilation, where clocks moving with respect to an observer at rest appear to run slower. To see how this comes about, using Lorentz transformations Eq. (3.9) we compare time intervals in S  and S frames as  t  (1 − v 2 /c2 ) = t. In other words, a time interval of t  in the moving frame S  will appear shorter to the S observer. That is, the moving clocks run slower. A dramatic application of both length contraction and time dilation came with the decay of μ-mesons, which are created in the upper atmosphere by high energy cosmic-ray particles. μ-mesons have typical velocities around 0.998 c and lifetimes around 2 × 10−6 s. Even at such high speeds, due to their short life span they should not be able to reach the ground level where they are observed. The solution comes from special relativity. From the μ-meson’s point of view, the distance they have to travel to the ground level is significantly shortened due to length contraction, thus making it possible for them to reach the ground. On the other hand, from the viewpoint of an observer on the ground, due to time dilation, the lifetime of the μ-mesons is significantly longer, giving them the time to reach the ground level.

3.8.3 Regions of Space-Time Minkowski’s geometric representation of space-time allows us to view the universe in terms of world events, where each event is represented by four numbers, three of which represent the location in space and the fourth number is giving the time of the event. Let us now consider a case where the fastest way to communicate between two cities, say I and II, is by mail and a mail from I to II takes two days. Let A be the event corresponding to the mailing of a letter from I and event B the reception of the mail at II. Now, all the events that took place before the event A are in the past of the event B. In other words, they are the events that could affect B. All the events that took place after the reception of the mail, that is, event B, are in the future of event A. These are the events that A could affect. Events that took place while the letter was in transit are indeterminate as far as their time order is concerned.

3.8 Minkowski Space-Time

51

Fig. 3.6 Regions of space-time and the worldline of a particle moving with respect to O

Similarly, we can also divide the Minkowski space-time manifold into four regions defined by the light cones c2 t 2 = x 2 . These are the events that are connected to O by a light wave (Fig. 3.6). Events in regions I and II are called time-like events with respect to O, where ds 2 > 0. Region I corresponds to all the events that lie in the future light cone of O. They are the events that O could affect. Region II corresponds to all the events that lie in the past light cone of O. They are the events that could have affected O. Events in regions III and IV are called space-like events, where ds 2 < 0. Trajectory of a particle moving with respect to O is called the worldline of P (Fig. 3.6). Since the speed of light is invariant, for O to effect an event in region III or IV it has to send a signal faster than the speed of light. If infinite signal speeds were possible, then the space-time picture would have only two regions. All the events in the upper half plane would be in absolute future of O and all the events in the lower half plane would be in the absolute past of O.

3.8.4 Geometric Representation Minkowski’s space-time representation allows us to view Lorentz transformations as rotations of the space-time axes. In Cartesian coordinate systems, rotations preserve the orthogonality of the Cartesian axes (Fig. 3.7). Due to the fact that space and time intervals transform differently in Lorentz transformations, the transformed axes are oblique and are rotated by the angle θ = tan−1 v/c (Fig. 3.8). Lorentz transformations also display the fact that simultaneous events in one reference frame are no longer simultaneous in the moving frame. Using the third Lorentz transformation Eq. (3.9),we see that space and time intervals are interconnected as t  = (t − xv/c2 )/ 1 − v 2 /c2 . Therefore, simultaneous events that are separated in space in the S frame, that is, t = 0, x = 0, are no longer simultaneous in the moving S  frame, that is, t  = 0. A point that Einstein emphasized in his 1905 paper and gave a definition of simultaneity. The geometric representation of the Lorentz transformations also shows us another important property. Even though space and time intervals measured in a moving frame are affected, as long as we are moving with speeds less than or equal to the speed of light, the order of events

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Fig. 3.7 Rotations between two Cartesian coordinates preserve orthogonality

Fig. 3.8 Geometric representation of Lorentz transformations

does not change. In other words, the cause-and-effect relation is preserved by the Lorentz transformations. In Galilean transformations space and time are absolute, hence simultaneous events in one inertial frame are also simultaneous in all other inertial frames.

3.9 An Iconic Formula Another groundbreaking consequence of Einstein’s special theory of relativity is the iconic formula

3.9 An Iconic Formula

53

E = mc2 ,

(3.13)

which in 1905 Einstein announced in a 3 page paper entitled “Does the Inertia of a Body Depend on its Energy Content?” as his fifth Miracle Year paper. Here, E is the energy, m is the mass, and c is the speed of light. The derivation was short and beautiful with just eight equations. It was almost as if Einstein pulled it out of a proverbial crystal ball. Its scientific and social reverberations were far-reaching and showed that mass is nothing but concentrated energy. Very few equations in science and engineering have been etched into so many brains. In the concluding remark of his derivation, Einstein said “If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2 , the fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that the mass of body is a measure of its energy content.” Einstein used L for E in his paper. Compared with the speed of sound, which is 343 m per second in dry air, the speed of light is 300,000 km per second, which is very large. This formula implied that even a tiny amount of matter corresponds to a huge amount of energy. Inside the sun, the amount of mass converted into energy through nuclear

Fig. 3.9 First Solvay Congress, Brussels, 1911, the theme was the theory of radiation and the quanta. L-R seated at table: Nernst; Brillouin; Solvay; Lorentz; Warburg; Perrin; Wien; Curie; Poincare. L-R Standing: Goldschmidt; Planck; Rubens; Sommerfeld; Lindemann; De Broglie; Knudsen; Hasenohrl; Hostelet; Herzen; Jeans; Rutherford; Kamerlingh-Onnes; Einstein; Langevin (Photographie Benjamin Couprie, Institut International de Physique Solvay, courtesy AIP Emilio Segrè Visual Archives)

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reactions is 6 million tons per second. In other words, every passing second, 6 million tons of matter is being converted into pure energy as radiation and lost from the sun. Mass of the sun is M = 2 × 1027 tons. This archetypal equation is the reason why stars shine, why life exists because the elements needed are synthesized in stars, and why nuclear reactors and nuclear power plants are possible (Fig. 3.9).

Chapter 4

Atoms and Radioactivity

During the last part of the nineteenth century, other important discoveries were also taking place. In 1896, Wilhelm Röntgen discovered X-rays and sent copies of his paper “A New Kind of Rays” written in German and the ghostly photographs of his wife’s hand showing her bones to leading scientists in Germany and around the world. With the press quickly catching up on this, his discovery became famous among scientists and public alike. It was later shown that X-rays are a form of electromagnetic waves. On the other hand, Henri Becquerel discovered that uranium compounds emit radiation, which he called “uranic rays.” The two types of radiation emitted by uranium compounds were called alpha and beta rays by Ernest Rutherford. Later, Gerhardt Schmidt discovered that thorium emits a new type of radiation called the gamma rays, which were shown to be another type of electromagnetic radiation. Marie Curie labeled substances that emit “Becquerel rays” as radioactive. With her husband, she also discovered that radioactivity is not confined to uranium alone and radium and polonium are also radioactive. In 1898, Curie’s first paper appeared in Paris. It was now established that radioactivity transforms one type of element into another. What exactly was an alpha particle? After it has been determined that beta particles are fast-moving electrons, this was the question that Rutherford wanted to answer. He had already found that they are fast-moving positively charged particles. With the help of Hans Geiger, in 1908 he determined that alpha particles are helium atoms that have lost all their electrons. Together with Geiger, they now design a simple experiment that involves passing alpha particles through a thin sheet of gold foil and then letting them hit a paper screen coated with zinc sulfide, where there they cause tiny flashes of light. Counting these flashes involved long hours of work in total darkness.1 As they suspected, alpha particles scattered most of the time by small amounts, while a few deflected by quite larger angles.2 However, when they checked if any were scattered backwards, to their surprise, they found that a few alpha particles scattered 1

Wilson (1983), p. 287. Pais (1986), p. 188. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_4

2

55

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Fig. 4.1 Ernest Rutherford (right) with Hans Geiger (left) who invented, between 1908 and 1913, a simple device for counting alpha particles one by one, conversing in Schuster Laboratory, University of Manchester, England (AIP Emilio Segrè Visual Archives, Physics Today Collection)

as if they were bounced off the gold foil. As Rutherford said “almost as incredible as if you had fired a 15-inch shell to a piece of tissue paper.”3 They published their results in 1909, while Rutherford pondered over its implications (Fig. 4.1). After Einstein’s work on the Brownian motion, reality of atoms was quickly gaining support in the scientific community. Structure of atoms was also becoming an increasingly active area of research. J. J. Thomson’s model of the atom was such that hundreds of electrons are distributed in a uniform sphere of weightless positive charge like raisins in a cake. Thomson’s model was not without problems. In 1910, Thomson was forced to rethink his model when experiments at Cambridge showed that he has grossly overestimated the number of electrons in an atom. The dramatic decrease in the number of electrons forced Thomson to assume that most of the mass of the atom is due to a diffuse positively charged sphere, in which the electrons are distributed in a way to cancel the electromagnetic forces to assure the stability of the atom. Rutherford was aware of the fact that even the modified version of Thomson’s model is not sufficient to explain his experiments. Rutherford’s data indicated that most of the atom is empty. It consisted of a very small positively charged central core, which bared almost all of the atom’s mass. Electrons are “like a fly in a cathedral” said Rutherford.4 According to Rutherford, their distribution was not important. After 3 4

Rhodes (1986), p. 49. Rowland (1938), p. 56.

4.1 Bohr at Cambridge

57

all, they were not responsible for scattering the alpha particles. Using his model, Rutherford was able to make definite predictions about the fraction of alpha particles that would scatter to any given angle. Geiger undertook the tedious task of checking Rutherford’s predictions and found complete agreement. Now, Rutherford was ready to announce his model of the atom in a 1911 paper, which he presented at a meeting of the Manchester Literary and Philosophical Society.5 Rutherford’s model was not free of problems either. The main problem was what would stop these electrons from falling on the positively charged nucleus. An obvious answer would be the centrifugal force that balances the sun’s attraction and keeps planets rotating in stable orbits. However, electromagnetic theory has one feature that separates it from gravity; that is, accelerating charges radiate electromagnetic waves. Therefore, electrons orbiting the nucleus like the “planets” would continuously lose energy by radiation and very quickly fall onto the nucleus, thus making Rutherford’s atom unstable. Rutherford was well aware of this problem, but in 1911 he chose to ignore it and thought that this is a problem that concerns the microstructure of the atom and that it is better left for future.6 In 1912, Rutherford’s laboratory performed a more complete check of the Rutherford’s scattering formula and also discovered that the charge of the nucleus is half of the atomic weight, with the exception of the hydrogen atom with the atomic weight one.7 This also determined the number of electrons in the helium atom as two.

4.1 Bohr at Cambridge At the end of September 1911, with a poor translation of his PhD thesis, Niels Bohr heads for Cambridge to work with J. J. Thomson, who had won the 1906 Nobel Prize for discovering the electron.8 Bohr considered Cambridge as an excellent center for physics, which it was, and also thought Thomson as a wonderful scientist to work with. Even though Bohr was always polite and courteous, his poor command of the English language often gave the wrong impression about him. In their first meeting, he entered Thomson’s office with a copy of his book on atomic structure and pointing to a specific place, and saying “this is wrong.”9,10 Of course, as a universally admired physicist, this didn’t bode well with Thomson. In fact, Bohr’s relation with Thomson never got better. While frustrated with the physics research he was doing in Thomson’s group, he tried to improve his English, he joined a local football club; he already had some success as a goalkeeper in Denmark. Bohr’s scholarship at Cambridge was for one year, and it was funded by the famous brewery Carlsberg. 5

Kumar (2014), p. 80. Rutherford (1911), reprinted in Boorse and Motz (1966), p. 709. 7 Kumar (2014), p. 81. 8 Baggott (2016), p. 26. 9 Baggott (2016), p. 26. 10 Pais (1991), p. 120. 6

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4.2 Bohr Meets Rutherford Bohr’s first meeting with Rutherford was during a visit to Manchester in early November 1911. Their second meeting was at the Cavendish research student’s annual dinner in early December. Impressed by the powerful personality of Rutherford, he decided to spend the last few months of his postdoctoral scholarship at Manchester with Rutherford. Bohr knew about the Ruthorford’s model of the atom, but his main interest was in radioactivity and the Manchester laboratory was the world’s leading center for it. Later that month, he expressed his interest in coming to Manchester to Rutherford, who agreed to accept him as long as he has the permission of Thomson at Cambridge. Thomson does not object and Bohr starts at Manchester in the middle of March 1912. Unlike Thomson, Rutherford was a charismatic leader who always took a keen interest in how all the people in his group are doing. Bohr started to work on experiments investigating the physical properties of metals. But he soon realizes that experimental physics is not his cup of tea and asks Rutherford if he could work on theoretical problems. Rutherford generally had a poor opinion of theoreticians. He once said “they play games with symbols, but we turn out the real solid facts of nature.”11 He somehow liked Bohr and saw something different in him. He once said “He is a football player.”12 Rutherford was not only an excellent charismatic leader, but also had an uncanny eye to spot talent around him. Eleven of his students eventually won the Nobel Prize, not to mention some of his close collaborators. Encyclopedia Britannica considers him as the greatest experimentalist since Michael Faraday. He was awarded the 1908 Nobel Prize “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances.”13

4.3 Bohr at Manchester At Manchester, Bohr meets Hungarian Georg von Hevesy, who won the Nobel Prize in 1943 for chemistry for developing the radioactive tracing technique, which became a powerful diagnostic tool in medicine as well as found widespread use in chemical and biological research. Both strangers in a foreign country with poor command of the English language, they bond very quickly. In particular, Hevesy helps Bohr to ease into the laboratory life. It was during conversations with Hevesy, that Bohr learns how so many radioactive elements have been discovered and the difficulties in classifying them in the periodic table. In 1910, Frederick Soddy proposed that chemically inseparable elements be called “isotopes”and hence should occupy the 11

Andrade (1964), p. 210. Rosenfeld and Rüdinger (1967), p. 46. 13 Encylopaedia Brittanica. 12

4.4 Germ of an Idea

59

same spot in the periodic table.14 This was strange because at that time elements were listed in order of increasing atomic weight. During chats with Hevesy, Bohr realizes that one has to differentiate between nuclear and atomic phenomena. At this point, he turns to Rutherford’s atom model. Ignoring its difficulties, he realizes that Rutherford’s model fixes the number of electrons in an atom, which for a neutral atom has to equal the nuclear charge. Therefore, hydrogen with the nuclear charge plus one must have a single electron. Helium with the nuclear charge plus two must have two electrons. At this point, it becomes clear to Bohr that it is the nuclear charge not the atomic weight that determines the location of an element in the periodic table. The essential point that Bohr captured was that radioactivity is a nuclear event, not an atomic phenomenon. When a radioactive element transmutes into another element by emitting alpha, beta, or gamma radiation, it is a nuclear event. This way, Bohr was able to move sideways or vertically in the periodic table. Rutherford approaches Bohr’s idea with a cold shoulder, pointing to the danger of “extrapolating from comparatively meager experimental evidence.”15 Even though Bohr told him that this would also be a proof of his atom, Rutherford didn’t budge. A part of the reason could be Bohr’s failure to express himself properly. Also, considering that at that time Rutherford was preoccupied with the writing of his book, he may not have been able to give his full attention to what Bohr was saying. Bohr made several other attempts to convince Rutherford, but his insistence in not to accept Bohr’s idea, eventually forced Bohr to let the matter rest. Frederick Soddy soon picks up the same “displacement laws” and eventually wins the Nobel Prize for chemistry in 1921. Despite losing a potential Nobel Prize due to his own lack of confidence in his idea and his supervisor’s lack of enthusiasm, Bohr’s respect and admiration to Rutherford never diminished and he always considered an approving statement from Rutherford as “the greatest encouragement for which any of us could wish.”16

4.4 Germ of an Idea After discouraged by Rutherford from pursuing his ideas on radioactivity, Bohr turns his attention to another problem, which was the distribution of electrons in an atom. In his scattering formula, Rutherford concentrated on the nucleus and ignored the effects of the electrons. A recent paper written by Charles Galton Darwin, grandson of the famous naturalist and the only theorist in Rutherford’s group, inspires Bohr. Darwin investigated the behavior of alpha particles as they pass through matter. In his scattering formula, Rutherford considered only the nuclei and ignored the effects of the electrons. On the other hand, ignoring the nuclei, Darwin assumed that as alpha particles passed through matter, they primarily lost energy due to collisions 14

Soddy (1913), p. 400. Niels Bohr, AHQP interview, 2 November 1962. 16 Pais (1991), p. 125. 15

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Fig. 4.2 Niels Bohr, 1885–1962 (Max-Planck Institute fur Physik, courtesy AIP Emilio Segrè Visual Archive, gift of Max-Planck-Institute via David C. Cassidy)

with the atomic electrons. Not knowing how the electrons are distributed, Darwin assumed that they are distributed evenly either inside, or on the surface of the atom. His results depended only on the size and the nuclear charge of the atoms. When his predictions differed with some of the experimental results, Bohr immediately noticed where the problem was (Fig. 4.2). Both Rutherford and Darwin in their calculations have ignored the interaction between the nucleus and the electrons. He was already approaching the end of his stay at Manchester. Even though he did not have anything concrete to present, he discusses his idea with Rutherford. This time Rutherford listens to him carefully and encourages him to continue. With this germ of an idea and the hope that he might convert Rutherford’s unstable atom into a stable quantum atom, he returns to Denmark.

4.5 Master Stroke Rutherford’s atom was successful in explaining the scattering of alpha particles from a gold foil. It implied that most of the atom is empty and almost all of its mass is at the center in the nucleus. However, Rutherford said nothing about the distribution of the electrons. Electrons could not be stationary, since they would be pulled onto the

4.5 Master Stroke

61

positively charged nucleus. Electrons could not be orbiting the nucleus like the planets either, since Maxwell’s theory predicts that they would lose energy continuously by emitting radiation, thus spiraling down onto the nucleus. For some, this was a fatal flaw for the Rutherford atom, but Bohr still thought he could find a way to stabilize the Rutherford atom. Since Newton’s dynamics and Maxwell’s theory put no restrictions on the orbits that the electrons could have, no choice but to think outside the box was left to Bohr. With Einstein’s and Planck’s examples in front of him, this was exactly what Bohr did. Since atoms exist and since they are stable, Bohr assumed that there must exist special stable orbits available to electrons. In these special orbits, electrons would not emit radiation and thus would not crash into the nucleus. It was a stroke of genius. Bohr knew that classical physics must not be valid in the atomic world, and hence these special quantized orbits should somehow involve the Planck constant. In September 1912, Bohr still did not have anything solid to present Rutherford and asks for additional time to complete the paper. Rutherford tells him not to rush since “it was unlikely anyone else was working along the same line.”17 In theory, it is easy to hypothesize, but one still needs a physical explanation, and of course, an experimental support at the end. Before Christmas, Bohr learns about a recent paper published in 1911 by John William Nicholson, who proposed a theory of atomic structure. For a moment, it gives Bohr the chills thinking that Nicholson may have already solved the problem. Soon he realizes that what Nicholson has done was not what he had in mind. Nicholson’s model was based on the idea that different elements are constructed from combinations of four primary atoms. These primary atoms are made up of a nucleus surrounded by a rotating ring of electrons. Nicholson showed that the angular momentum of a rotating electron ring could only be integer multiples of h/2π, where h is the Planck constant. Even though Rutherford’s comment on Nicholson’s model was “Nicholson has made an awful hash of the atom,”18 this was all that Bohr needed. Linear momentum of a particle moving along a straight line is its mass times velocity. Angular velocity of a particle moving in a circle of radius r is L = mvr.

(4.1)

Bohr applied this to hydrogen, which has only one proton and a single electron revolving around it. Quantizing the angular momentum of the electron as L n = mvn rn = nh/2π, n = 1, 2, . . . ,

(4.2)

where m is the mass, and v is the orbital velocity of the electron, gave Bohr a way to calculate the radius of his special orbits. Equating the attractive Coulomb’s force on the electron to the outward centrifugal force:

17 18

BCW, Vol. 2, p. 578, Letter from Ernest Rutherford to Bohr, 11 November 1912. Kumar (2014), p. 98.

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k

e2 v2 , = m r2 r

(4.3)

where k is a known constant from Maxwell’s theory, gave the orbital velocity as v 2 = ke2 /mr. Using this in the quantization rule Eq. (4.2), Bohr obtained the radius of his special orbits as  rn =

h2 4π 2 mke2

 n 2 , n = 1, 2, 3, . . . .

(4.4)

Using the radius of the lowest orbit, allowed Bohr to calculate the size of the hydrogen atom as 5.3 nm, where a nanometer is a billionth of a meter (10−9 m)—in close agreement with the experimental results of that day. Keep in mind that despite Einstein’s papers on Brownian motion, still there were a lot of scientists who were skeptical of the existence of atoms; hence, this alone was a very significant accomplishment. Using classical physics, Bohr also wrote the total energy E, that is, the kinetic energy plus the electrostatic energy of the electrons, mv 2 /2 − ke2 /r, as E = −ke2 /2r, which after substituting Eq. (4.4), allowed him to calculate the energy of the special orbits as En = −

2π 2 k 2 me4 h2



1 n2

 , n = 1, 2, 3, . . . .

(4.5)

Here, E 1 corresponds to the lowest energy state called the ground state. All the other states with n > 1 are the excited states. When an electron in an excited state drops to a lower energy state, it emits radiation with an energy equal to the difference of the energies of the two states. Bohr kept on working on his quantum atom throughout the rest of 1912 and into early 1913. But for his hypothesis about the special orbits that were stable to be accepted, he needed more results that can be justified with new or recent experiments. The breakthrough he needed came in February 1913, when Hans Hansen, a friend of him from their student days in Copenhagen, brought to his attention something called the Balmer series. Spectroscopy is the study of emission or absorption of electromagnetic radiation by atoms or molecules. Spectra of each element produced a unique set of lines like the fingerprints, where each line corresponds to a certain wavelength. However, the spectra of an element look in general too complicated to hope to find any pattern among its lines. The simplest atom is hydrogen, which also has the simplest spectra. In 1885, a Swiss mathematician Johann Jacob Balmer, interested in numerology, finds that the wavelenghts of the four consecutive lines in the spectra of hydrogen, whose wavelengths were measured accurately by Anders Angström, follow a simple pattern given by the formula: 1 = RH λ



1 1 − 2 2 n1 n2

 ,

(4.6)

4.5 Master Stroke

63

where R H is a constant, later called the Rydberg constant and equal to 1.097 × 107 m−1 . The first four lines were obtained when n 1 is fixed at 2 and n 2 is allowed to run through 3, 4, 5, 6. He also predicted a fifth line corresponding to n = 7, which he did not know at that time, but was already verified by Angström. This series was called the Balmer series. “As soon as I saw Balmer’s formula, the whole thing was immediately clear to me”19 said Bohr. It was the emitted radiation by the electrons when they drop from an excited state to a lower energy state with the energy given by the Planck-Einstein formula, E = hν, where ν is the frequency (ν = c/λ) of the emitted radiation. Using the energies of his special orbits Eq. (4.5), Bohr was not only able to obtain Balmer’s formula, but he was also able to write the Rydberg constant in terms of the fundamental constants as 2π 2 k 2 me4 . (4.7) RH = ch 3 Similarly, by fixing the lower state that the excited electrons drop, Bohr was able to drive the Lyman and the Pashen series, where n 1 = 1 and 3, respectively. Bohr sends the first of his trilogy of 1913 papers to Rutherford on March 6, 1913, and asks him to communicate it to the British Journal Philosophical Magazine. He also adds that he is anxious to hear what Rutherford thinks about it. He was particularly interested in what Rutherford thinks about him mixing quantum and classical concepts. It does not take long for Rutherford to respond with “Your ideas as to the mode of origin of spectra in hydrogen are very ingenious and seem to work out well; but the mixture of Planck’s ideas with old mechanics make it very difficult to form a physical idea of what is the basis of it all.”20 Rutherford’s response was encouraging, but he had also marshaled some serious criticisms. He was still troubled by what also made his atom unstable. In Bohr’s quantum atom, electrons could not be in the space between the special orbits. Electrons had to jump miraculously from one state to another instantaneously; otherwise, electrons would radiate energy continuously. Rutherford also raised another equally serious criticism. An electron jumping from an excited state behaves as if it knows beforehand where to stop. For example, an electron in the excited state n = 4 could jump down to n = 3, 2, or 1, but how does it know where to stop? The final comment of Rutherford was about the length of the paper. After revising the manuscript, Bohr sends even a longer paper. Rutherford agrees with the changes but still thinks that the manuscript has to be shortened. But before Bohr received this letter, he was already on his way to Cambridge. After long deliberations, an exhausted Rutherford finally caves in and remembers the incident as “although I first thought that many sentences could be omitted, it was clear, when he explained to me how closely knit the whole was, that it was impossible to change anything.”21 Bohr’s first paper of his trilogy appeared virtually unchanged in the Philosophical Magazine as On the Constitution of Atoms 19

Bohr (1963). Eve (1939), p. 221. 21 Rosenfeld (1967), p. 54, also Kumar (2014), p. 106. 20

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and Molecules. It was dated April 5, 1913, and appeared in July. The second and third parts were about possible arrangements of electrons inside atoms. They appeared in September and November issues, respectively. It is worth mentioning that Bohr in his first 1913 paper, instead of quantizing the angular momentum as suggested by Nicholson Eq. (4.2), he quantized the orbital energy of the electron as ν E = nh , n = 1, 2, 3, . . . , 2

(4.8)

where ν is the orbital frequency ν = v/2πr. Russel McCormmach, who discussed Nicholson’s atomic theory and Bohr’s paper in a 1966 paper, mentions that “It must strike many readers as odd on first looking at Niels Bohr’s famous 1913 series of articles on the quantum theory of atoms and molecules that a name as unfamiliar as Nicholson turns up so frequently. Bohr’s stressing of Nicholson’s work was purposeful and revealing, and it is part of the plan of this paper to show that Nicholson’s theory was very probably an important motivating influence on the direction of Bohr’s own developing notions of atomic structure.”

4.6 First Reactions At the 83rd annual meeting of the British Association for the Advancement of Science in September 12, 1913, along with Bohr, among the audience were J. J. Thomson, Rayleigh, Rutherford, Jeans, Lorentz, and Curie. The reaction was mixed, mostly silent. Thomson’s reaction was as totally unnecessary, James Jeans begged to differ. Rayleigh did not believe that nature behaved this way. Elsewhere in Europe, Max von Laue in a private discussion, stressing the validity of Maxwell’s theory, said “An electron in a circular orbit must emit radiation”. Paul Ehrenfest confides in to Lorentz that Bohr’s atom “has driven me to despair.”22 The heuristic arguments that Bohr used to build his quantum atom were apparently found by some prominent physicists as too audacious and risky that needed more backing. The first support came from the spectrum of the sun, where Bohr had predicted that some of the lines attributed to hydrogen were actually belonging to helium ions that lost one electron. The controversy was solved by one of Rutherford’s team. After a detailed investigation of the spectrum, they proved that the so-called Pickering-Fowler lines actually belong to helium. Thus, proving Bohr right. When Einstein heard the news, he was in Vienna. Bohr’s friend Georg von Hevesy wrote to Bohr saying that Einstein’s reaction to the news was “then it is one of the greatest discoveries.”23

22 23

Mehra and Rechenberg (1982), Vol. 1, p. 236. Eve (1939), p. 226.

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In 1914, James Franck and Gustav Hertz bombarded mercury atoms with electrons. They thought that the 4.9 eV of energy lost by the electrons is the energy needed to rip an electron from the mercury atom. The correct interpretation came from Bohr; when the energy of the electrons fired at the mercury atoms was less than 4.9 eV nothing happened, but when electrons with energies greater than 4.9 eV were fired, mercury atoms emitted ultraviolet light. Bohr argued that the 4.9 eV corresponds to the energy difference between the first excited state and the ground state of the mercury atoms. When an electron drops from its first excited state to its ground state, it emits light of wavelength 253.7 nm. Franck-Hertz experiment provided the direct evidence for Bohr’s atom and the existence of atomic energy levels. In 1914, Einstein’s was busy with his general theory of relativity, but still he was one of the first to recognize the importance of the Franck-Hertz experiment as a striking verification of the existence of the energy levels in atoms.24 In July 1913, Bohr was appointed to a lectureship position at the University of Copenhagen. However, his quantum atom was still not free of problems. Beyond hydrogen it hardly worked for the helium atom with two electrons. In the meantime, spectroscopic techniques were improving and people were finding that some of the lines in the hydrogen atom could be doublets. Bohr did not agree, but in early 1915 he had to change his mind when it was established that the red, blue, and violet lines in the hydrogen spectrum were indeed doublets. Bohr’s atomic model could not explain this; however, Arnold Sommerfeld from the University of Munich had a solution (Fig. 4.3). Bohr’s special orbits were circular and quantized in terms of the quantum number n. Sommerfeld assumed that these orbits can be elliptical, thus needed a second quantum number k to describe its eccentricity, such that for n = 1, k = 1, for n = 2, k = 1, 2, for n = 3, k = 1, 2, 3. When n = k, the orbit was circular. Sommerfeld showed that these elliptical orbits have slightly different energies. For example, for n = 2, k = 1, 2, the two orbits corresponding to k = 1 and k = 2 have slightly different energies yielding a doublet instead of a single line as predicted by Bohr. However, Bohr-Sommerfeld model also failed in explaining the Zeemann and the Stark effects. In 1897, Pieter Zeeman showed that in an external magnetic field, single lines in a spectrum split into several lines. When the magnetic field is turned off, the lines become single again. Sommerfeld noticed that the elliptical orbits are still in a plane. In the presence of a magnetic field, orientation of the orbits in space become important. Thus, one needs another quantum number m, called the magnetic quantum number that quantizes the orientation of the orbit with respect to the magnetic field, which can take the values from −k to k : m = −k, −k + 1, . . . , 0, . . . , k − 1, k. Sommerfeld’s extra energy states labeled in terms of the three quantum numbers, n, k, and m, allowed explanation of the extra lines in Zeeman splitting. Similarly, following Sommerfeld, Stark effect has also been explained. In March 1916, Bohr writes to Sommerfeld “I do not believe ever to have read anything with more joy than your beautiful work.”25 Orientations of the orbits, also called space quantization, were 24 25

Eve (1939), p. 218. Kumar (2014), p. 114.

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Fig. 4.3 Arnold Sommerfeld, 1868–1951 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

experimentally verified five years later in 1921. Due to his growing recognition, Bohr finally gets the appointment he has been awaiting for the newly created professorship position in Copenhagen in May 1916. In March 3, 1921, the Universitetets Institut for Theoretisk Fysik,26 commonly known as the Bohr Institute, was formally opened. Established in the wake of the First World War and the troubled years that followed, it quickly became an international center and a magnet for the aspiring physicists.

4.7 The Demon in Quantum The quantum revolution was on its way, but even its discoverers were having trouble with the demon they have created. Einstein was having difficulty in accepting the dual nature of light. He told Lorentz “I wish to assure you in advance that I am not the orthodox light-quantizer for whom you take me.”27 After he returned from the first Solway Conference in November 1911 on The Theory of Radiation and the Quanta, Einstein decided to put the quantum problem aside, at least for a while and decided to concentrate on his general theory of relativity. In summer 1916, starting with a simple Bohr atom with two energy levels, Einstein investigated ways in which an electron could jump from one energy state to another. When an electron jumps from a higher energy state to a lower energy state 26 27

Kumar (2014), p. 115. CPAE, Vol. 5, p. 175. Letter from Einstein to Hendrik Lorentz, 27 January 1911.

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by emitting a light-quanta, it is called a spontaneous emission. A second type of quantum jump occurs when an electron in an atom absorbs a light-quanta and jumps to a higher energy state. Einstein now introduced a third type of quantum jump called the stimulated emission, which occurs when a light-quantum hits an electron already in an excited state and stimulates it to jump to a lower energy state and emit a lightquanta. Today, stimulated light emission is the basis of LASERs, which is the short for Light Amplification by Stimulated Emission of Radiation. However, this discovery also brought with it something that deeply disturbed Einstein. His calculations clearly showed that the exact time and the direction of the emitted radiation were entirely random. One can only calculate the probability of when an atom will emit radiation, but not its exact time. The exact details were left to pure chance with no connection to cause and effect. Finding chance and probability at the hearth of quantum mechanics was deeply disturbing to Einstein. Even though he already accepted the reality of light-quanta, In January 1920, he wrote to Max Born “that business about causality causes me a lot of trouble.”28 An electron in an excited state will definitely fall onto the more stable ground state by emitting a light-quanta, but its exact time and the direction of the emitted light are left to chance. One can only calculate their probability. In 1924, Einstein was still troubled by this; “I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist.”29 These strong words show Einstein’s strong conviction to causality. Special relativity has shown that absolute simultaneity does not exist and due to the speed of light being the maximum signal speed, there will always be a delay between the cause and effect. However, in quantum mechanics, the connection between cause and effect, if not broken altogether, was left to pure chance. This was deeply disturbing to Einstein. According to him, this was not how the universe worked. Even though he was still willing to accept it for a while, he was hoping that it is temporary and when a mathematical theory of quantum mechanics is discovered, what he called “this business of causality” will be resolved. When Bohr introduced his quantized model of the atom in 1913 and explained the mystery of the atomic spectra, Einstein’s response was “like a miracle.”30 Bohr like most physicists, at that time did not believe in the existence of light-quantum. Like Planck, to him radiation was absorbed and emitted in quanta, but light itself was continuous. In April 27, 1920, Bohr gave a talk in Berlin on the quantum atom and the theory of atomic spectra. With Einstein and Planck among the audience, he did not address the problem of the nature of light, but he was deeply impressed by Einstein’s spontaneous and stimulated emission of radiation. Where he stopped, Einstein has picked up by showing that it is all a matter of chance and probability. Einstein believed that in due time the road to causality will be restored, but Bohr in his lecture emphasized that no exact determination of the time and direction of the 28

Born (2005), p. 22. Letter From Einstein to Max Born, 27 January 1920. Born (2005), p. 80. Letter From Einstein to Max Born, 29 April 1924. 30 Einstein (1949), p. 47. 29

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emitted quanta will ever be possible. Einstein was not happy with this. During Bohr’s stay in Berlin, they had several chances to discuss physics. Both tried to convince the other, but they were on opposite sides and will remain to be so until the end of their lives.

4.8 Shell Model and Bohr Festspiele Despite the tumultuous years following the First World War, in March 1922 Bohr publishes an important paper in the Zeitschrift für Physik entitled The Structure of Atoms and the Physical and Chemical Properties of Elements, where he constructs his new atom model one shell at a time, layer by layer, like an onion. In June 1922, Bohr goes to Germany to give a series of lectures over a period of eleven days in Göttingen University to explain his new theory about the distribution of electrons inside atoms. To these celebrated lectures known as the Bohr Festspiele, more than hundred physicists attended from all over Germany. Among the audience were 22year-old Wolfgang Pauli, Werner Heisenberg, who was barely 20-year old, and young Pasqual Jordan. Pauli was working toward a PhD under Sommerfeld at the University of Munich. Bohr argued that electrons are arranged in orbital shells around the nucleus, like the layers of an onion. Each shell was also made up of a set of electron orbits that could accommodate only a certain number of electrons. Elements that shared the same chemical properties are the ones that had the same number of electrons in their outermost shell. At the core of Bohr’s approach was what Bohr called the correspondence principle. Bohr believed that quantum rules apply on the atomic scale; however, when extrapolated to the classical realm, they should not conflict with the well-established results of classical physics. Correspondence principle allowed him to eliminate any results that showed conflict with classical physics when extrapolated to the classical domain. Bohr with his new model, not only explained the position of the elements in the periodic table, but he also predicted a new element with the atomic number 72, which would be chemically similar to zirconium with the atomic number 40 and titanium with the atomic number 22. Almost simultaneously with Bohr’s Göttingen lectures, his old friend Georg von Havesy and Dirk Coster designed an experiment to check Bohr’s prediction. In 1922, Bohr was awarded the Nobel Prize. On his return to Copenhagen, one of the telegrams that he found on his desk that delighted him most was the one from none other than his old friend and mentor Rutherford, which said it all; “I knew it was merely a question of time, but there is nothing like the accomplished fact. It is well merited recognition of your great work and everybody here is delighted in the news.”31 On December 1922, the Nobel Prize presentation ceremony was held in Stockholm, the Swedish capital. On his way to Stockholm, Havesy and Coster have already concluded their investigation. Before his lecture, Coster calls and tells him that they have accumulated a sufficient amount of element 72 and they 31

Moore (1966), p. 116.

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will start checking its chemical properties. Soon they verify that element 72, later called hafnium, has chemical properties similar to those of zirconium, and unlike the opinion of the rival group in France, has properties markedly different from those of rare elements. In his Nobel lecture, Bohr concludes his talk with this dramatic announcement.

4.9 De Broglie and Particle-Wave Duality Einstein’s E = mc2 connected mass to energy and Einstein’s light-quantum connected energy to frequency. To the French Physicist Prince Louis de Broglie, this begged the question could these equations be combined so that mass is also associated with a wave, hence a frequency? After all, from E = mc2 , light-quanta also possesses a certain amount of mass and momentum, which is mass times velocity. De Broglie continued, if electromagnetic waves with a certain frequency are endowed with particle-like properties like momentum, then particles, such as electrons, could also have wave-like properties. Equating the energy of a particle to hν :

which could be rewritten as

mc2 = hν,

(4.9)

c (mc)c = h , λ

(4.10)

where (mc) is the momentum p, de Broglie obtained the simple relation λ=

h , p

(4.11)

which was later called the de Broglie wavelength for the wave nature of particles. Due to the presence of the Planck constant, which is a very small number, for macroscopic objects the wave nature of particles would not be observable. Like light, which has dual wave-particle nature, particles would also have dual nature as particle-wave. Prince Louis de Broglie was the younger son of Victor, fifth duc de Broglie (Fig. 4.4). When he thought of combining the particle and wave properties in 1923, he was thirty-one-year old and he had joined a private lab run by his brother specializing in the study of X-rays. He later wrote “After long reflection in solitude and meditation, I suddenly had the idea, during the year 1923, that the discovery made by Einstein in 1905 should be generalized by extending it to all material particles and notably to electrons.”32 This eureka moment immediately led de Broglie to realize that he could explain Bohr’s stable orbits in the hydrogen atom as standing waves of the electrons. Standing waves are like the oscillations of a violin string. They occur when a string of fixed 32

Pais (1991), p. 240.

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Fig. 4.4 Louis de Broglie, 1892–1987 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

length oscillates with its end points fixed. Since the end points cannot move, only certain wavelengths are allowed. For a string of length L, the allowed wavelengths are λ2 λ3 λn λ1 = L , 2. = L , 3. = L, . . . , n = L. (4.12) 1. 2 2 2 2 In other words, only the waves whose integer multiples of half wavelengths fit into the length L of the string are allowed. For circular orbits in the hydrogen atom, the standing electron wave could only have wavelengths whose integer multiple fits into the circumference of the orbit as 2πr = nλ, n = 1, 2, . . . .

(4.13)

Substituting the wavelength of the electron Eq. (4.11), de Broglie obtained the radii of the stable orbits as h 2πr = n , p h =n . mv

(4.14) (4.15)

Using the velocity of the electron from classical physics Eq. (4.3): v = (ke2 /mr )1/2 , de Broglie obtained Bohr’s quantized orbits Eq. (4.4):   rn = h 2 /4π 2 mke2 n 2 , n = 1, 2, . . . .

(4.16)

4.10 Einstein Was Right

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Now, Bohr’s quantum number n, which he introduced to quantize angular momentum Eq. (4.2), can be thought of as the number of electron wavelengths that fit into each orbit. Bohr’s condition for stable orbits now becomes the condition for the standing electron waves. De Broglie published his results as three short papers in the Comptes Randus (proceedings) of the Paris Academy in September and October 1923. He presented expanded versions of these papers as his thesis in 1924 to the Faculty of Science of the Paris University. He personally hands in an early copy of his thesis to the prominent French physicist Paul Langevin, who was also in his committee as the external examiner and asks his opinion about his conclusions. Langevin finds de Broglie’s ideas interesting but far-fetched.33 Not knowing what to make of it, Langevin asks de Broglie for an extra copy of his thesis to send it to none other than Einstein at the University of Berlin. Einstein returns with the comment “He has lifted a corner of the great veil.”34,35 This was sufficient for Langevin and the rest of the committee. In November 1924, he was now Dr. de Broglie. His thesis was published in the French journal Annals de Physique in 1925. In 1924, in a letter to Lorentz, Einstein wrote “... de Broglie has undertaken a very interesting attempt to interpret the Bohr-Sommerfeld quantum rules.”36 Not everybody was sympathetic to de Broglie’s theory. In fact, some even called it la Com˙edie Française.37

4.10 Einstein Was Right About the same time with de Broglie’s theory, Sommerfeld was in the USA in Madison Wisconsin for a year escaping from the turbulent days ahead in Germany. In February 1923, Sommerfeld wrote to Bohr, bringing to his attention the discovery that Arthur Compton had made. Since Compton’s paper had not been published yet, he wrote cautiously “I do not know if I should mention his results, I want to call your attention to the fact that eventually we may expect completely fundamental new lesson.”38 What Compton did was to shoot a beam of X-rays at graphite (carbon) atoms and measure the scattered wave that he was interested in. It was already established that X-rays are electromagnetic waves. According to the wave theory, the scattered waves should have the same frequency, or wavelength, as the incident wave. However, Compton found that the scattered waves always had frequencies less than the incident wave. Unable to explain this with the wave theory, he turns to the Einstein’s light-quantum hypothesis. It immediately becomes clear to him that the frequency and the intensity of the scattered X-rays are what one expects from them 33

Abragam (1988), p. 30. Abragam (1988), p. 30. 35 Moore (1989), p. 187. 36 Pais (1982), p. 436. 37 Gamow (1966), p. 81. 38 Stuewer (1975), p. 241. 34

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when light-quanta scatters off from the electrons of the graphite atoms like billiard balls bouncing off each other. In other words, X-rays behaved like the light-quanta that Einstein had predicted with energies equal to hν. Any energy transferred to the electrons appeared as a decrease in the frequency of the scattered X-ray. For anybody who still did not believe in the light-quanta, this was an irrefutable evidence that X-rays, hence light, are composed of packets of energy with the energy hν and travel in a definite direction with the momentum h/λ. Compton also gave a detailed mathematical analysis of his data in terms of how the light-quanta loses energy, hence the decrease in frequency, as a function of the scattering angle. Compton presented his discovery in November 1922 at a conference in Chicago. Sommerfeld described Compton’s discovery as “probably the most important discovery that could have been made in the current state of physics.”39 However, Bohr still refused to accept Compton’s result that light is quantized as Einstein predicted. Along with his collaborators, Hendrick Kramers and John C. Slater, he argued that energy and momentum may not be precisely conserved in the atomic realm, thus hoping to explain the decrease in the frequency of the scattered X-ray within the wave theory. It was early/mid-1925 when experiments eventually confirmed that indeed the collisions between the X-rays and the electrons conserve energy and momentum, thus proving Bohr wrong and Einstein right. This established the dual nature of light. Compton’s results were published in May 1923 in the American journal Physical Review and Compton received the Nobel Prize in 1927.

4.11 Electron Waves Having an interesting idea was one thing, but having solid experimental support for it was another. De Broglie was aware of this and tried to convince experimentalists in his brother’s lab to put his theory to test. He argued that a group of electrons, when pass through a small aperture should show effects of diffraction. However, he failed to convince his brother and eventually stopped trying. In the meantime, thirty-four-year-old Clinton Davisson of the Western Electric Company in New York, later became Bell Labs, had been shooting a beam of electrons into various metal targets. To avoid collision of the electrons with other atoms, experiments were conducted in a vacuum chamber. In 1925, by accident air had entered into the chamber, which oxidized the surface of the nickel target. To remove the oxide, Davisson and Germer heated the specimen to high temperatures, which caused the polycrystalline structure of the nickel to form large single crystal areas over regions of the width of the electron beam, which caused diffraction. Unaware of this, they wrote up their data and published it. Lester Germer was a graduate student at the Columbia University, working under Davisson’s supervision at the Western Electric Company.

39

Pais (1991), p. 234.

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In the summer of 1926, Davisson attends the Oxford meeting of the British Association for the Advancement of Science. There he was shocked to learn the importance of their experiment and that some physicists believe that their data supported de Broglie’s theory. On his return to New York, Davisson and Lester Germer immediately checked whether electrons really diffracted. It was January 1927 before they already had conclusive evidence that electrons indeed diffracted, thus verifying the wave nature of the electrons as de Broglie predicted. At the same time, George Paget Thomson, son of J. J. Thomson, in Aberdeen, Scotland was carrying out his own experiments on electron diffraction. In 1929, de Broglie was awarded the Nobel Prize in physics for his discovery of the wave nature of electrons. In 1937, Thomson and Davisson were awarded the Nobel Prize for physics for their experimental discovery of the diffraction of electrons by crystals. Even though the experiment is known as the Davisson-Germer experiment, it is a mystery why the Nobel Committee ignored Germer. However, he was nominated 26 times and on December 10, 1937, he was mentioned in the award presentation speech by Professor H. Pleijel, Chairman of the Nobel Committee for Physics of the Royal Academy of Sciences as “Davisson and his collaborator Germer were able to present the incontestable evidence, reached by the experiments, of the mechanical waves and of the correctness of the theory of de Broglie.” He was also mentioned as “For their experiments Davisson and Germer availed themselves of a cubic nickel crystal” and “Davisson and Germer examined the reflection of the electronic beams in various directions and obtained results which agreed with the wave theory.”

Chapter 5

Revolution Within Revolution

During the first quarter of the twentieth century, incredible developments had taken place in quantum physics—from Planck’s blackbody radiation law to Einstein’s lightquanta, from Bohr’s quantum atom to de Broglie’s electron waves. However, it was still a hodgepodge of quantum and classical ideas. Despite the success of Bohr’s quantum atom in explaining the hydrogen atom, it failed for the next simplest atom; helium. There were even more problems with the spectra of atoms like sodium and the atoms of the rare-earth elements. Even when they worked, Bohr-Sommerfeld had to introduce ad hoc selection rules to explain the presence, or the absence, of certain lines in the spectra of atoms. Despite the success of de Broglie waves in explaining Bohr’s stable orbits in the hydrogen atom as the standing electron waves, and their subsequent observation by Davisson and Germer, their physical nature was still obscure. As mentioned in the Nobel Prize presentation speech by Prof. H. Pleijel, people thought of them as mechanical waves. If they were mechanical waves, then what was the wave equation they satisfy. Bohr’s quantum atom still had problems with causality that caused Bohr and Einstein to clash. “The more success the quantum theory enjoys, the more stupid it looks” said Einstein in May 1912.1 Everybody saw that what is now called the old quantum theory, has to give way to a new theory— a theory that would be mathematically and conceptually coherent and complete, a theory that would be called quantum mechanics.

5.1 Footsteps of Quantum Mechanics Pauli goes to Munich in 1918 to work with Arnold Sommerfeld toward a PhD degree (Fig. 5.1). Sommerfeld was in charge of the theoretical physics at the Munich University. He was an excellent teacher, who already successfully supervised several 1

CPAE, Vol. 5, p. 299. Letter from Einstein to Heinrich Zangger, May 12, 1912.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_5

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Fig. 5.1 Wolfgang Pauli, 1900–1958 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

young talented students. His aim was to create an institute that would be a breeding ground for theoretical physics. Since its beginning in 1906, it was a modest institute with four rooms: a lecture theater, a seminar room, a small library, and his office. The large laboratory at the basement, where in 1912 Max Laue’s theory that X-rays are electromagnetic waves was tested and confirmed, had already brought a quick recognition to the institute. Pauli was quickly noticed for his sharp and biting criticisms of new and speculative ideas, who once said to a colleague: “I do not mind if you think slowly, but I do object when you publish more quickly than you can think.”2 Another time his comment about a paper he has read was “it is not even wrong.”3 But history would prove that he was not always right either. Nobody, including Einstein, was immune form his sharp comments. Sommerfeld’s trust and high regard for Pauli’s talent and knowledge of relativity were evident when he asked Pauli to help him on a major article he was asked to write on relativity for the Encyclopedia der Mathematischen Wissenschaffen. After seeing the article, Sommerfeld said “it proved so masterly that I renounced all collaboration.”4 This encyclopedia article was so brilliantly written and at a time when most people were still trying to understand relativity that it quickly became a classic reference on special and general theories of relativity, which also drew Einstein’s praise. This article was published two months after Pauli received his PhD in 1921. As his PhD thesis, Sommerfeld had asked Pauli 2

Cropper (2001), p. 257. Cropper (2001), p. 257. 4 Mehra and Rechenberg (1982), Vol. 1, Pt. 2, p. 384. 3

5.2 Pauli and the Exclusion Principle

77

to apply Bohr-Sommerfeld quantum rules to ionized hydrogen molecule that had lost an electron. Pauli’s treatment was flawless, but with one problem: It did not agree with the experimental results. As shocking as it was to Pauli, his thesis demonstrated that the Bohr-Sommerfeld model had reached its limit. In October 1921, with his PhD in his pocket, Dr. Pauli leaves for Göttingen to start his new post as assistant to Max Born. “W. Pauli is now my assistant, he is amazingly intelligent and very able”5 wrote Born to Einstein. As happy as Born was to recruit him as his assistant, he was sad when Pauli in April 1922 left to become an assistant in Hamburg. In 1922, Pauli was back in Göttingen to attend Bohr’s seminars on his new shell model of the atom, where he would meet Bohr for the first time. He quickly impresses Bohr, and Bohr invites him to Copenhagen for a year as his assistant. First, he was afraid of the problems that learning a new language like Danish would cause him, but he accepts the offer and goes to Copenhagen for a year in the fall of 1922.

5.2 Pauli and the Exclusion Principle At Copenhagen, while Pauli helped edit Bohr’s paper in German, he also got involved in the problem of anomalous Zeeman Effect. In strong magnetic fields, spectral lines of atoms split into doublets or triplets. This is known as the normal Zeeman effect, which Bohr’s model could not explain. At this point, Sommerfeld came in and introduced two new quantum numbers, l and m, in addition to the principal quantum number n. These new quantum numbers quantized the shape and the spatial orientation of Bohr’s circular orbits. But soon the problem aggravated, when it was discovered that some of the lines in the hydrogen spectrum actually split into a quartet or more, instead of a doublet or a triplet. To Pauli, this meant a fundamental problem buried deep in the heart of old quantum physics, which was a mixture of classical and quantum ideas. Not being able to come up with an answer, Pauli, in despair, wrote to Sommerfeld in June 1923: “up until now I have gone thoroughly wrong.”6 Another day as he was strolling along the streets, a colleague asked “you look very unhappy” Pauli answered “how can one look happy when he is thinking about the anomalous Zeeman effect.”7 Pauli needed a more fundamental understanding of the quantum atom. Guess work and ad hoc rules used by Bohr to explain complex atomic spectra were too much for him. He did not believe that Bohr’s theory of the periodic table could explain the distribution of electrons in the atom. In the Bohr-Sommerfeld quantum atom model, electrons were assumed to move in orbits in three dimensions quantized in terms of three quantum numbers (n, l, m), which specified the angular momentum, shape, and the spatial orientation of the orbit:

5

Born (2005), p. 56. Letter from Born to Einstein, October 21, 1921. Pais (2000), p. 221. 7 Pauli (1946), p. 213. 6

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n = 1, 2, 3, . . . , l = 0, 1, 2, . . . , n − 1, m = −l, −l + 1, . . . , 0, 1, . . . l. When n = l + 1, orbits were circular and when l + 1 is less than n, orbits were always elliptical. Whether the orbits are circular or elliptical, they lied on a plane. To explain the normal Zeeman effect, Sommerfeld had introduced the third quantum number m, which specified the orientation of the orbit with respect to the magnetic field. In June 1922, during the Göttingen lectures of Bohr, Pauli learned about the shell model for the first time. Due to their stability, nobel gases: helium, neon, argon, krypton, …, with atomic numbers 2, 10, 18, 36, . . . , required high energies to remove an electron from them. Bohr had argued that they consisted of closed shells. On the other hand, the elements that preceded the noble gases in the periodic table, the halogens group: hydrogen, flourine, chlorine, bromine,…, with the atomic numbers 1, 9, 17, 35, . . . , formed chemical compounds by picking an electron, thereby filling their vacancy in their outermost shell. Similarly, the alkalis group that followed the nobel gases in the periodic table: lithium, sodium, potassium,…, were easily able to form stable compounds by loosing an electron. Since only the valence electrons which are the electrons outside the closed shells participate in chemical reactions, Bohr argued that elements with the same number of valence electrons shared similar chemical properties and occupied the same column in the periodic table. All the halogens that had seven electrons in their outermost shell acquired stable noble gas status by picking an electron and alkalis all had a single electron in their outer most shell were quick to loose an electron to form stable compounds. Sommerfeld hailed the shell model as “the greatest advance in atomic physics since 1913.”8 Sommerfeld also said that if one could derive the atomic numbers 2, 8, 18, 32, . . . of the elements in the rows of the periodic table, which corresponded to the number of electrons that each shell could accommodate, it would be “the fulfillment of the boldest hopes of physics.”9

8 9

BCW, Vol. 4, p. 740. Postcard from Arnold Sommerfeld to Bohr, March 7, 1921. BCW, Vol. 4, p. 740. Letter from Arnold Sommerfeld to Bohr, April 25, 1921.

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In a 1924 paper written by a Cambridge graduate student Edmund Stoner, The Distribution of Electrons Among Atomic Levels was published in the Philosophical Magazine. Stoner was still working under Rutherford and argued that electrons in the shell model were assumed to move in three-dimensional shells. Actually, these shells were energy states that were nested within each other. Energy of each shell was determined by the quantum numbers (n, l, m), where each unique set corresponded to a distinct energy level. In other words, for a given n, there were n l-states, and for each l-state, there were (2l + 1) m-states. For example, for n = 1 there was only one state: (1, 0, 0), for n = 2 there were four states: (1, 0, 0), (1, 1, −1), (1, 1, 0), (1, 1, 1). Similarly, for n = 3 there were nine states: (3, 0, 0), (3, 1, −1), (3, 1, 0), (3, 1, −1), (3, 2, −2), (3, 2, −1), (3, 2, 0), (3, 2, 1), (3, 2, 2). In general, for the nth shell, there were n 2 states or orbits. Stoner argued that the stability and the inertness of the noble gases, helium, neon, argon, krypton, xenon, and radon, could be due to each of these elements having filled or closed shells. This meant that for a given n = 1, 2, 3, . . . , the number of electrons that each shell could accommodate was 1, 4, 9, 16, 25, 36, . . . . Since the number of protons in the nucleus is equal to the number of electrons, these numbers should also correspond to the atomic numbers. However, the periodic table said otherwise, hence Stoner argued that the number of electrons that each shell could accommodate should be doubled as 2n 2 to yield the numbers 2, 8, 18, 32, 50, 72, . . . . There was no mathematical reasoning behind Bohr’s shell model. Including Rutherford, nobody was sure how Bohr reached his conclusions.10 Stoner’s derivation of the number of electrons that each shell could accommodate as 2n 2 yielded the correct series for the elements in the rows of the periodic table as 2, 8, 18, 32, . . . .

10

Pais (1991), p. 205.

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Even though Stoner’s result could be understood as twice the number of the energy states for a given n, the mysterious factor of two remained unexplained. It appeared that each energy state could accommodate one more electron instead of just one. Pauli was aware of the fact that to make Bohr’s model work, there had to be some restriction on how the electrons are distributed among the shells. Otherwise, there seemed nothing to stop electrons from crowding into the ground state with the lowest energy. When Sommerfeld brought Stoner’s paper to Pauli’s attention, he immediately realized that in addition to the quantum numbers (n, l, m), there had to be a fourth quantum number that could take only two values a and b. The extra states that resulted from this two-valued quantum number, not only explained the additional splitting in the spectral lines, called the anomalous Zeeman effect, but also allowed Pauli to introduce the exclusion principle: No two electrons in an atom can have the same set of four quantum numbers. Although the origin and the physical nature of the fourth quantum number remained elusive, it had explained the location of the elements in the periodic table and why the electrons in multi-electron atoms do not collapse into the ground state. Pauli’s paper entitled On the Connection Between the Closing of Electron Groups in Atoms and the Complex Structure of Spectra was published in Zeitschrift für Physik on March 21, 1925. As for the exclusion principle, Pauli concluded his paper with “we can not give a more precise reason for this rule.”11

5.3 White Dwarfs and Neutron Stars With the Pauli’s exclusion principle, a new statistics emerged called the Fermi-Dirac statistics, which was named after Enrico Fermi and Paul Dirac. They derived the distribution independently in 1926. In the Fermi-Dirac distribution, derivation of the most probable number of particles, n k , in the kth bundle with gk levels proceeds exactly the same way as in the Bose-Einstein distribution (Sect. 2.9.3). However, there is the exception that due to the Pauli exclusion principle, each level can be occupied only by one particle. We now distribute n k particles among the gk levels of the kth bundle. For the first particle, there are gk levels available, which leaves only (gk − 1) levels for the second particle and so on, thus giving the number of arrangements as gk ! . (5.1) gk (gk − 1) · · · (gk − n k + 1) = (gk − n k )! Since the particles are indistinguishable, n k ! arrangements among themselves have no significance; hence for the kth bundle, this gives the number of possible arrangements as 11

Pais (2000), p. 223.

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gk ! . n k !(gk − n k )!

(5.2)

For the whole system, this gives the number of microstates as W =

 k

gk ! . n k !(gk − n k )!

(5.3)

For the equilibrium distribution, we find the occupation numbers, n k , that maximize entropy S = k log W , subject to the conditions that the total number of particles and the total energy are fixed. This yields the Fermi-Dirac distribution as gk . (5.4) e−α−βεk + 1   Using the total number of particles N = k n k and the total energy U = k n k εk , one can obtain α and β, where β is again equal to −1/kT . With respect to the BoseEinstein distribution Eq. (2.12), the change in sign in the denominator is crucial. It is the source of the enormous pressures due to the exclusion principle that hold up white dwarfs and neutron stars. Due to their low luminosity, white dwarfs were discovered in 1910. The nearest white dwarf is Sirius B, 8.6 light years away—the smaller companion of the Sirius binary star. White dwarfs are remnants of stars, primarily supported by electron degeneracy pressure resulting from the Pauli exclusion principle. Since there is no fusion taking place in white dwarfs, their low luminosity is due to their residual thermal energy. Their mass is comparable to the mass of the sun, while their radius is comparable to that of earth with one sugar cube of white dwarf matter weighing several tons. Chandrasekhar mass limit is the maximum mass of a stable white dwarf, which is usually quoted as 1.44M . Above the Chandrasekhar limit, gravitational collapse forces the stellar remnant to evolve into a neutron star or a black hole. The mass limit is named after Subrahmanyan Chandrasekhar, who in a series of papers published between 1931 and 1935 improved on the earlier calculations by Edmund Stoner and Wilhelm Anderson in 1929. Existence of a mass limit was based on a breakthrough of combining relativity with Fermi-Dirac statistics. Existence of the mass limit for the white dwarfs was the source of one of the greatest controversies in science. A 19-year-old Subrahmanyan Chandrasekhar, a child prodigy and cousin of the Nobel Laureate Chandrasekhara Raman, arrived at the University of Cambridge on a PhD scholarship from the Indian government. He also brought with him some calculations indicating that a white dwarf much heavier than the sun could not exist and should collapse to a point of infinite density until it becomes a black hole. Black hole solution of the Einstein’s general theory of relativity was known, but they were no more than a mathematical curiosity. nk =

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Expecting a warm welcome to his idea at Cambridge, Chandrasekhar was ignored and disheartened by the cold shoulder he received. A disappointed Chandrasekhar still pushed on to receive his PhD in the summer of 1933. On October 1933, as the second Indian elected after Srinivasa Ramanujan, he was the recipient of the Prize Fellowship at Trinity College for the duration of the years 1933−1937. With his thesis behind, Chandrasekhar returned to his research on the fate of stars. At the time, Arthur Eddington was at the peak of his career as a scientist, philosopher, and popularizer of science. He had made Einstein’s general theory of relativity known by the masses by organizing the dramatic expedition to the west coast of Africa during the 1919 solar eclipse that detected the deflection of starlight by the sun. He was also one of the key figures that established the field of astrophysics almost single handedly. In 1930, he was also busy with his ambitious theory that would combine quantum theory and general relativity, which he considered as the culmination of his life’s work that he called the fundamental theory. As Chandrasekhar was working on his mass limit, to his delight, Eddington, a doyen of the field, appeared to be interested in his work and frequently checked up on him to see how he is progressing. With Eddington’s apparent approval, an elated Chandrasekhar followed his advice to present his results at a meeting of the Royal Astronomical Society at London. As Chandra was preparing his talk, he found out that the speaker after him is none other than Eddington and on the same topic. He was puzzled, but paid no more attention to it (Fig. 5.2). On January 11, 1935, with all the leading figures of astrophysics in the audience, Chandrasekhar presented his theory that concluded with the all important result that stars beyond a certain mass cannot die quietly and that they must continue to collapse to a point and beyond. Expecting that Eddington would endorse his results, young Chandra was sure of a triumphant end. But to his dismay, a bossy Eddington with the full force of his command of the English language was out there to destroy him. Chandrasekhar’s ideas to Eddington were merely a game playing. How could a huge star disappear at a point? He said “The star has to go on radiating and radiating and contracting until, I suppose, it gets down to a few km radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. … I think there should be a law of Nature to prevent a star from behaving in this absurd way.”12 Even though Eddington’s objections were dubious with no solid foundation, his reputation was such that no body dared to speak. The talk ended without even Chandrasekhar being given a chance to defend his ideas. Even though Bohr, Fowler, and Pauli favored Chandrasekhar’s mass limit, due to Eddington’s status they refrained from stating their opinion publicly. Eddington maintained his stance till the rest of his life. Chandrasekhar was awarded the 1983 Nobel Prize with William Fowler for “…the theoretical studies of the physical processes of importance to the structure and evolution of the stars.”13 This conflict had a long-lasting effect not just on Chandrasekhar but also on astrophysics. For decades, nobody bothered to follow the path opened by Chandrasekhar’s work on compact objects. 12 13

Miller (2007). Also The Observatory 58, 33–41 (1935). Miller (2007). Also The Observatory 58, 33–41 (1935).

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Fig. 5.2 L-R: Albert Einstein, Hendrik Lorentz and Arthur Eddington (AIP Emilio Segrè Visual Archives)

For stellar remnants with masses above the Chandrasekhar limit, collapse continues until the neutron degeneracy pressure stops the collapse and supports the star for masses below the Tolman-Opponheimer-Volkov (1939) limit 0.7M . In addition to the degeneracy pressure of neutrons, repulsive nuclear interactions push this limit to around 2 solar masses. The most massive neutron star detected so far is PSR J040+6620, whose mass is estimated to be 2.14 solar masses. Neutron stars are the smallest densest currently known objects. They are mostly formed from the remnants of supernova explosions. Neutron stars are almost entirely composed of neutrons. A 1.4 solar mass neutron star has a radius on the order of 10 kilometers, roughly the size of Mount Everest. One sugar cube of neutron star matter weighs close to several hundred million tons. Some neutron stars emit beams of electromagnetic radiation, which makes them observable as pulsars. The first pulsar (PSR B1919+21) was observed by Jocelyn Bell Burnell and Antony Hewish in 1967, which was also the

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Fig. 5.3 Subrahmanyan Chandrasekhar wins Nobel Prize in Physics 1983 (photograph by Dorothy Davis Locanthi, courtesy AIP Emilio Segrè Visual Archives, Locanthi Collection)

first observational evidence for the existence of neutron stars. As the stellar remnant collapses, to conserve angular momentum, like an ice skater, it spins up. For a solar mass neutron star, this corresponds to rotation periods up to several hundreds per second. Sun’s rotation period is 27 days. The most well-known supernova remnant is the Crab nebula, first observed by the Chinese astronomers in July 4, 1054, and remained visible until April 6, 1056. Chinese called it the guest star. Crab nebula harbors a pulsar (PSR B0531+21), which is a relatively young neutron star discovered in 1968. It is one of the few pulsars identified optically, and also, it was the first to be connected to a supernova remnant. Beyond the neutron star mass limit, there is nothing that can stop the collapse; hence, the stellar remnant continues to collapse to become a black hole (Fig. 5.3).

5.4 Spin Without Spinning A German-American Ralph Kronig, after receiving his PhD from the Columbia University, was traveling on a fellowship from the Columbia University. On his way to spend ten months at the Bohr institute, Kronig arrives at Tubingen on January 1925 to spend several weeks working with Landé, Gerlach, and Back. After greeting him, Arthur Landé informs Kronig that Pauli would be visiting them the next day and shows him a letter that Pauli had sent him the previous November, where Pauli wrote about the anomalous Zeeman effect and the fourth quantum number that could only take two values. Excited by the news, Kronig begins to think about a possible interpretation of this fourth quantum number in terms of a spinning electron. Finding the idea fascinating, he spends the rest of the day developing the theory and doing the mathematics. When he showed his calculations to Landé, both men anxiously began to wait Pauli’s arrival to hear what he would say about it. Kronig’s theory could explain the anomalous Zeeman effect and also was able to qualitatively explain the doublet splitting in the spectra of the alkali atoms by means of the spin-orbit effect.

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Fig. 5.4 Paul Ehrenfest (right) walking out from a building in Ann Arbor, Michigan (photograph by Samuel Goudsmit, courtesy AIP Emilio Segrè Visual Archives, Goudsmit Collection)

When Pauli heard Kronig’s theory, he ridiculed the classical spinning electron idea. He praised it as a clever flash of wit but rejected it as having no basis in reality: “That is surely quite a clever idea, but nature is not like that”14 said Pauli. Thrown off by Pauli’s rejection, Kroger hopes to get a better reception at Copenhagen. Unfortunately, Bohr and others at the institute also give Kronig’s idea the cold shoulder on the same grounds as Pauli. Feeling completely dejected, Kronig drops the idea. On the other hand, in autumn 1925 at Leiden Holland, two Dutch graduate students of Paul Ehrenfest, Samuel Goudsmit (23) and George Uhlenbeck (24), were more fortunate (Fig. 5.4). According to Uhlenbeck’s recollection, he and Goudsmit unaware of Kronig’s ideas essentially reinvented the idea in one afternoon.15 They argued that in Bohr-Sommerfeld model the quantum numbers (n, l, m) specified the angular momentum, shape, and the spatial orientation of the electron orbits. Hence, 14 15

Ralph Kronig, AHQP interview, December 11, 1962. Commins (2012), pp. 135–136.

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Fig. 5.5 (L-R): Uhlenbeck, Kramers, Goudsmit (AIP Emilio Segrè Visual Archives, Goudsmit Collection)

one would need a fourth quantum number if the electron had a spin. This again implied the mental image of electron as a tiny classical spinning sphere. When they talked to Ehrenfest, he told them that either it is nonsense or something very important and asked them to write a short paper and also recommended them to consult Lorentz, who listened to them carefully and told them that he will return with his comments. Soon he returned with a handwritten manuscript with some serious objections based on his deep knowledge of classical electrodynamics. Discouraged by Lorentz’s letter, they rush to Ehrenfest to withdraw the paper. Ehrenfest tells them that it is too late, since he had already sent the paper to the publisher, but he tried to comfort them by saying that they are young enough to be forgiven for their stupidity!.16 Their short paper appeared in Naturwissenschaften in November 1925, 13, 953, and even a shorter version in English in Nature in February 1926, 117, 264. When Bohr and Heisenberg became aware of Uhlenbeck and Goudsmit’s papers, they began to have second thoughts about it, thinking that despite its problems, after all, it might have something to it (Fig. 5.5). Pauli was still adamant about his objections. He thought that the new quantum number represented something that did not have a classical counterpart.17 Besides, along with Lorentz, he also thought that if the electron was a sphere of classical 16

Uhlenbeck (1976), Personal Reminiscences, Physics Today June. Reprinted in Wearth and Phillips (1985). Also in Pais (1989). 17 Pais (2000), p. 222.

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radius, r0 = e2 /m 0 c2 , its surface velocity would exceed the speed of light, a flagrant contradiction with the special theory of relativity. By March 1926, objections that Pauli had to reject electron spin had all been gone. He wrote to Bohr “Now there is nothing else I can do than to capitulate completely”18 Despite feeling bitter about loosing the spin discovery, in later years, Kronig described himself as I was “very inexperienced at age 20, was undoubtedly crestfallen by Pauli’s rejection.”19 Because of the Pauli-Kronig affair, the Nobel Committee probably shied away from awarding them the Nobel Prize. Pauli always felt guilty for discouraging Kronig, just as he did for receiving the Nobel Prize in 1945 for discovering the exclusion principle, when Uhlenbeck and Goudsmit were denied. He said “I was so stupid when I was young.”20 Uhlenbeck and Goudsmit received their doctorates the same day on July 7, 1927. Ehrenfest also helped them secure positions at the University of Michigan. With faculty positions hard to come by, Goudsmit later said “the post in America was for me a far more significant award than the Nobel Prize.”21

5.5 Emergence of the New Theory Eighteen-year-old Heisenberg’s career officially started in 1920, when Sommerfeld allowed him to attend a research seminar aimed at more advanced students. One of the students in the seminar was none other than Pauli. When Sommerfeld was taking the young Heisenberg around the institute, he told him that Pauli was his most talented student. Heisenberg immediately sat next to Pauli, which started a lifelong professional relationship. As Pauli was busy writing his relativity article, he steered Heisenberg toward the quantum atom, which he considered to be the more fertile ground with plenty of uninterpretted results and with no coherent theory in sight. In June 1922, Heisenberg was one of the young scientists that attended Bohr’s festival. He was impressed by Bohr’s carefully selected words in his arguments: “Each one of his carefully formulated sentences revealed a long chain of underlying thoughts, of philosophical reflections, hinted at but never fully expressed.”22 It was evident that Bohr reached his conclusions by his deep intuition and ad hoc assumptions based on inspiration, rather than detailed calculations. At the end of the of the third seminar, Heisenberg pointed some weaknesses in one of his key papers. After the seminar, Bohr invites him for a walk later that day. About that 3 h hike, Heisenberg later mentions as “my real scientific career only started that afternoon.”23 In that hike, Heisenberg also noticed the deep cracks in the theory that worried Bohr himself; 18

Enz (2002), p. 115. Commins (2012), pp. 135–136. 20 Enz (2002), p. 117. 21 Goudsmidt (1976), p. 248. 22 Heisenberg (1971a), p. 38. 23 Heisenberg (1971a), p. 38. 19

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Fig. 5.6 Werner Heisenberg, 1901–1976 (AIP Emilio Segrè Visual Archives, Segrè Collection)

one of the founding fathers of the quantum theory. Impressed by the twenty-year-old Heisenberg, Bohr invited him to Copenhagen for a term. But since Sommerfeld was about to leave for America, Copenhagen had to wait (Fig. 5.6). During his absence, Sommerfeld had arranged Heisenberg to work with Max Born in Göttingen. Along with Bohr’s institute at Copenhagen, Sommerfeld’s institute at Munich and Born’s group at Göttingen acted as the cradle of talented quantum theorists for years to come. When Born met Heisenberg, he immediately noticed that there was something special about this young man. He wrote to Einstein “easily as gifted as Pauli.”24 When Heisenberg returned to Munich, he finished his doctoral thesis in July 1923 on turbulence, a topic that Sommerfeld assigned him to improve his basic physics knowledge. His oral examination did not go well. His stumbling on some simple experimental physics questions like the resolving power of a telescope almost cost him his PhD. Especially, when he struggled to explain how a storage battery worked, an argument broke out between the head of the experimental physics department, Wilhelm Wien and Sommerfeld. Wien wanted to fail him, but eventually Wien caved in and he passed with the lowest of the three passing grades.25 Pauli had passed with the highest grade. Used to being at the top of his class, a disheartened and humiliated Heisenberg decides that he could no longer stay at Munich. Packing up and taking 24 25

Born (2005). Letter from Born to Einstein, p. 73, April 7, 1923. Cassidy (1992), pp. 149–154. APS News (1998, Vol. 7, no. 1).

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the night train, he fled to Göttingen, where he had a standing offer from Max Born waiting for him. After what happened in his oral exam, he was afraid that Born may no longer be interested in him as his assistant. However, after seeing that the questions were tricky, Born’s confidence in his talents remains intact and he decides to let the offer stand. He also comforted him by saying that he will soon bounce back. Born believed that the new quantum theory that would be called the quantum mechanics had to be built from scratch. The hotch potch of classical and quantum ideas supplemented by ad hoc rules that lied at the hearth of Bohr-Sommerfeld atom eventually had to give way to this new theory. Unable to explain the anomalous Zeeman effect, Pauli was already declaring openly that it is a strong evidence that the old quantum physics has reached its limits. Like Pauli, Heisenberg had also been working on the anomalous Zeeman effect. Before Christmas 1923, he wrote to Bohr about his recent work and received an invitation to spend a few weeks in his institute. Accepting this offer, Heisenberg arrived at Copenhagen in March 15, 1924. He was hoping to spend some time with Bohr, but his first few days were frustrating. He had hardly seen Bohr. Besides, Bohr’s group of brilliant young scientists looked intimidating to him. Each speaking several languages, appeared to have a much deeper command of atomic physics. While he was struggling with these emotions, Bohr walks into his room. After apologizing, Bohr offers him to go on for a short hiking tour. During this three-day tour, they not only talked about physics, but also Heisenberg got to know and admire Bohr as a man. He wrote to Pauli, “I am of course, absolutely enchanted with the days I am spending here.”26 Pauli and Heisenberg always remained in contact. Especially Pauli was always interested in what Heisenberg was doing. When he learned that Heisenberg was going to spend a few weeks in Copenhagen, he wrote to Bohr describing him as a “gifted genius who would one day advance science greatly.”27 During these two weeks, Bohr and Heisenberg mostly talked about physics and the conceptual problems that underlie quantum atom. Years later, Heisenberg would talk about these two weeks as a “gift from heaven.”28 Shortly after Heisenberg returned to Göttingen, he received another invitation from Bohr to extend his stay in Copenhagen . Since Born was going to America for the upcoming winter semester, he gladly gave his blessing. Heisenberg returned to Bohr institute on September 17, 1924. He was 22 years old with his name on a dozen quantum physics papers, but he still had a lot to learn from Bohr. Later, he would say: “From Sommerfeld I learned optimism, in Göttingen mathematics, from Bohr physics.”29 By the end of April 1925, he was back in Göttingen, fully convinced that the new quantum theory had to be constructed from the ground up.

26

Mehra and Rechenberg (1982). Vol. 2, p. 140. Letter from Heisenberg to Pauli, March 26, 1924. Mehra and Rechenberg (1982). Vol. 2, p. 133. Letter from Pauli to Bohr, February 11, 1924. 28 Mehra and Rechenberg (1982). Vol. 2, p. 142. 29 Mehra and Rechenberg (1982). Vol. 2, p. 3. 27

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5.6 Heisenberg the Magician During his second visit, he closely worked with Hendrik Kramers, whose work helped convince Heisenberg that the new theory could only be constructed by abandoning the classical view of the atom as little planetary systems. It was about this time that Heisenberg decided to concentrate on what was observable. Secrets of the atomic structure were unveiled in the atomic spectra. Frequencies of the individual lines in the hydrogen spectrum were successfully accounted by the Bohr-Sommerfeld model, but the long-standing unsolved problem was the brightness of the individual lines, in other words, their intensities. Heisenberg now stopped visualizing what was happening inside the atom. Anything that was not observable in laboratory, like the idea of electrons orbiting the nucleus, had to be discarded, and attention had to be focused on quantities like the frequencies and the intensities: quantities that could actually be measured. In June 1925, Heisenberg was still clueless about how to construct this new theory and how to calculate the intensities of the spectral lines of hydrogen. On top of this, a severe case of hay fever did not help his morale either. Unable to concentrate on anything, he had to get away. Sympathetic to Heisenberg’s condition, Born grants him a two-week holiday. Heisenberg immediately takes the night train on June 7, 1925 to the port of Cuxhaven on the coast and then the ferry to the island of Helgoland. Free from distractions and the fresh North Sea air, along with some hiking activity, quickly helped him regain his health and improved his spirits. Once again, he was able to concentrate on the frequencies and the relative intensities of the spectral lines. In terms of observables, they were the only data available about the inside of the atom. As a way of book-keeping of all possible transitions that could occur between different energy states of the hydrogen atom, he wrote the following array: v11 v12 v13 · · · v1n v21 v22 v23 · · · v2n .. .. .. .. .. . . . . . . vm1 vm2 vm3 · · · vmn

This array represented all the possible frequencies that could be emitted by an electron jumping from the mth state to the nth state with the energies E m and E n , respectively. When an electron jumps from an energy state with the energy E m to E n , where m > n, a line in the emission spectrum corresponding to vmn would appear. When m < n, a corresponding line appears in the absorption spectrum. Terms with m = n would all be zero, since a transition from one energy state E m , to the same energy state E m , would not result with with any emission or absorption. Heisenberg now needed a second array, amn , representing the transition rates between the various energy states. If the probability of a particular transition from the energy state E m to E n is high, then the corresponding spectral line, vmn , would be brighter. Heisenberg had two main starting points; the first one was the recognition that at the quantum level classical equations are no longer valid. Second, the correspon-

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dence principle must hold. For example, if we consider electron orbits in the classical limit as the principal quantum number n becomes very large, then the frequency of the radiation emitted by the electron as it spirals toward the nucleus approaches the orbital frequency. Heisenberg was aware of the fact that electrons are like harmonic oscillators that are oscillating with the orbital frequency of the electrons, thus could produce all the frequencies of the spectrum. Since transitions between energy states manifest themselves as spectral lines, Heisenberg then constructed an abstract model consisting of an infinite series (Fourier series) of harmonic oscillators, each with an amplitude and a frequency. His starting point was that in the quantum realm, only transitions between states are observable; hence, physically observable quantities should be associated with two (states) indices, rather than one. He therefore identified each term, each oscillator, in the series with a quantum jump from one energy state m to another state n. The result was an array amn , where the intensities of the spectral lines could be calculated as the squares of the amplitudes that appeared in the array. For example, for the transition from the state n to n − 2, it was necessary to multiply the amplitude of the term corresponding to the transition from n to n − 1 with the amplitude of the term corresponding to the transition from n − 1 to n − 2, etc. After some hefty calculations, he was able to calculate the mechanical properties of the oscillators like their displacement from their equilibrium position and momentum. That evening everything began to fall into their places. The new theory built out of observables appeared to predict the right frequencies and the intensities of the spectral lines. But did it satisfy the conservation of energy principle? It was almost three before morning that he proved that the theory was indeed consistent and that it did not violate the energy conservation law. Later, he wrote in his book Der Teil und das Ganze: “It was about three o’clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and waited the sunrise on the top of a rock.” In the morning, some of the initial euphoria had faded. Algebra of the new theory implied when two observables, X and Y, are multiplied, their order mattered. In other words, X times Y was not equal to Y times X. Mathematicians called such algebra non-commutative. As this strange algebra continued to baffle him, he hastily wrote his calculations and on Friday, June 19 returned to the main land. First he went to Hamburg to see Pauli. After receiving encouraging words from Pauli, he continued to Göttingen, where he would write the refined version of his new theory. After a little more than a week, he finished the paper and sent a copy to Pauli, requesting his comments within two or three days. The reason for the rush was that he was going to give a seminar at the Cambridge University on July 28. He was still afraid that Pauli might find a fatal flaw and that he may have to scrap it.30 Pauli greeted the paper with joy and wrote to a colleague “a new hope, and a

30

Mehra and Rechenberg (1982). Vol. 2, p. 291.

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renewed enjoyment of life”31 and “although it is not the solution of the riddle” Pauli added “I believe it is now once again possible to move forward.”32 Heisenberg had also given a copy to Born commenting that “he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advice him on it.”33,34 Born was enthusiastic, but confused about Heisenberg’s arguments and in particular, about the strange multiplication that Heisenberg had used. However, he submitted the paper without delay. Heisenberg’s paper was received by Zeitschrift für Physik on July 29, 1925. Years later in 1992, Nobel Laureate Steven Weinberg would write about Heisenberg’s paper as “If the reader is mystified at what Heisenberg was doing, he or she is not alone. I have tried several times to read the paper that Heisenberg wrote on returning from Helgoland, and although I think I understand quantum mechanics, I have never understood Heisenberg’s motivations for the mathematical steps in his paper. Theoretical physicists in their most successful work tend to play one of two roles: they are either sages or magicians... . It is not difficult to understand the papers of sage-physicists, but the papers of magician-physicists are often incomprehensible. In that sense, Heisenberg’s 1925 paper was pure magic.”35 Born was perplexed by the strangeness of Heisenberg’s calculations. In particular, what was the meaning of the mysterious multiplication rule he used? But then there was something strangely familiar about it. One morning on July 10, Born suddenly remembers a lecture that he had attended years ago as a student. The strange multiplication that Heisenberg used was nothing but multiplication of two matrices, where columns are multiplied into rows. A matrix is an array of numbers organized in terms of rows and columns. In contrast to numbers, multiplication is not commutative. In other words, when two matrices, A and B, are multiplied, AB, in general, is not equal to BA. Although the theory of matrices was well established by mathematicians, at the time it was not included in the toolbox of theoretical physicists. Now, with the source of the strangeness in Heisenberg’s paper identified, Born turned to Pauli for help. But Pauli refused by saying “You are only going to spoil Heisenberg’s physical ideas by your futile mathematics.”36 Unable to make progress on his own, Born called one of his students, Pascual Jordan, who was well versed in matrix algebra due to his background in mathematics. By the end of September, Born and Jordan had already laid the foundations of the new quantum mechanics in terms of matrices and Jordan was preparing the manuscript to be sent for publication. They sent a copy to Heisenberg who was at Copenhagen at the time. When Heisenberg received the paper, he said to Bohr “It is full of matrices, and I hardly know what they are.”37 Soon Heisenberg caught up 31

Cassidy (1992), p. 204. Kumar (2014), p. 193. 33 Pais (1991), p. 278. 34 Kumar (2014), p. 193. 35 Aitchison et al. (2004), p. 2. 36 Born (1978), p. 218. 37 Greenspan (2005), p. 127. 32

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Fig. 5.7 Paul Dirac, 1902–1984 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

and when he returned to Göttingen in the middle of October, three of them wrote the final version of what became known as the three-man paper, where the first logically complete and consistent formulation of quantum mechanics was laid out. One of the highlights of the new theory was that in quantum mechanics observables are represented by matrices with two indices, i and j, as xi j , which labeled the rows and the columns of the matrix, respectively. The two indices were the consequence of the fact that in quantum mechanics, only the transitions between states are observable, not the individual states. Since matrix multiplication, in general, does not commute, position x and momentum p matrices in quantum mechanics satisfy the commutation relation ih I, (5.5) [x, p] = x p − px = 2π where I is the identity matrix. In classical mechanics, position and momentum are ordinary numbers, which commute. In the above commutation relation, as the Planck constant goes to zero, the commutation relation vanishes, thus yielding the classical result x p = px, in accordance with the correspondence principle. In November, while the three-man paper was still being written, Pauli was also working on the new theory and beats them to the punch by reproducing the line spectrum of the hydrogen atom. Heisenberg, in particular, was upset. He would later say “I myself had been a bit unhappy that I could not succeed in driving the hydrogen

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spectra from the new theory.”38 On top of this, Pauli had also calculated the Stark effect, which was the effect of external electric fields on the atomic spectra. Pauli submitted his paper to Zeitschrift für Physik on January 17, 1926. These were the first solid successes of the new theory, which Heisenberg later recalled as “Pauli had provided the first concrete vindication of the new quantum mechanics.”39

5.7 Dirac Enters the Picture In one of visits to Cambridge, Heisenberg had discussed his ideas with the prominent British physicist Ralph Fowler, who asked for a copy when the paper was done. Heisenberg sends him a proof copy of his paper. Finding the paper difficult to read, unimpressed Fowler passes the paper to one of his young students, 23-year-old Paul Dirac (Fig. 5.7). Dirac also finds the paper difficult to understand, but eventually realizes that at the heart of Heisenberg’s theory lies the fact that observables are represented by matrices that in general do not commute. When he was done, he had independently arrived at the Heisenberg’s multiplication rule and the commutation ih I. He had also proved the energy conservation principle and relation [x, p] = 2π obtained the frequencies emitted by transitions between Bohr states. Dirac’s paper was submitted to the British journal Proceedings of the Royal Society 5 days later than Pauli’s paper. In May 1926, Dirac received his PhD, which was the first-ever thesis written on quantum mechanics.

38 39

Pais (2000), p. 224. Pais (2000), p. 224.

Chapter 6

Schrödiger’s Wave Mechanics

Born in Vienna on August 1887, Erwin Schrödinger received his PhD from the University of Vienna in May 1910. The title of his thesis was On the Conduction of Electricity on the Surface of Insulators in Moist Air. On October 1, 1910, he reported for his obligatory military service, which lasted a little more than a year. After he returned to the university in 1914, he qualified as a privatdozent (lecturer). During the First World War, he served in the army for four years. After the war, he returned to Vienna in 1917 and became an associate professor in 1920. By now, he had already established a solid reputation. Hence, he secured a professorship position in theoretical physics at the University of Zurich in October 1921. Soon after he started his position at the University of Zurich in October 1921, he was diagnosed with bronchitis and possibly tuberculosis. On doctors orders, Schrödinger and his wife Annemarie (Anny) retreated to a villa in the Alpine resort of Arosa near Davos. During the nine months they spent there, Anny nursed him back to his health. As he was recovering, he managed to write two scientific papers. On November 1922, he returned to his heavy teaching duties at the university, which left him little time for research. As he felt left behind by the young generation, in 1924 he was invited to the IV. Solway Conference, even though he was not asked to present a paper (Fig. 6.1). During this period, his marriage with Anny was in shambles, both indulging in extra-marital affairs. Schrödinger would frequently take refuge in the Bohemian lifestyle of the city with friends like Pieter Deby and Herman Weyl. These turbulent times also showed its effect on his research. He published no papers in 1923. In October 1925, a footnote in Einstein’s paper that mentioned de Broglie’s paper as “a very notable contribution”1 caught his eye. Intrigued by Einstein’s seal of approval, he acquired a copy of de Broglie’s thesis and on 3rd of November wrote to Einstein

1

Pais (1991), p. 241.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_6

95

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Fig. 6.1 Erwin Schrödinger, 1887–1961 (photograph by Francis Simon, courtesy of AIP Emilio Segrè Visual Archives)

“A few days ago I read with the greatest interest the ingenious thesis of de Broglie, which I finally got hold of.”2 On November 23, in a joint seminar held by the physicists at the University of Zurich and the ETH, Schrödinger was asked by Deby to present a talk on de Broglie’s thesis. Among the audience was a young Swiss student Felix Bloch, who 50 years later remembered the seminar as “Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle and how he could obtain the quantization rules of Niels Bohr and Sommerfeld by demanding that integer number of waves should be fitted along a stationary orbit.”3 At the time, there was no experimental confirmation of the de Broglie matter waves, which would come in 1927. Deby found the whole thing far-fetched and “rather childish.”4 He argued that for all waves from electromagnetic to sound waves and to waves on a violin string, there exists a wave equation called the wave equation that describes it. In what Schrödinger had presented, he saw no wave equation describing de Broglie’s matter waves. May be the rest of the audience did not pay much attention to this comment, but Schrödinger was affected. He realized that Deby was right—if there was a wave, then there had to be a wave equation describing it—and almost immediately, he decided to look for the missing wave equation for the de Broglie matter waves. 2

Mehra and Rechenberg (1987), Vol. 5, Pt. 2, p. 412. Letter from Schrödinger to Wilhelm Wien, 21 October 1926. 3 Bloch (1976), p. 23. 4 Bloch (1976), p. 23.

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Just before Christmas 1925, Schrödinger left Zurich for a short vacation in the Swiss Alps to a little villa in Arosa, where he had stayed and vacationed before with Anny. Since his relation with Anny was at all time low, leaving her in Zurich, this time he chose to invite an old girlfriend. He also took with him his calculations on the de Broglie’s matter waves. No body knows the name of this mysterious lady, but when he returned on January 8, 1926, he had discovered the wave equation. It is not possible to derive Schrödinger’s equation starting with classical physics. Schrödinger did not drive it either. He was quite vague about the steps he followed, but probably he started constructing it with de Broglie’s hypothesis that linked the wavelength associated with a particle to its momentum, λ = h/ p, and the relativistic equation that linked the frequency of a particle to its energy, ν = E/ h. Even though for clarity we are going to write a few of the essential equations in Schrödinger’s arguments, if the reader prefers, one can simply skip them and just read through the text. Schrödinger first wrote the classical expression for a plane wave: (x, t) = 0 ei(kx−wt) , k =

2π , ω = 2π ν, λ

(6.1)

where (x, t) represents the space and time dependence of any one-dimensional wave. Using λ = h/ p and ν = E/ h, he connected (x, t) to the momentum and the energy of the particle: p

E

(x, t) = 0 ei(  x−  t) ,  =

h . 2π

(6.2)

This allowed Schrödinger to write the following partial derivative: p2 ∂ 2 (x, t) = − 2 (x, t). 2 ∂x 

(6.3)

Schrödinger now wrote the classical energy conservation equation for a particle moving under the influence of a conservative potential V (x): p2 + V (x) = E, 2m

(6.4)

p2 (x, t) + V (x)(x, t) = E(x, t), 2m

(6.5)

and multiplied it with (x, t):

Substituting p 2 from the above equation he wrote −

2 ∂ 2 (x, t) + V (x)(x, t) = E(x, t). 2m ∂x2

(6.6)

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Finally, separating the spatial and the time dependences in (x, t) as (x, t) = (x)T (t),

(6.7)

Schrödinger obtained his famous time-independent Schrödinger wave equation: −

2 d2 (x) + V (x)(x) = E(x). 2m dx 2

(6.8)

As simple as this equation looks, it had sprang from Schrödinger’s true genius and deep insight. Of course, its true justification could only come from its applications, in particular, to the hydrogen atom, and that was exactly what Schrödinger had done next.

6.1 Wave Equation and the Hydrogen Atom Schrödinger rewrote the above equation as   2 d2 − + V (x) (x) = E(x), 2m dx 2

(6.9)

where the quantity inside the square brackets is called a differential operator. When a differential operator acts on a function (x) and produces the same function multiplied by a constant E, it is called an eigenvalue equation, where (x) is an eigenfunction and the constants E are the eigenvalues. That is why Schrödinger chose the title of his paper as Quantization as an Eigenvalue Problem. His paper was received by Annalen der Physik on January 27, 1926, and published on March 13. The paper consisted of the application of his wave equation to the hydrogen atom. The rest of the paper was more technical. He solved his wave equation in three dimensions using spherical polar coordinates (r, θ, φ), where r is the distance of the electron from the nucleus, and the angles θ and φ correspond to the latitude, and the longitude, respectively. He found that the wavefunction (r, θ, φ) could be separated as a function entirely in terms of r, and a function of the angular coordinates θ and φ as (r, θ, φ) = R(r )Y (θ, φ). The result was a set of highly complex differential equations. He could solve the angular part of the wavefunction in terms of spherical harmonics with two quantum numbers l and m, which can only take the values l = 0, 1, . . . , and m = −l, . . . , 0, . . . , l. However, he was at a loss when he came to the final differential equation for R(r ). On December 27, in a letter to Weyl, a frustrated Schrödinger wrote “I am struggling with a new atomic theory. If only I knew more mathematics! I am very

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optimistic about this thing and expect that if I can only ... solve it, will be very beautiful.”5 Schrödinger was right in being optimistic. He had already obtained the quantization rules for the orbitals of the Bohr-Sommerfeld theory for l and m. The radial function R(r ) depended on two quantum numbers n and l. Today, this equation is solved in textbooks by using the Fröbenius method of infinite series. In his paper, Schrödinger used the Laplace technique, which is a powerful tool for solving arbitrary order ordinary differential equations, but rarely taught and one that heavily relies on complex contour integrals.6 When Schrödinger returned to Zurich on January 8, he had immediately sought help from his friend Weyl. Soon he had the solution for the radial function that depended on the principal quantum number n, and l, which could only take the following values: n = 1, 2, . . . , and l = 0, 1, . . . , n − 1. By solving his wave equation in three dimensions and by fitting integer number of electron waves around the nucleus, Schrödinger had obtained the Bohr-Sommerfeld quantum numbers and also the energy levels of the Bohr-Sommerfeld quantum atom that depended on the principal quantum number n as 1/n 2 . In obtaining the solutions of his wave equation, all Schrödinger had assumed was the wavefunction to be single valued, finite (no infinities) and no sudden breaks or discontinuities. All the rest— the quantum numbers, Bohr-Sommerfeld energy levels, and Balmer’s formula— followed by the solution of his wave equation. Between January and June 1926, Schrödinger had published six highly creative papers that his mathematician friend Weyl called the “late erotic outburst in his life.”7 Solution of the Schrödinger equation for the hydrogen atom has always been considered as one of the most successful applications of Schrödinger’s wave mechanics. During this time, Schrödinger had also solved a range of problems in atomic theory with his wave mechanics. He also introduced the time-dependent version of this theory that opened the path to the investigation of processes like the emission, absorption, and scattering of radiation by atoms. A problem that perplexed Schrödinger was that how could there be two grossly different descriptions of the quantum world? One, in terms of particles, and the other in terms of waves. Yet, the two theories gave the same answer when applied to the same problem. In one of these six papers, Schrödinger had also demonstrated the equivalence of his wave mechanics and Heisenberg’s matrix mechanics, thus putting his mind to rest. Schrödinger’s theory had replaced the mysterious quantum jumps between the electron states of Heisenberg’s theory by the smooth and continuous transitions from one allowed three-dimensional standing wave pattern of the wavefunction to another. Besides, in contrast to the Heisenberg’s highly abstract matrix mechanics, which looked confusing to physicists at that time, Schrödinger’s wave mechanics 5

Moore (1989), 196. Galler et al. (2020). 7 Moore (1989), p. 191. 6

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was formulated in terms of differential operators that were already in physicist’s toolbox. Schrödinger’s wave mechanics not only offered an alternative that is easy to work with, but also one that can be visualized. In Heisenberg’s theory, observables were represented by matrices, whose multiplication was not commutative. In Schrödinger’s theory, observables are represented by differential operators, which are also non-commutative. For example, in the configuration space, product of the momentum operator,  p = −i ddx ,  = h/2π, and the position operator,  x = x, do not commute. As in the Heisenberg theory, the position and the momentum operators satisfy the commutation relation [ x,  p ] = iI, where I is the identity operator. Another advantage of Schrödinger’s theory was that a bridge between the classical mechanics and the quantum mechanics was established through the classical Hamiltonian: H (x, p) =

p2 + V (x) = E, 2m

(6.10)

where E is the total energy. Multiplying both sides by the wavefunction (x) and  replacing x and p with their operator counterparts yields  p 2 /2m + V ( x ) (x) = E(x), which is nothing but the one-dimensional time-independent Schrödinger equation: 2 d2 (x) + V (x)(x) = E(x). (6.11) − 2m dx 2

6.2 First Reactions to Wave Mechanics As Schrödinger and Heisenberg tried to understand each other’s theories, at first, there was no animosity between them. However, as emotions went rampant, both had difficulty in restraining their feelings. In a footnote of his paper, Schrödinger wrote “I was absolutely unaware of any generic relationship with Heisenberg.”8 He also said “I naturally new about his theory but because of to me very difficultappearing methods of transcendental algebra and because of the lack of visualibility, I felt deterred by it, if not repelled.”9,10 On the other hand, Heisenberg wrote to Pauli “The more I think about the physical portion of Schrödinger theory, the more repulsive I find it... what Schrödinger writes about the visualibility of his theory is [paraphrasing Bohr] probably not quite right, in other words, it is crap.”11,12 As Schrödinger’s theory gained popularity because of its mathematical accessibility,

8

Moore (1989), p. 211. Moore (1989), p. 211. 10 Moore (1989), p. 211. 11 Cassidy (1992), p. 215. Letter from Heisenberg to Pauli, 8 June 1926. 12 Baggott (2016), p. 67. 9

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Heisenberg’s frustration grew. At one point, he lost control and even called Born a traitor when he started using Schrödinger’s wave equation. As for the other reactions, after receiving his complementary copy from Schrödinger, Planck returned with the comment that he had read the paper “like an eager child hearing the solution to a riddle that plagued him for a long time.”13 Einstein wrote “the idea of your work springs from true genius.”14 Initially, Sommerfeld thought that the wave mechanics was totally crazy, but soon he changed his mind. Uhlenbeck wrote “The Schrödinger equation came as a great relief” and continued “now we did not any longer have to learn the strange mathematics of matrices.”15 In June, Born declared wave mechanics “as the deepest form of the quantum laws.”16 Pauli was astonished and impressed by the relative ease that Schrödinger was able to solve the hydrogen atom problem with his wave mechanics. A problem that Pauli was able to solve before Heisenberg and Born with matrix mechanics. Pauli had submitted his paper to Zeitschrift für Physik on January 17, 1926, ten days before Schrödinger had sent his paper to Annalen der Physik on January 27, 1926. Pauli told Pascual Jordan “I believe that the work counts among the most significant recently written.”17

6.3 Heisenberg, Schrödinger, and Bohr Encounter Heisenberg was in panic. His theory and his personal glory of discovery at Helgoland was about to be overshadowed, if not overthrown by Schrödinger’s mathematically appealing theory. The problem was not just the looks of the two theories, and Schrödinger was attempting to eliminate particles and quantum jumps by his classical space-time perspective. As Born said, if Schrödinger’s theory were to be adopted as “the deepest form of the quantum laws,” then Heisenberg’s theory would face the risk of being left on the dusty shelves of libraries. Heisenberg had to do something to regain the advantage. First, in April 1926 he had turned down assistant professorship in theoretical physics at the University of Leipzig, in favor of returning back to Copenhagen as Bohr’s assistant. There he was also hoping to join forces with Bohr. He arrived at Copenhagen in May 1926. During the summer of 1926, there was a high demand among physicists to hear Schrödinger’s theory from Schrödinger himself. In this regard, Sommerfeld and Wien had invited Schrödinger to give two seminars in Munich, which Schrödinger readily accepted. The two seminars were held on July 21 and July 23. This was an excellent opportunity for Heisenberg to meet Schrödinger and to hear his theory from himself. 13

Moore (1989), p. 209. Letter from Planck to Schrödinger, 2 April 1926. Moore (1989), p. 209. Letter from Einstein to Schrödinger, 16 April 1926. 15 Pais (2000), p. 306. 16 Cassidy (1992), p. 213. 17 Mehra and Rechenberg (1987), Vol. 5, Pt. 1, p. 1. Letter from Pauli to Pascual Jordan, 12 April 1926. 14

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Both seminars were jam-packed, and Heisenberg had attended both of them. He was silent most of the time, but at the end of the second seminar, during the questions and answers section, he could no longer keep quiet. As he rose to talk, all heads were turned at him. He mentioned that Schrödinger’s theory could not explain Planck’s radiation law, Franck-Hertz law, the Compton effect, and the photoelectric effect— all of which demanded discontinuity and quantum jumps—the very concepts that Schrödinger’s wave mechanics aimed to eliminate. From the looks of the people, the response was not what Heisenberg had hoped for. In particular, the older generation was not happy. Even before Schrödinger had attempted to answer his questions, Heisenberg’s old professor Wien stood up. He had tried to fail Heisenberg in his thesis defense for not being able to answer some simple experimental physics questions. During his thesis defense, Sommerfeld and Wien had an argument and finally he was allowed the pass by the skin of his teeth. Apparently, Wien had not forgotten him. Annoyed with the 24-year-old Heisenberg’s comments, the old professor, with a commanding voice said “Young man, Professor Schrödinger will certainly take care of all these questions in due time.”18 and then motioned for him to sit down. He also added “You must understand that we are now finished with all that nonsense about quantum jumps.” Heisenberg was once again humiliated and embarrassed by his old professor at Munich. Sommerfeld already enticed by Schrödinger’s mathematics, did not come to rescue this time, which also left Heisenberg disappointed. Heisenberg later would describe Wien’s behavior to Pauli as “he almost threw me out of the room.”19,20 Shaken and a disheartened Heisenberg was convinced that Schrödinger’s physical interpretation of quantum mechanics could not be right. After receiving a full account of what happened in Munich from Heisenberg, Bohr decided to invite Schrödinger to Copenhagen for a seminar and discussions in a narrower circle of physicists at his institute. Schrödinger arrived by train on October 1, 1926. Bohr met him at the train station. It was the first time they met each other. Confrontation began almost immediately and continued daily from early morning to late night. Even though Bohr was always kind and considerate to his guests, his strong conviction to show Schrödinger that he was the one in error, apparently had affected his this side as well. Bohr appeared even to Heisenberg as a “remorseless fanatic, one who was not prepared to make the least concession or grant that he could ever be mistaken.”21 Both man were determined to defend his interpretation of quantum mechanics passionately, while attacking the other’s weaknesses ferociously. Schrödinger thought that the “whole idea of quantum jumps (is) a shear fantasy,”22 Bohr responded “But it does not prove that there are no quantum jumps.” and continued as “You can’t seri-

18

Moore (1989), p. 222. Moore (1989), p. 222. 20 Heisenberg (1971b), pp. 73–76. 21 Heisenberg (1971a), p. 73. 22 Kumar (2014), p. 222. 19

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103

ously be trying to cast doubt on the whole basis of quantum theory.”23 Schrödinger tried to concede by saying that there were still so much to be understood. As Bohr pressed, Schrödinger finally lost his cool and bursted; “If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.” Bohr realizing that he may have pushed too far, tried to calm him; “But the rest of us are grateful that you did.” and continued that his mathematical theory added so much to the clarity and the simplicity of quantum mechanics.24 As the discussions raged day and night both men had exhausted so much intellectual energy that eventually Schrödinger got weak and succumbed to a severe case of cold. As Bohr’s wife Margrethe was trying to nurse him with hot beverages and cake, Bohr was sitting on his bedside and still pressing “But surely Schrödinger you must see...” Schrödinger certainly saw, but only in the eyes of his wave mechanics, where transitions are continuous. According to Bohr, on the other hand, these transitions were all taking place in discrete quantum jumps: The two diametrically opposing views seemed impossible to reconcile. Both men were convinced that his side was the right side. But no one was able to convince the other that he was the one in error. Schrödinger was not willing to accept that quantum theory represented a clean break from the classical concept of reality, Bohr and Heisenberg were convinced that there was no way of going back to the classical view of orbits and continuous paths in the quantum realm. As soon as Schrödinger returned to Zurich, he wrote to Wien describing his visit to the Bohr Institute as “In spite of all theoretical points of dispute, the relationship with Bohr, and especially Heisenberg, both of whom behaved toward me in a touchingly kind, nice, caring, and attentive manner, was totally cloudless, amiable, and cordial.”25,26 In the same letter, he also wrote “He [Bohr] is completely convinced that any understanding in the usual sense of the word is impossible.”27 Despite the fact that neither man had budged, there was no winner either. However, both men were deeply affected by each other’s views and by the intensity of the discussions. Bohr and Heisenberg had abandoned using any classical concept of space-time, but the very fact that wave mechanics had found such great support with its accessibility and successful applications, also made them think that maybe both views are needed to reconcile the particle and the wave properties together.

6.4 Born and the Meaning of  (Psi) While the controversy about the interpretation of quantum mechanics was raging at Copenhagen, Born had his own ideas about the meaning of Schrödinger’s wavefunc23

Heisenberg (1971a), pp. 73–75. Heisenberg (1971a), pp. 73–75. 25 Baggott (2016), p. 84. 26 Mehra and Rechenberg (1987), Vol. 5, Pt. 2, p. 826. Letter from Schrödinger to Wilhelm Wien, 21 October 1926. 27 Moore (1989), p. 228. Letter from Schrödinger to Wilhelm Wien, 21 October 1926. 24

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Fig. 6.2 Max Born, 1882–1970 (AIP Emilio Segrè Visual Archives, Physics Today Collection)

tion  (Fig. 6.2). Bohr and Heisenberg had already given up any classical concept of space-time to describe atoms and paths and orbits of particles. For them, atomic transitions took place in discontinuous discrete quantum jumps. For Schrödinger, these transitions were taking place smoothly, described by the solutions of his wave equation. He viewed  as a real physical wave. In classical wave theory,  represents the displacement of a medium from its equilibrium shape. For example, when we shake one end of a rope, the disturbance travels along the length of the rope, where (x, t) represents the displacement of a small mass element from its undisturbed position as a function of position and time. In this case, rope is the medium. When we drop a pebble into a lake, ripples spread out on the surface, where  represents the displacement of the surface of the lake from its undisturbed position. Now, water is the medium. In the case of sound waves, when we shout, disturbances in the form of pressure waves travel in air. In all these cases, the exact form of  is found from the solution of the wave equation and depends on the details of the source causing the disturbance and the physical characteristics of the medium. With these examples behind, nineteenthcentury physicists were puzzled by light and had to introduce a mysterious medium called ether as the necessary medium for light to travel. The confusion ended when Maxwell showed that light is nothing but an electromagnetic wave, that consists of oscillating electric and magnetic fields that are perpendicular to each other. When one of the fields decreased, it generated the other and vice versa, thus traveling billions of light years in vacuum unattenuated without ever needing a medium. In wave mechanics, neither Schrödinger’s equation nor its solutions looked like the classical wave equation and its solutions. However, Schrödinger still thought  as a real physical wave and wanted to understand everything, including particles, in terms of it. For example, he wanted to represent particles as wave packets, where a collection of waves when superimposed yield a localized wave that maintains its shape as it moves. However, Lorentz showed that an electron wave packet cannot maintain its shape for long. Waves that make up the wave packet would travel at different speeds, thus eventually making the wave packet diffuse and disappear into nothingness. Despite its successes, Schrödinger’s troubles did not end there. He was

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105

also troubled by the fact that his wavefunction in general would contain the imaginary number i, which is the square root of − 1. In mathematics, it is possible to define hybrid numbers called complex numbers that have both real and imaginary parts like z = a + ib, where a and b are real numbers. In physics and engineering, we often make use of complex techniques for convenience in real problems. But at the end, we always take either the real or the imaginary part of the solution as the physical solution. To maintain his interpretation of  as physical waves, Schrödinger tried to interpret ||2 as the smeared-out charge density of the electron wave packet. However, spreading of the wave packet did quickly put an end to this. Imaginary number i in Schrödinger’s wave mechanics was appearing as an essential part of the physical theory, not for convenience. This alone was perplexing. Furthermore, for the helium atom with two electrons, the wavefunction could not be interpreted as two three-dimensional waves. Instead, the wavefunction appeared as a single wave existing in a six-dimensional space. For systems that contain N particles, the wavefunction  would depend on 3N position coordinates, thus making it impossible to visualize in this abstract multidimensional space. When Schrödinger’s first paper appeared in March 1926, Born was almost at the end of his five-month stay in America. He was one of the founders of matrix mechanics with Heisenberg and Jordan. He was the one who first noticed that the strange multiplication that Heisenberg used was matrix multiplication. Born immediately realized the importance and the power of Schrödinger’s wave mechanics. He was particularly impressed with the ease and the elegance of the way Schrödinger had solved the hydrogen atom problem. A problem that required immense effort and Pauli’s genius to solve with matrix mechanics. However, Born was also dismayed by Schrödinger’s attempt to eradicate particles and quantum jumps altogether. In late 1926, Born wrote to describe his aim as “to drop completely the physical pictures of Schrödinger which aim at a revitalization of the classical continuum theory, to retain only the formalism and to fill that with a new physical content.”28 Born was aware of the fact that both the matrix mechanics and the wave mechanics were successful in describing transitions within stationary orbits, or states, of the electrons bound in atoms and, the position and the intensity of the spectral lines. For the new physical content, Born needed a quantum theory of collisions between atoms and electrons. He hoped that this would lead him to a theory of interactions between light-quanta and matter and also to a proper interpretation of . Since Heisenberg’s matrix mechanics had been designed to describe transitions between stationary states of electrons in atoms, it was not suitable for describing collisions. Thus, he turned to Schrödinger’s wave mechanics to introduce quantum jumps. Born treated collisions between an electron and an atom as an electron plane wave scattering from an atom in a stationary state with its characteristic frequency. The wavefunction of the scattered electron would be a superposition of all the possible angles that it could scatter. Unlike in colliding billiard balls, where the angle of scattering can be predicted from the masses and the velocities prior to the collisions, in quantum mechanics, the causal connection before and after the collision was lost. Born said 28

Pais (1986), p. 256.

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that the wavefunction represented the probabilities that the electron wavefunction would scatter into certain angles. This was Born’s new physics content. In the quantum realm, when it comes to collisions, quantum physics could not tell the state after the collision, but only “how probable is a given effect of the collision.”29 Born argued that the square of the modulus of the wavefunction ||2 , which is a real number, represented the probability that the electron would scatter into a particular direction. Since  could contain the imaginary number i, ||2 means  ∗ , where  ∗ is the complex conjugate of , where i in  is replaced by −i. Born’s hastily written first paper entitled Quantum Mechanics of Collision Phenomena was only four pages long and published on July 10 in Zeitschrift für Physik. Ten days later, it was followed by a more detailed paper bearing the same title. In these seminal papers, Born laid down the modern interpretation of the wavefunction that challenged the classical concept of determinism. Born considered a system whose wavefunction  could be represented as a superposition of the discrete eigenstates of the system {1 , 2 , . . . , n } , where n could be any number including infinity. The probability of the system being found in the ith eigenstate i is given by |ci |2 , which is a number between 0 and 1, and where ci is the amplitude or the coefficient of the eigenstate i in the expansion  = c1 1 + c2 2 + · · · . In August 1926, Born delivered a talk at Oxford. By this time, his views about the probabilistic interpretation of the wavefunction were a lot clearer. For the first time, he openly stated the clear distinction between the probabilities in quantum mechanics and the statistical probabilities. In classical physics, probabilities arise because we are either ignorant or not interested in learning the details of the microscopic properties of the system. We are perfectly satisfied by the averages taken over the microscopic variables hidden from the macroscopic view. For example, in statistical mechanics, temperature is related to the average velocity of the molecules. Even though the velocities of the individual molecules exist, knowing them not only does not add anything useful to our knowledge of the system, but trying to find them is not practical. Similarly, pressure is the average force per unit area imparted on the walls of the container by the impinging molecules. Born made it clear that in the quantum world, there are no hidden variables. The wavefunction contained the complete information about the system. The probabilistic interpretation of the wavefunction was the only interpretation. When an observation was made on a system, the wavefunction would collapse into one of the eigenstates of the system with the probability |ci |2 , where ci is the amplitude of the eigenstate i . Since the system would surely fall onto one of the eigenstates, conservation of probability demanded that the amplitudes satisfy the relation |c1 |2 + |c2 |2 + · · · + |ci |2 + · · · = 1,

(6.12)

which could be satisfied by normalizing the wavefunction, that is, by assuring that the integral of ||2 overall space is one.

29

Pais (1986), p. 257.

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107

By the collapse of the wavefunction, Born had succeeded in incorporating quantum jumps into Schrödinger’s wave mechanics. Born later said that in his interpretation of the wavefunction as probability, he was influenced by Einstein’s mention of the de Broglie waves as ghost waves in one of his unpublished papers. Unlike Schrödinger’s interpretation of the wavefunction as real matter waves, Born had apparently envisioned the wavefunction as a ghost wave that materialized only when an observation is made. Classical determinism was tied to causality that every effect has a cause. When an electron slams into an atom, unlike the case of colliding billiard balls, the electron can scatter into any one of the infinitely many possible angles, including backwards. It was as if the electron did not exist at all, or existed in some kind of suspended animation, until it was observed. Born said that we can only predict the probability of the electron scattering into a particular direction. Solutions of Schrödinger’s equation gave a probability wave, thus breaking the classical link between the cause and effect. As Born had admitted “I myself tend to give up determinism in the atomic world.”30 In the atomic world, causality was re-established at a new level, where the probability wave was evolving according to the Schrödinger’s wave equation. Born had pointed this out as “(the) probability itself propagates according to the law of causality.”31

6.5 Does God Play Dice? Born was appalled when he learned that Schrödinger wanted to eliminate particles and quantum jumps with his wave mechanics. Now, Schrödinger was shocked when he learned that Born wanted to give up determinism. In a letter to Wien, Schrödinger wrote “From an offprint of Born’s last work in Zeitsch. f. Phys. I know more or less how he thinks of things: The waves must be strictly causally determined through field laws, the wavefunctions on the other hand have only the meaning of probabilities for the actual motions of light or material particles.I believe that Born thereby overlooks that... it would depend on the taste of the observer which he now wishes to regard as real, the particle or the guiding field. There is certainly no criterion for reality if one does not want to say: the real is only the complex of sense impressions, all the rest are only pictures.”32 Einstein was as unhappy as Schrödinger with Born’s probabilistic interpretation of the wavefunction. In December 1926, he responded to a letter from Born, where Born had acknowledged his debt for the inspiration of the idea about ghost waves. Einstein wrote “Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring

30

Pais (1986), p. 257. Pais (2000), p. 39. 32 Moore (1989), p. 225. Letter from Schrödinger to Wilhelm Wien, 26 August 1926. 31

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us any closer to the secret of the old one. I, at any rate, am convinced that He is not playing at dice.”33,34 Born was disappointed by Einstein’s response. Einstein is usually quoted as “God does not play dice.” To understand what Einstein had meant by this, we have the dwell into his basic philosophy of nature and reality. Sometimes Einstein’s objection to Born’s probabilistic interpretation of the wavefunction is interpreted as his reluctance to give up determinism—may be due to the existence of some kind of hidden variables. However, Einstein’s understanding of reality was based on his unshakable belief that nature exists (runs) independent of whether there is somebody out there observing or not. He once asked Abraham Pais “Does the moon exist only when you look at it?”35 Even though the moon is a classical macroscopic object, Einstein had used this example to point out the absurdity of this line of thinking. In the case of an electron scattering from an atom, according to Born’s interpretation, until an observation is made, the electron is in a mixed state—a superposition of all the possible outcomes. Only with the observation, wavefunction of the electron collapses to a particular state. Until the wavefunction collapses, no body can tell, not even nature knows to what state the collapse will take place. Only the probabilities could be calculated. If you are confused, or if you think you understood it, I remind you what the American Nobel laureate Richard Feynman said ten years after Einstein’s death “I think I can safely say that nobody understands quantum mechanics.” This debate that had started in 1926 is still going on, though somewhat quietly, with the difference that most physicists have learned to live with it.

33

Baggott (2016), p. 78. Born (2005), p. 88. Letter from Einstein to Born, 4 December 1926. 35 Bernstein (1991), p. 42. 34

Chapter 7

Complementarity Embraces Particles and Waves

In June 1926, Max Born was eager to embrace wave mechanics, since it was easier to apply to the scattering problems. However, after witnessing the potency of the particle concept, he had also realized that particles simply cannot be eradicated. In fact, the tracks left in a cloud chamber provided irrefutable evidence for the particlelike properties of the electron. Cloud chamber was invented by C.T.R. Wilson in 1911 at Cambridge, where he had managed to create a small glass chamber containing air supersaturated with water vapor. Energetic alpha and beta particles, when passed through the chamber, left a trail of tiny water droplets in their wake by ripping electrons from the atoms in the air. Wilson had given a valuable tool to physicists to observe trajectories of alpha and beta particles. Besides the tracks in the cloud chamber, Schrödinger’s hopes to interpret particles as wave packets had already been shown to be untenable. By November, Born’s objections to Schrödinger’s interpretation of the wavefunction as real physical waves were sharp enough to cause him to reverse his position. On November 6, 1926, Born wrote to Schrödinger “It would have been beautiful if you were right. Something that beautiful happens, unfortunately, seldom in this world.”1 Even though the wave mechanics continued to be favored by physicists for its simplicity, Born had now turned back to support matrix mechanics. On the other hand, after the fight that Schrödinger put on to support his wave mechanics at Copenhagen, Bohr and Heisenberg were determined to find a resolution. Heisenberg would have preferred just to continue with his matrix mechanics. However, the persistence of Bohr and not to mention the undeniable advantages of the wave mechanics, and the discussions they had with Schrödinger when he was at the Bohr Institute, had also convinced him that may be a fresh approach is needed. Heisenberg’s work was cut out for him. The problem was not simply matrix versus wave mechanics. He had to think once again outside the box. He drew his clues from 1

Beller (1999), p. 36. Max Born, letter to Erwin Schrödinger, November 6, 1926.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_7

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the conversations he had with Dirac, who was at Copenhagen in September 1926 for a six month visit and Pascual Jordan who was responsible in converting Heisenberg’s Helgoland work into matrix mechanics. Pauli, as usual, also provided the inspiring key ideas. In October 1926, in a letter to Heisenberg, Pauli proposed an alternate interpretation of the wavefunction, where ||2 would be the probability (density) of finding the electron at a specific position in its orbit. In contrast to Born’s interpretation of ||2 as the probability of the electron being in a specific (eigen)state, this was introducing the space-time description by fusing the Schrödinger’s wavefunction and Born’s probability interpretation. Pauli was now convinced that what was needed is a logically consistent amalgamation of both approaches, rather than the elimination of one for the other. In his letter, Pauli had also made an important observation. In the collision between two electrons, when the electrons are sufficiently far from the collision center, they could be represented as plane waves with definite position q and momentum p. However, near the collision point, Pauli noticed that there is a dark point, or a blind spot, where things become fuzzy. If the positions were controlled than one would loose the control over the momentum and vice versa. Pauli wrote “One may view the world with the p-eye and one may view it with the q-eye, but if one opens both eyes at the same time one becomes crazy.”2 Particles followed well-defined paths, while waves were spread out and did not. How to reconcile them along with the Wilson cloud chamber results looked like an insurmountable task. Heisenberg tried to connect matrix mechanics with observable data. When he attempted to describe the clearly visible paths of electrons in the cloud chamber, he immediately ran into trouble as Pauli said. Then, he remembered a conversation he had with Einstein in April 28, 1926, when he presented his new, still considered as unconventional, theory at the Berlin University. After the talk, Einstein had shown personal interest in the 25-year-old Heisenberg and wanted to probe into the philosophical foundations of his theory. Einstein said “You assume the existence of electrons inside the atom, and you are probably right to do so. But you refuse to consider their orbits, even though we can observe electron tracks in the cloud chamber.I should very much like to hear more about your reasons for making such strange assumptions.”3 Heisenberg replied “we cannot observe electron orbits inside the atom”4 and added that he thought that a good theory should be based on observables like the frequencies of the spectral lines and their intensities, which were manifestations of the electron orbits. Einstein disagreed, “But you don’t seriously believe that none but observable magnitudes must go into a physical theory.”5 Heisenberg responded “Isn’t that precisely what you have done with relativity.” Einstein smiled; “A good trick should not be tried twice.”6 Einstein summed his view as “It is quite wrong to try founding a theory on

2

Enz (2002), p. 141. Wolfgang Pauli, letter to Werner Heisenberg, October 19, 1926. Heisenberg (1971a), p. 62. 4 Heisenberg (1971a), p. 63. 5 Heisenberg (1971a), p. 63. 6 Werner Heisenberg, AHQP interview, November 30, 1962. 3

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observable magnitudes alone. In reality the very opposite happens. It is the theory which decides what we can observe.”7 What Einstein meant was observation is a complex process that involves assumptions about the physical processes described in our theories. These physical processes produce a chain of effects in the apparatus which eventually lead to sense impressions that register in our mind. Einstein continued “You quite obviously assume that the whole mechanism of light transmission from the vibrating atom to the spectroscope or to the eye works just as one has always supposed it does, that is, essentially according to Maxwell’s law. If that were no longer the case, you could not possibly observe any of the magnitudes you call observable.”8 Einstein pushed his argument further “Your claim that you are introducing none but observable magnitudes is therefore an assumption about a property of the theory that you are trying to formulate.”9 With these arguments marshalled by Einstein, Heisenberg said “I was completely taken aback by Einstein’s attitude, though I found his arguments convincing.”10 Heisenberg was disappointed, since he was not able to persuade Einstein. Over time, their difference of opinion would grow. In February 1927, Bohr decided to take a four-week holiday. Heisenberg was happy that he would be alone for a while to concentrate on the problem of waveparticle duality. Even though it had been shown that matrix and wave mechanics were mathematically equivalent, the question that “is electron a wave or a particle?” was unanswered. Working late one evening, he was pondering over the meaning of the tracks left by the electrons in a cloud chamber, Einstein’s words “It is the theory that decides what we can observe.”11 started reverberating in his head. It was well past midnight. To clear his head, he went for a walk in the neighboring park. He sensed that he was onto something.

7.1 Uncertainty Principle With the late night chill, his thoughts started clearing. The tracks in the cloud chamber were not as sharp as they looked. They were visible because of the condensation of water droplets around the atoms that were ionized by the electrons passing through the chamber. Since electrons were much smaller than the droplets, the instantaneous position and the velocity of the electrons can only be known approximately from the tracks. He thought “May be those tracks represented much less than what he and Bohr have taken them for.” There was no continuous, unbroken path as they believed. May be the correct question that they should have asked was “Can quantum mechanics represent the fact that an electron finds itself approximately in a given place and that 7

Heisenberg (1971a), p. 63. Heisenberg (1971a), p. 64. 9 Heisenberg (1971a), p. 64. 10 Heisenberg (1971a), p. 64. 11 Heisenberg (1971a), p. 77. 8

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it moves approximately with a given velocity, and can we make these approximations so close that they do not cause experimental difficulties.”12 He rushed back to his room and began to do the mathematics with his new mechanics. Quantum physics was apparently putting restrictions on what could be known and observed. But how could the theory tell what can and cannot be observed and known. The answer was the uncertainty principle, or as he called the indeterminacy principle. He had discovered that at a given moment quantum mechanics forbids the precise determination of both the position and the momentum, hence the velocity, of a particle. It was possible to measure either the position or the momentum of the particle precisely, but not both simultaneously. Heisenberg had shown that mathematically this was stated as the product of the uncertainties in position, q, and momentum, p, cannot be less than h/4π, that is, qp  h/4π.

(7.1)

Having deduced the uncertainty principle, now Heisenberg had to show that this is not about experimental uncertainties due to the imperfections and the limitations of the equipment used, and that it is a fundamental principle of nature that holds in all physical processes. If we know the position at a given time precisely, q = 0, than we loose all information about the momentum and vice versa. The more precisely we know q, or p, the less we will know about the other. In classical physics h = 0, hence there was no such restriction, where both variables can be measured precisely at the same time.

7.2 Heisenberg’s Microscope Heisenberg believed that it was the act of measurement that introduced the unavoidable uncertainty in the position and momentum of the particle. As we tried to measure one, the other was disturbed irreversibly beyond control. To understand his new principle, Heisenberg constructed one thought experiment after another, where he questioned and deliberately tried to violate the uncertainty principle. Thought experiments are imaginary experiments that are in principle possible to do, but for the time being not doable because of technical difficulties. However, they are very useful in dwelling into the basics of a theory and to explore its bounds. No matter what he tried, after calculations he found that it was not possible to beat the uncertainty relation. It appeared to be a deep fundamental rule of nature that all processes obeyed. Than he remembered something, when he was a student at Göttingen, he had discussed with a friend the difficulty in explaining the electron orbits in atoms, his friend told him that it should be possible to build a microscope to see the electron. An ordinary microscope works by shining light on an object, where the reflected light 12

Heisenberg (1971b), p. 78.

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forms an image. To see an electron in orbit, one needs a microscope that worked with gamma rays that has much smaller wavelengths, hence much higher frequencies than the visible light. In 1923, Compton scattering of electrons with X-rays had decisively proven Einstein’s light-quanta (photons). Now, Heisenberg imagined photons scattered from the electrons in orbit into his gamma-ray microscope. Since the resolution of a microscope increases with frequency, in order to locate the electron in its orbit he had to use sufficiently high frequencies. But high frequencies also meant high energies for the photons, which in turn meant a bigger jolt for the electron when hit by the photon. As one tried to pin down the position of the electron, the higher became the jolt, thus the disturbance of its momentum. The more precisely Heisenberg wanted to know the location of the electron, the less accurately he was going to know its momentum, or velocity. To lessen the jolt on the electron, he tried reducing the frequency, but this time, he was going to know the position less precisely. Heisenberg explained “the more precisely the position is determined the less precisely the momentum is known, and conversely. In this circumstance we see a direct physical interpretation of the equation pq − qp = −i h I /2π .”13 The fact that pq is different from qp meant that the order in which these variables are measured matters. The first measurement disturbs the second variable irreversibly, thus yielding a completely different result when the variables are measured in different order. Heisenberg first noticed in Helgoland that in his new mechanics qp is different from pq. At first, not knowing what it means, he was worried. When he returned to Göttingen, Born was the one who first noticed that this was a consequence of the matrix mechanics that Heisenberg’s theory was based on. Now Heisenberg had discovered that the commutation relation is a direct consequence of the uncertainty principle, thus making its physical implication clear. In classical physics, h is zero; hence, the order of the measurements is irrelevant. Heisenberg also discovered another uncertainty relation between the conjugate variable energy E and time t as Et  h/2π,

(7.2)

thus extending similar arguments to the energy and time measurements. This relation is usually interpreted to signify the lifetime of an emission. In February 1927, Heisenberg wrote a long letter to Pauli. It was the draft of his paper on the uncertainty principle. Once again, he was relying on Pauli’s critical assessment before Bohr returned from his ski trip. He explained; “Because I felt that when Bohr comes back he will be angry about my interpretation. So I first wanted to have some support and see whether somebody else liked it.”14 Pauli’s response was exactly what he was hoping to hear: “Day is dawning in quantum theory.”15 This removed any additional question marks that Heisenberg might have had. He immediately started working on the final version of his paper that would be ready 13

Wheeler and Zurek (1983), pp. 62–84. Pais (1991), p. 304. 15 Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 93. 14

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for Bohr’s eyes. By March 9, Heisenberg had completed the paper. It was only then he informed Bohr his work and wrote to Bohr in Norway: “I believe that I have succeeded in treating the case where both p and q are given to a certain accuracy... I have written a draft of a paper about these problems which yesterday I sent Pauli.”16 However, he had neither sent a copy to Bohr nor given any more details about what he had done. This was how much he was afraid of Bohr’s first reaction, who wanted him to concentrate on wave-particle duality, while Heisenberg had focused on particles and discontinuity. Time would soon prove how right he was to be afraid.

7.3 Bohr Returns from the Ski Trip One day before Bohr returned from his four-week vacation on March 23, 1927, Heisenberg had already submitted his paper entitled On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics to Zeitschrift für Physik. After his trip, an invigorated Bohr first took care of some administrative duties waiting for him at the institute and then read Heisenbergs’s paper carefully. In their first meeting, Bohr’s first words were “not quite right.”17 Heisenberg was stunned, Bohr not only did not agree with his interpretation, but also had found a fatal error in his analysis of the gamma-ray microscope. Heisenberg had traced the origin of the uncertainty to the Compton effect, where the photon disturbed the momentum of the electron in a sudden and unpredictable way. Bohr disagreed by saying that what prohibits the precise measurement of the electron’s momentum was the impossibility of knowing the exact position of the photon when it enters the microscope. He continued that in Compton effect, it is possible to calculate the change in momentum precisely, as long as one knows the angle that the photon has scattered after going through the aperture of the microscope. Position of the electron when it collides with the photon was uncertain because of the finite aperture of the microscope, which restricts its ability to locate the position of the electron exactly. According to Bohr, the origin of this uncertainty was the wave nature of the gamma rays. The resolution of a microscope is restricted by diffraction, which causes blurring of the image, that is, inability to distinguishing objects closer than a minimum resolvable distance. Since the resolution of a microscope increases with shorter wavelengths, Heisenberg was right in using gamma rays. However, he had ignored the fact that the finite length of the aperture also introduced a fundamental limit on the resolution of a microscope. Heisenberg was once again haunted by his thesis defense, when Wien wanted to fail him because he was not able to answer some basic experimental physics questions like the resolving power of a microscope, or a telescope, satisfactorily. Bohr had firmly believed in that the wave-particle duality was at the core of the uncertainty principle.To show this, he resorted to the Schrödinger’s wave packet idea. 16 17

Pais (1991), p. 304. Letter from Heisenberg to Bohr, March 10, 1927. Cassidy (1992), p. 241. Letter from Heisenberg to Pauli, April 4, 1927.

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Fig. 7.1 Uncertainties in x and p satisfy the relation xp  h/4π, where x and p are the spreads of the wavefunctions in the configuration and the momentum spaces, (x) and ( p), respectively

A wave packet, (x), centered at x = 0 is a superposition of a set of waves with an amplitude distribution given as a function of momentum, ( p), which is peaked around the momentum p = 0. For a localized electron in space at x = 0, (x) is narrow and not spread out. To localize the electron further, one has to add more waves to the superposition, thus making the amplitude distribution in the momentum space broader, thus increasing the uncertainty in momentum (Fig. 7.1). For a precisely localized electron, we have to add all the waves with the same weight, thus loosing all information about the momentum. Bohr had also shown that the uncertainties x and p, that is, the spreads in (x) and ( p), respectively, are related and satisfy the Heisenberg uncertainty relation xp ≥ h/4π. The relation between (x) and ( p) is such that ( p) is the Fourier transform of (x). What troubled Bohr was that Heisenberg had fixated exclusively on particles and discontinuity. He was not willing to include any kind of wave interpretation into his arguments. Bohr, on the other hand, believed that the wave interpretation of quantum mechanics cannot be ignored. Bohr considered Heisenberg’s refusal to include waveparticle duality into his understanding of quantum mechanics as a deep conceptual flaw in his approach. Later, Heisenberg would say “I did not know exactly what to say to Bohr’s arguments, so the discussion ended with the general impression that Bohr has again shown that my interpretation is not correct.”18 They were both upset, and Heisenberg was raged. Both avoided each other for a couple of days. When they eventually decided to discuss the uncertainty principle again, Bohr had hoped that Heisenberg has seen that his approach is the way to go and had already decided to rewrite his paper. But Heisenberg was adamant. Bohr tried to convince him again that he should not publish the paper. Young Heisenberg later said “I remember that it ended by my breaking out in tears because I just couldn’t stand this pressure from Bohr.”19 From Heisenberg’s point of view, it was his discovery at Helgoland that was at stake. As Schrödinger’s wave mechanics was becoming the theory of choice by most theoretical physicists, he was afraid that his theory will eventually be forgotten. He wanted to publish as soon as possible to confront Schrödinger’s claim that matrix mechanics was untenable because of its unvisualizability. Heisenberg abhorred Schrödinger’s wave mechanics and his physics of continuity as much as Schrödinger had loathed his discontinuities and particles. Heisenberg thought that with his uncertainty principle, finally he had the 18 19

Werner Heisenberg, AHQP interview, February 25, 1963. Werner Heisenberg, AHQP interview, February 25, 1963.

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right tool to prove that his theory was the correct interpretation of quantum mechanics. To this affect, he added a long footnote to his paper: “Schrödinger describes quantum mechanics as a formal theory of frightening, indeed repulsive, abstractness and lack of visualizability. Certainly one cannot overestimate the value of the mathematical (and to that extent physical) mastery of the quantum-mechanical laws that Schrödinger’s theory has made possible. However, as regards questions of physical interpretation and principle, the popular view of wave mechanics, as I see it, has actually deflected us from exactly those roads which were pointed out by the papers of Einstein and de Broglie on the one hand and by the papers of Bohr and by quantum mechanics [i.e. matrix mechanics] on the other hand.”20 Two weeks after he had submitted his paper on March 22, 1927, he wrote to Pauli that he had quarreled with Bohr.21 On May 31, 1927 Heisenberg’s paper appeared in print. He had added a note to correct his misinterpretation of the gamma-ray microscope and acknowledged his debt to Bohr for bringing to his attention some critical points that he had overlooked. The same day, he wrote to Pauli “In the ardour of this struggle I have often criticized Bohr’s objections to my work too sharply and, without realizing or intending it, have in this way personally wounded him. When I now reflect on these discussions, I can very well understand that Bohr was angry about them.”22

7.4 Complementarity While Heisenberg was working on the uncertainty principle at Copenhagen, Bohr at the ski slopes in Norway was pondering over the complementarity principle. Bohr believed that he could reconcile these mutually exclusive wave and particle properties of nature through this principle. He thought both properties were essential for a complete description of nature. However, there were restrictions. Depending on the experiment, observers can only look into either the wave or the particle property. No experiment can reveal both wave and particle properties at the same time. Bohr argued that “Evidence obtained under different conditions cannot be comprehended within a single picture, but must be regarded as complimentary in the sense that only totality of the phenomena exhausts the possible information about the objects.”23 Unlike Heisenberg, who was under the spell of his intense dislike of waves and continuity, Bohr saw support for his complementarity in the uncertainty relation. Einstein’s lightquanta with energy E = hν and its momentum from the special relativity, E = pc, and through the de Broglie formula λ = h/ p, had already embodied the wave-particle duality.

20

Heisenberg (1927), p. 82. Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 182. Letter from Heisenberg to Pauli, April 4, 1927. 22 MacKinnon (1982), p. 258. Letter from Heisenberg to Pauli May 31, 1927. 23 Bohr (1949), p. 210. 21

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According to Bohr, the question of whether light was a wave or a particle had no meaning. In quantum mechanics, there was no way to answer this question. The only meaningful question was “does light behave like a wave or a particle?” The answer depended on the choice of the experiment. Bohr was not saying that during an observation, an irreducible and uncontrollable disturbance does not happen, but he was saying that the disturbance does not lie in the act of measurement as Heisenberg had believed, but in the observer’s choice of which side of the wave-particle duality to make the measurement.

7.5 Copenhagen Interpretation In early June 1927, Pauli had finally found time to visit the institute. Heisenberg was full of remorse. He wrote to Bohr “I am very ashamed to have given the impression of being quite ungrateful.”24 Bohr was disappointed at the reaction of his young protégé. Scars were left in both men, but Pauli’s presence had helped to heal the wounds. Eventually, a consensus was reached about the interpretation of quantum mechanics. The key points were the • • • •

Complementarity of waves and particles, Uncertainty principle, Quantum probabilistic interpretation of the wavefunction, Correspondence principle, where the transition from the classical to the quantum pictures take place in the limit of very large quantum numbers.

After several drafts, Bohr changed the title from The Philosophical Foundation of the Quantum Theory to the Quantum Postulate and the Recent Development of Atomic Theory. The final version had to wait, but it was ready to be presented in the International Physics Conference in September 1927 at Como, Italy. Among the participants were Born, Compton, de Broglie, Heisenberg, Lorentz, Fermi, Debye, Pauli, Planck, Rutherford, Sommerfeld, and others. Almost everybody was there except Schrödinger, who just succeeded Planck at the University of Berlin, and Einstein, who refused to come to fascist Italy. On September 16, Bohr presented the new interpretation of quantum mechanics for the first time that would later be known as the Copenhagen interpretation. During their many and long discussions with Bohr, Heisenberg had often wondered “Can nature be as absurd as it seemed to us in these atomic experiments?”25 Bohr’s reply was always a resounding “YES!”. The Copenhagen interpretation was pointing to the fact that in quantum mechanics we have reached the limits of what we can know scientifically. To try to go beyond this was pointless. Although we may talk about the existence of the position and

24 25

Pais (1991), p. 309. Letter from Heisenberg to Bohr, June 18, 1927. Heisenberg (1989), p. 30.

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momentum of the electron as if they are independently existing properties, they become real only when the electron interacts with the instrument designed to reveal the particular property. Bohr’s talk was the crystallization of the many discussions he had with Heisenberg. He had carefully stitched all the elements of the new interpretation of quantum mechanics−the complementarity, the uncertainty principle, Born’s probabilistic interpretation of the wavefunction, and the role of measurement. This was a view shared by Bohr, Heisenberg, Born, and Pauli. Bohr’s soft-spoken words and mostly wordy presentation was vague and did not leave much of an impression on the audience. Hungarian physicist Eugine Wigner said to Rosenfeld “This lecture will not induce any one of us change his own [opinion] about quantum mechanics.”26

7.6 The Measurement Problem A quantum system is by definition a microscopic system, where the classical laws of physics break down and the laws of quantum mechanics have to be used. The most rudimentary difference between the classical physics and the quantum mechanics is the effect of measurement on the state of the system. Measurement always involves some kind of interaction, hence a disturbance between the measuring device and the system. For example, in order to measure the temperature of an object, we may use a mercury thermometer. During the measurement process, the thermometer and the object come to thermal equilibrium at a common temperature by exchanging a small amount of heat. At this point, the mercury in the thermometer rises to a new level, thus allowing us to read the temperature from a scale. At the end of the measurement process, neither the mercury in the thermometer nor the object will be at their initial temperatures.However, we can always assure that the amount of mercury inside the bulb is sufficiently small so that only a tiny amount of heat exchange is sufficient for it to come to thermal equilibrium with the object. Hence, we can measure the temperature of an object without significantly altering its temperature. This shows that even in classical physics, measurement effects the state of a system. However, what separates classical physics from the quantum mechanics is that in classical physics these effects can either be minimized by a suitable choice of instrumentation or algorithms can be designed to take corrective measures. In summary, in quantum physics, it is no longer possible to separate the observer and the observed and the equipment used to make a measurement and what is being measured. Determinism and causality were the central pillars that Newton’s physics was built on. Even after Einstein’s relativity, given the exact values of the position and velocity of an object at a given time, it was possible to know the position and velocity at any other time. Elevating measurement and observation, hence the observer, to a key role, quantum mechanics had severed the link between the past and the future with the Heisenberg’s uncertainty principle. According to quantum mechanics, we cannot 26

Pais (1991), p. 315.

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determine present precisely. Not being able to know both position and velocity exactly simultaneously, opened up an infinitude of possibilities for the future. Therefore, in quantum processes, nature only allowed us to calculate probabilities exactly. As stated in the last paragraph of Heisenberg’s uncertainty paper, “it follows that quantum mechanics establishes the final failure of causality.”27

7.7 Fifth Solvay Conference The first Solvay Conference in 1911 was supported by the wealthy Belgian industrialist Ernest Solvay.It was a summit meeting on quantum, and it was the first invitationonly international conference on physics. Along with Einstein, Lorentz, Wien, Marie Curie, Poincare, Planck, Sommerfeld, and de Broglie were among the participants. Einstein was given the honor of presenting the final talk. The fifth Solvay Conference was scheduled for October 1927 (Fig. 7.2). After the first one, it was the first Solvay conference that Einstein had attended. It was chaired by Lorentz and the invitation made clear that the “conference will be devoted to the new quantum mechanics and to questions connected with it.”28 Even though Einstein had been asked to present a talk, he declined the offer and wrote to Lorentz “I have not been able to participate as intensively in the modern development of quantum theory as would be necessary for that purpose.” He also continued “This is in part because I have on the whole too little receptive talent for fully following the stormy developments, in part because I do not approve of the purely statistical way of thinking on which the new theory is founded.”29 Bohr was also not scheduled to present a talk. His contribution to the theoretical development of quantum mechanics, unlike Heisenberg, Pauli, Born, and Dirac, was somewhat indirect. After the Copenhagen interpretation of quantum mechanics at Como, it was the first time that Einstein and Bohr would meet each other. Bohr was anxious to learn what Einstein had thought about the recent interpretation of quantum mechanics. Bohr would later say “Several of us came to the conference with great anticipations to learn his reaction to the latest state of the development which, to our view, went far in clarifying the problems which he had himself from the outset elicited so ingeniously.”30 Certainly, a word of approval from Einstein would have meant a lot to Bohr, but soon he would find how wrong he was. What had added to his anxiety was in the middle of April 1927, as he was working on the new interpretation of quantum mechanics, at Heisenberg’s request, he had sent a copy of Heisenberg’s uncertainty paper to Einstein. In the accompanying letter, Bohr had praised Heisenberg’s work

27

Heisenberg (1927), p. 83. Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 232. 29 Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 241. Letter from Einstein to Hendrik Lorentz, June 17, 1927. 30 Bohr (1949), p. 212. 28

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Fig. 7.2 ’Institut international de physique Solvay, cinquieme conseil de physique fifth Solvay congress, Brussels, 1927.’ (Fifth Solvay Congress, Brussels, Belgum) The theme was electrons and protons. Left to right, back row: A. Piccard; E. Henriot (Brussels); P. Ehrenfest; E. Herzen; T. de Donder (Brussels); E. Schrodinger; J. E. Verschaffelt (Ghent); W. Pauli; W. Heisenberg; R.H. Fowler (Cambridge); L. Brillouin. Middle row: P. Debye; M. Knudsen; W. L. Bragg; H.A.Kramers; P.Dirac; A.H. Compton; L. deBroglie; M. Born; N. Bohr. Front Row: I. Langmuir; M. Planck; M. Curie; H.A. Lorentz; A. Einstein; P. Langevin; C. Guye (Geneva); C.T.R. Wilson; O.W. Richardson. ABSENT: Sir W. H. Bragg; H. Deslandres; E. Van Aubel (Ghent). (Photograph by Benjamin Couprie, Institut International de Physique Solvay, courtesy AIP Emilio Segrè visual archives)

and also outlined his emerging ideas.31 For some unknown reason, Einstein had not replied Bohr’s letter.32 On Monday October 24, after a brief welcome from Hendrik Lorentz, opening the conference was left to William L. Bragg, who was now 37 and was only 25 when he was awarded the Nobel Prize in physics in 1915.On Wednesday morning, Born and 31

Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 187.Letter from Bohr to Einstein, April 13, 1927.,Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 187.Letter from Bohr to Einstein, April 13, 1927.,BCW, Vol. 6, p. 418. Letter from Bohr to Einstein, April 13, 1927. 32 Kumar (2014), p. 246.

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Heisenberg had presented what would eventually be called the Copenhagen interpretation. Their conclusion provocatively ended with the statement “We consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification.”33 Something a revolutionary like Einstein could never accept. However, Einstein did not fall into their trap and did not took part in the discussions. Noticing Einstein’s disbelief at the boldness of their attitude, Ehrenfest passed a note to Einstein “Don’t laugh!” Einstein replied “I laugh only at their naivete. Who knows who would have the [last] laugh in a few years.”34 After lunch, Schrödinger made his presentation on wave mechanics. He concentrated on the aspects of wave theory that had caused concern, such as its interpretation in the configuration space. In the subsequent discussion, Heisenberg objected to Schrödinger’s suggestion that future developments may lead to a more conventional interpretation of the wavefunction. Heisenberg said “I see nothing in Mr. Schrödinger’s calculations that would justify this hope.”35 In August 1927, Prof. Schrödinger had moved to Berlin as Planck’s successor, a position he had secured for the discovery of wave mechanics. Friday morning Hendrik Lorentz opened the meeting for a general discussion.After expressing his views, he made a plea for causality and determinism. Lorentz would die a few months later in February 1928. After Lorentz, Bohr was asked to address the audience to describe his complementarity principle and the new interpretation of quantum mechanics. He did not say anything different than what he had already said in Como, but this time he was directly addressing Einstein and was hoping to convince him about the correctness of the Copenhagen interpretation.

7.8 Double-Slit Experiment and Einstein—Bohr Debate At first Einstein was quiet, but eventually stood up to make a comment. He described a thought experiment involving diffraction of a beam of electrons through a narrow slit and a photographic plate to collect the diffracted electrons. Einstein argued that if quantum mechanics was a complete theory, then each electron would be represented by a wavefunction. It would be this wavefunction that would produce the diffraction pattern. However, as soon as the wavefunction impinges on the screen, it would collapse and produce a single dot on the photographic plate. Einstein continued, if that dot had appeared at point A, then it was certainly absent at any other point B on the plate. Before the collapse, the wavefunction was smeared out over the entire plate. How could the wavefunction smeared out over the entire screen be localized instantaneously at one point by the act of measurement? As usual, Einstein was after the juggler. He felt that the act of measurement had changed the physical state of the 33

Bacciagaluppi and Valentini (2009), p. 437. Mehra (1975), p. xvii. 35 Bacciagaluppi and Valentini (2009), p. 472. 34

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Fig. 7.3 Einstein’s single slit thought experiment

system over the entire plate, which implied action at a distance, thus violating the postulate of relativity (Fig. 7.3). Einstein also offered a solution. He argued that if the wavefunction  represented an ensemble of particles, rather than a single electron, then each member of the ensemble passing through the slit would hit a different point on the plate with the probability ||2 , thus producing a diffraction pattern. According to Einstein, ||2 did not represent the probability of finding the electron at a specific point on the photographic plate, but it was the probability of finding a member of the ensemble at that point. Like Bohr, Heisenberg, Pauli, and Born were all confused. They were not sure where Einstein was going with this. Certainly, the wavefunction was collapsing, but it was an abstract probability wave, not a real physical wave. Bohr said “I feel myself in a very difficult position because I don’t understand what precisely is the point which Einstein wants to [make].”36 Incredible as it sounds, Bohr then said “I do not know what quantum mechanics is. I think we are dealing with some mathematical methods which are adequate for [a] description of our experiments.”37 According to Bohr, Einstein was trying to cling on to the classical concept of causal space-time description, where the screen and the photographic plate had well-defined positions, which was not tenable in quantum mechanics. Instead of responding to Einstein’s objections, Bohr had repeated his own views. Einstein was not impressed. When the discussions continued in the evening at dinner, Einstein had advanced his thought experiment by adding a second screen with a double slit between the first screen and the photographic plate (Fig. 7.4). Einstein now assumed that the intensity of the electron beam was adjusted such that only one electron at a time passes through both screens and hits the photographic plate. He said that if one could find out in which direction the electron was deflected at the first screen, one would be able to tell through which one of the two slits it subsequently passed through in the second screen. Using the point where the electron hits the photographic plate, he then argued that one would be able to trace its trajectory, thus observing the particle property of 36 37

Bacciagaluppi and Valentini (2009), p. 489. Bacciagaluppi and Valentini (2009), p. 489.

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Fig. 7.4 Einsteins’s double-slit thought experiment

Fig. 7.5 Bohr’s response to Einstein’s double-slit thought experiment

the electron. Einstein continued, if we repeat the experiment for a large number of electrons, then we would see the double-slit interference pattern, thus displaying both the particle and the wave properties simultaneously−in flagrant contradiction to Bohr’s complementarity principle. This time, Bohr had seen the weakness in Einstein’s argument. Einstein had assumed the screen to be infinitely massive; thus when the electron recoiled, the screen remained fixed. This left no room for the uncertainty principle. Bohr now replaced the first screen with a realistic one, that is, one with a movable screen connected to two weak springs and a pointer to observe the direction in which the screen recoiled. Controlling and observing the momentum transfer from the electron to the first screen required a screen that could move in the vertical direction (Fig. 7.5). Observing in which direction the screen recoiled, Einstein had argued that one could tell in which slit the particle went through in the second screen. Bohr responded by saying that it was not possible to control the momentum transfer at the first screen precisely as Einstein suggested. If one could measure the momentum transfer at the first screen precisely, then from the uncertainty principle,

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one would have no information about the position of the first screen. Bohr then argued that in order to read the pointer, one would have to illuminate the scale. Thus, photons scattering from the scale would disturb the screen, which would cause an uncontrollable momentum transfer to the screen. From the uncertainty principle, this would in turn imply uncertainty in the position of the screen. Bohr then showed that as one tried to find the slit in which the electron passed through in the second screen, uncertainty in the position of the first screen destroyed the interference pattern. If one tried to minimize the disturbance on the first screen by lowering the intensity of the light and by using low energy photons, then the uncertainty in the position would be so large that one would not be able to tell which slit the electron passed through in the second screen. Now, the result would be the double-slit interference pattern. Hence, Bohr had demonstrated both the particle and the wave properties of the electron, but not at the same time. Einstein presented more thought experiments as he forced Bohr to defend the uncertainty principle and the Copenhagen interpretation of quantum mechanics. Bohr rebuked them all, but Einstein’s insistence in questioning the conceptual foundations of quantum mechanics had also left question marks in the minds of some of the participants. May be the apparent winner was Bohr, but Einstein was not convinced and his challenge of the Copenhagen interpretation was just beginning.

7.9 After the Fifth Solvay Conference The double-slit thought experiment was an ingenious example for demonstrating complementarity by using two mutually exclusive experimental conditions. According to quantum mechanics, electron, or a photon, was neither a particle nor a wave. Depending on the experimental setup, it sometimes behaved like a wave, and sometimes like a particle. If the experiment was designed to determine which slit of the second screen the electron went through, then the particle property appeared with no interference pattern. If the experimental setup was not able to reveal the slit in which the electron went through, then the wave property showed up with an interference pattern. It was the experimenter who designed the experiment, and it was the experimenter who decided which property of the electron he/she wanted to see. It was the loss of independent reality, not the probability, that bothered Einstein. According to Einstein, nature should run its course according to the laws of nature independent of the presence of an observer making an observation. Later, Bohr would remember their debate with Einstein as “Einstein’s concern and criticism provided a most valuable incentive for us all to re-examine the various aspects of the situation as regards the description of atomic phenomena.”38 Copenhagen interpretation said that the measuring device and the act measurement were unavoidably connected with the object investigated with no separation possible. In the double-slit thought experiment, while the electron was subject to the laws of 38

Bohr (1949), p. 218.

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quantum mechanics, the equipment obeyed the classical laws. Yet Bohr had applied the uncertainty principle to a macroscopic object, the first slit, without establishing the boundary between the macroscopic (classical) and the microscopic (quantum) worlds. In this sense, it was a questionable move that Bohr had used to fend off Einstein’s criticism. de Broglie was disheartened by the poor reception that his pilot wave theory received, eventually converted to the Copenhagen interpretation.Einstein when reached Berlin wrote to Sommerfeld that quantum mechanics “may be a correct theory of the statistical laws, but it is an inadequate conception of individual elementary processes.”39 Heisenberg thought that they were the winners. When asked who were “they”, he said Bohr, Pauli, and myself. With Bohr’s institute quickly becoming a world center for quantum mechanics, the Copenhagen interpretation quickly became synonymous with the quantum mechanics. Immediately after the conference, Heisenberg had accepted the post of Professor and the director of the Institute for Theoretical Physics at Leipzig. April 1928, Pauli had moved to a professorship position at ETH in Zurich. Pascual Jordan, who was instrumental in converting Heisenberg’s theory into matrix language, succeeded Pauli at Hamburg. Bohr and his younger collaborators always acted together against the challenges to the Copenhagen interpretation. However, Dirac was an exception. He was appointed as Lucasian Professor of Mathematics at Cambridge University in September 1932, a chair once occupied by none other than Isaac Newton. Dirac always considered himself as mathematical physicist. He was never interested in the interpretation problem of the quantum mechanics. To him, this was an unproductive preoccupation that would not lead to new equations. It is told that when he once met young Richard Feynman in a conference, after a long silence, he said “I have an equation. Do you have one too?” With its supporters in key positions in prominent institutes, quantum mechanics soon became the quantum dogma. Anybody who came against it either faced the risk of not being able to find a job or risked an upcoming promotion. Bohr’s paper finally appeared in three languages: English, German, and French. The title of the English version was The Quantum Postulate and the Recent Developments of Atomic Theory appeared on April 14, 1928. Advocates of the Copenhagen interpretation were still hoping to convince Einstein and Schrödinger. Bohr sends a copy of his paper to Schrödinger, who replied as “if you want to describe a system e.g. a mass point by specifying its [momentum] p and [position] q, then you find this description is only possible with a limited degree of accuracy.”40 He continued to argue that introduction of new concepts was needed where this limitation no longer exists. He also pointed to the difficulty to invent this conceptual scheme by saying, since—as you emphasize so impressively—the new fashioning required touches upon the deepest levels of our experience: space, time, and causality. Schrödinger’s letter had delicately expressed his reservations. Bohr replied by thanking him and restating his position. 39

Cassidy (1992), p. 253. Letter from Einstein to Arnold Sommerfeld, November 9, 1927. Mehra and Rechenberg (2000), Vol. 6, Pt. 1, p. 266. Letter from Schrödinger to Bohr, May 5, 1928.

40

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During the Easter of 1929, Pauli visited Einstein in Berlin. He found Einstein adamant about his objections to the Copenhagen interpretation. His belief in the existence of reality independent of an observer was unshaken. Shortly after Pauli’s visit, Einstein received the Planck Medal from Planck himself. During the ceremony, he restated his criticisms openly to the audience as “I admire to the highest degree the achievements of the younger generation of physics which goes by the name quantum mechanics and believe in deep level of truth of that theory, but I believe that the restriction to statistical laws will be a passing one.”41 It was clear that the debate between Einstein and Bohr was not over.

41

Pais (1982), p. 31.

Chapter 8

Sixth Solvay Conference and Titans Meet Again

During the fifth Solvay conference, Einstein had presented several thought experiments, and each was designed to demonstrate the inconsistency of the quantum mechanics. Bohr had found flaws in each one of the Einstein’s arguments and successfully defended the Copenhagen interpretation. Bohr had also designed thought experiments of his own and also found no inconsistencies. Three years later, starting on October 20, 1930, the six-day sixth Solvay Conference was commissioned to be on the topic of magnetism. With twelve and future Nobel laureates present, and Dirac, Heisenberg, Pauli, Kramers, and Sommerfeld among the 34 participants, the setting for the second round of Einstein-Bohr debate was ready and it was about to begin (Fig. 8.1).

8.1 Einstein’s Light Box Stuns Bohr Einstein was prepared this time and dropped a bombshell on unsuspecting Bohr after one of the formal sessions. Einstein started the discussion by describing a new thought experiment. With a soft voice, he asked Bohr to imagine a box full of light with a hole equipped with a shutter mechanism that could be opened at a specific time only to let a single photon out. He also asked the clock inside the box to be synchronized with the clock outside. Now he said, weigh the box and set the clock to a specific time to let only one photon out. One would now know precisely at what time the photon had left the box. At first, Bohr was not concerned, since the uncertainty principle was about conjugate variables like the position and momentum, or the energy and time. Hence, it was possible to measure any one of the complementary variables precisely. When Einstein uttered the words, now weigh the box again, Bohr was stunned. He immediately realized where Einstein was going with this. From the last of his miracle year papers, Einstein had proven that mass and energy are equivalent, thus by weighing the box one would be able to find the precise amount of the energy © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_8

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Fig. 8.1 Sixth solvay conference, October 20–26,1930; Seated L-R: Th. De Donder, P. Zeeman, P. Weiss, A. Sommerfeld, Marie Curie, P. Langevin, A. Einstein, D. Richardson, B. Cabrera, N. Bohr, W. J. De Haas; Standing L-R: E. Herzen, E. Henriot, J. Verschaffelt, Manneback, A. Cotton, J. Errera, O. Stern, A. Piccard, W. Gerlach, C. Darwin, P.A. Dirac, H. Bauer, P. Kapitza, L. Brillouin, H. A. Kramers, P. DeBye, W. Pauli, J. Dorfman, J. H. VanVleck, E. Fermi, W. Heisenberg. (Photo by Benjamin Couprie, Institut International de Physique Solvay; courtesy AIP Emilo Segrè visual archives)

that the escaping photon had carried away. Einstein had shown that it was possible to know both the time and the energy precisely simultaneously with this ingenious thought experiment. Copenhagen interpretation was in deep trouble. Bohr was shell-shocked. Including Bohr, nobody saw a solution in the horizon. Heisenberg and Pauli tried to comfort Bohr by saying “Ah well it will be all right, it will be all right.”1 All evening Bohr was in panic, he went from person to person trying to convince people that it could not be true, if Einstein was right, then it would be the end of physics, but he was not able to present any convincing arguments either.2 That evening as they were returning to Hotel Metropole, one picture said it all. An ecstatic Einstein with a victory cigar in his hand and a mocking smile on his face, probably still thinking about how he had put an end to this quantum reality nonsense, while Bohr looked defeated and as Einstein was trying to shake him off, he was trying to keep up with him and still trying to explain something (Fig. 8.2). 1 2

Pais (2000), p. 225. Rosenfeld (1968), 232.

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8.2 Bohr Could Not Believe His Eyes Bohr could not sleep that night. He reconsidered every aspect of Einstein’s light box, hoping to find a hole in his arguments. He even drew, what he called a pseudorealistic diagram to help him visualize the experiment. He even included bolts and nuts in the diagram. Since the box had to be weighed, he mounted the light box on an arm like gallows with a spring. He included a pointer and a scale to read the position of the box. For calibration, he used a weight hang at the bottom of the box. He adjusted the weight so that the scale would read zero. After the photon escaped the box was lighter, hence to reposition the pointer at zero he had to add more weight. Since he could calculate the energy lost due to the escaping photon precisely by using Einstein’s iconic formula E = mc2 , he assumed no time limit for doing this. From the arguments he had used in the previous Solvay Conference three years ago, he had to illuminate the scale so that it can be read. Now, the photons colliding with the pointer would cause an uncontrollable amount of momentum transfer to the light box. From the uncertainty relation between the momentum and position, this would imply an uncertainty in the position of the light box inversely proportional to the uncertainty in momentum. The more one tried to control the momentum transfer, the higher the uncertainty in the position of the light box would be and vice versa. At this point, it dawned onto Bohr that Einstein had ignored an important effect in his own general theory of relativity. He could not believe his eyes, Einstein was so obsessed with destroying the Copenhagen interpretation that in desperation he had forgotten his own theory. According to his general theory of relativity, clocks closer to earth surface ran slower. This new time dilation effect was extremely small, but in principle it was calculable and measurable. This effect meant that the clock inside the light box and the one outside could not remain synchronized after the photon had left the box. Unlike Einstein’s claim, uncertainty in measuring the position of the light box had made it impossible to know the exact time that the shutter had opened. That night, Bohr also showed that greater the accuracy in measuring the energy of the photon by E = mc2 , the greater the uncertainty in measuring the position of the box. Using this combination of uncertainties, Bohr also verified the energy and time uncertainty relation. The uncertainty principle, hence the Copenhagen interpretation of quantum mechanics, was saved.

8.3 Bohr Turns the Tables Completely Next morning, when Bohr came down for breakfast, he had completely turned the tables. He was now the victor, and it was Einstein who was shocked and silenced. Some still criticized Bohr for applying the uncertainty principle to macroscopic objects like the pointer, scale, and the light box without making the boundary between the classical and the quantum worlds clear. Whatever reservations some may still had, Einstein had accepted Bohr’s arguments.This would be the last time Einstein

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Fig. 8.2 L-R: Albert Einstein and Niels Bohr (Photograph by Paul Ehrenfest, courtesy AIP Emilio Segrè Visual Archives, Ehrenfest collection)

challenged quantum mechanics publicly. In September 1931, Einstein once again nominated Heisenberg and Schrödinger for the Nobel Prize. In his nomination letter, he said “In my opinion, this theory contains without doubt a piece of the ultimate truth.”3 Apparently, the two debates that Einstein had with Bohr had not changed his opinion about quantum mechanics being incomplete. Even though majority of the physics community had accepted Bohr as the winner and went on with successful applications of quantum mechanics, this debate was not going to be over.

3

Pais (1982), p. 515.

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8.4 Tumultuous Years in Europe After the sixth Solvay conference as a sign of things to come, on December 7, 1930, Sigmund Freud said “we are moving towards bad times.”4 Indeed, Europe was entering into stormy waters. In January 1933, Nazis seized power in Germany. At the time, Einstein was in America as a visiting professor at the California Institute of Technology (Caltech), which was becoming one of America’s leading centers for scientific excellence. When he returned to Belgium on March 28, 1933, he immediately resigned from the Prussian Academy of Science and dissolved all of his official ties with the German institutes. He also surrendered his German passport to the German embassy and renounced his German citizenship. In March 1933, Einstein also made it public that he would not return to Germany.5 In April 7, 1933, Nazis passed the Law for the Restoration of the Career Civil Service applying to almost two million government employees. In particular, paragraph 3 contained the infamous Aryan clause: “Civil servants not of Aryan are to retire.”6 Non-Aryan was defined as a person who had one parent or grandparent who was not Aryan. Once again, Jews were on target. In May 10, 1933, swastica-clad students and academicians marched down the main entrance of the Berlin University and burned 20,000 books, including the works of Marx, Brecht, Freud, Zola, Proust, Kafka, and Einstein. Forty thousand people watched. Similar bonfires followed throughout Germany. Since German universities were government institutions, this immediately triggered evaporation of German universities. Soon, more than thousand academicians, including 313 professors, were dismissed or resigned. Within three years, more than 1600 scholars including twenty who had been or would be awarded Nobel Prize were expelled. Some like Alfred Erich Frank in medicine, who set a milestone in diabetes melitus with Synthalin, the first oral anti-diabetic drug, and Fritz Georg Arndt, who co-discovered the Arndt-Eistert synthesis in chemistry, found a welcoming home in the University of Istanbul-Turkey. Frank, Arndt, and friends, and their students were instrumental in converting Istanbul University into a modern leading university in the world and helped build the foundations of the young Turkish Republic. While Gottingen, one of the cradles of quantum mechanics was relegated to a second-class institution. When the Nazi minister of education asked David Hilbert, one of the most influential mathematicians of his time, whether it was true that his institute suffered from the departure of Jewish scientists and their friends, Hilbert answered, “Herr Minister−No it didn’t suffer−It just does not exist anymore.”7 October 1933, Einstein returned to America to spend five months at the Institute for Advanced Study, but he would never return to Europe. In November 1933, Heisenberg received his deferred 1932 Nobel Prize, while Dirac and Schrödinger shared the 1933 prize. Dirac first wanted to refuse it because of the publicity it would cause. However, when Rutherford convinced him that not 4

Brian (1996), p. 199. Letter From Sigmund Freud to Arnold Zweig, December 7, 1930. Kumar (2014), p. 291. 6 Friedlander (1997), p. 27. 7 Jungk (1960), p. 44. 5

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accepting would cause even more publicity, he decided to accept it. Born was deeply hurt when the Swedish Academy ignored him. In response to a congratulation letter that Born sent, Heisenberg wrote “The fact that I am to receive the Nobel Prize alone, for work done in Göttingen in collaboration−you, Jordan and I−this fact depresses me and I hardly know what to write to you.”8 Born had every right to be hurt. After all, when Heisenberg had no knowledge of matrices, it was him who noticed that the strange multiplication that Heisenberg used was matrix multiplication. First Born and Jordan, and then eventually Heisenberg, had converted Heisenberg’s Helgoland work into matrix language together. Twenty years later, Born would admit to Einstein “For the last twenty years I have not been able to rid myself of a certain sense of injustice.”9 Born would finally receive the Nobel Prize in 1954 for his fundamental work in quantum mechanics and especially for his statistical interpretation of the wavefunction. After some initial difficulties with the administration at the Institute of Advanced Studies,10 in April 1934 Einstein made it public that he would stay in Princeton indefinitely. Einstein was eventually going to be happy at Princeton and would enjoy the peaceful environment that he found there for the rest of his life. During the past three years after the sixth Solvay Conference, a note worthy scientific development was the publication of von Neumann’s book The Mathematical Foundations of Quantum Mechanics (1932) in German. In this book, von Neumann had presented his famous impossibility proof, where he showed that no hidden variables theory can reproduce the predictions of quantum mechanics.This theorem certainly put the damper on the philosophical concerns over the interpretation of quantum mechanics. Majority of the scientific community used this theorem to ignore Einstein and Schrödinger’s concerns over the nature of reality and preferred to move forward with the practical successes of the quantum mechanics. The younger generation widely shared Dirac’s view that quantum mechanics explained “most of physics and all of chemistry.”11 In January 1935, Robert Oppenheimer described Princeton as a “madhouse” and Einstein as “completely cuckoo.”12 Even Einstein admitted his new role in quantum mechanics when he said “in Princeton I am considered an old fool.”13 They would soon find out how wrong they were.

8

Greenspan (2005), p. 191. Letter From Heisenberg to Born, November 25, 1933. Born (2005), p. 200. Letter from Born to Einstein, November 8, 1953. 10 Kumar (2014), pp. 297–299. 11 Bernstein (1991), p. 49. 12 Smith and Wiener (1980), p. 190. Letter From Robert Oppenheimer to Frank Oppenheimer, January 11, 1935. 13 Born (2005), p. 128. 9

Chapter 9

Einstein at Princeton

When Einstein arrived at Princeton in October 1933, he was asked what he needed (Fig. 9.1). He replied “A desk or table, a chair, paper and pencils” and then he continued “Oh yes, and a large wastebasket, so I can throw away all my mistakes.”1 At Princeton Einstein devoted most of his time to unified field theory. A theory that would unify Maxwell’s electromagnetic theory and his theory of gravitation. He was hoping that such a theory would also contain the new physics that would remove all his objections to the Copenhagen interpretation of quantum mechanics. As soon as Einstein settled in Princeton, he started working with his assistants, a Russian, Boris Podolsky, and an American, Nathan Rosen. Rosen had arrived from MIT in 1934 and Einstein had met Podolsky at Caltech in 1931, where they have collaborated on a paper. Einstein already had a new idea, and he would soon unleash his new attack on the Copenhagen interpretation of quantum mechanics. To show the inconsistency of quantum mechanics, in the fifth and the sixth Solvay conferences, Einstein had targeted the uncertainty principle by using single particle systems. In each case, using the disturbance that the act of measurement would cause on the particle and the experimental setup, Bohr had managed to find holes in Einstein’s arguments and had successfully defended the Copenhagen interpretation. In conclusion, Einstein admitted that quantum mechanics is a logically consistent theory but still insisted that it could not be the final theory that Bohr claimed.

9.1 Einstein-Podolsky-Rosen Argument Einstein’s new strategy was a new thought experiment, where it would be possible to find the state of a quantum system without disturbing it in any way. In their seminal paper, Einstein-Podolsky-Rosen (EPR) first prepared the philosophical groundwork 1

Brian (1996), p. 251.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_9

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Fig. 9.1 Robert Oppenheimer was the director of the Institute for Advanced Study at Princeton, 1947–1966. (AIP Emilio Segrè visual archives, physics today collection)

about what they mean by objective reality. According to EPR, any physical theory has to make the distinction between the objective (physical) reality, which is independent of our theories, and the physical concepts that theories are built on. These concepts are naturally expected to correspond to the objective reality. Through these concepts, we picture the physical reality to ourselves.2 In judging the success of a physical theory, EPR argued that one should answer two questions with an unequivocal “Yes”: Is the theory correct? and is the description given by the theory complete? The correctness of a theory is determined by its degree of compliance with the experimental results. As far as quantum mechanics is concerned, it appeared to be a correct theory. So far no conflict with experiments performed in laboratory had been found. However, according to EPR, this was not enough for a theory to be complete. They defined completeness as “every element of the physical reality must have a counterpart in the physical theory.”3 EPR also defined an element of reality as “If, without in anyway disturbing a system, we can predict with certainty (with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”4 For a theory to be complete, what EPR had demanded was a tall order. Every element of reality had to have a counterpart in a complete theory. If there existed one or more elements 2

Einstein et al. (1935), p. 138. References to paper reprinted in Wheeler and Zurek (1983). Einstein et al. (1935), p. 138. 4 Einstein et al. (1935), p. 138. 3

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of reality not accounted by the theory, then according to EPR that theory must be incomplete. From here, Einstein had shifted his point of attack from the internal consistency of quantum mechanics to the nature of reality. To move further along this line of thinking, EPR had to define what elements of reality that they think were not incorporated in quantum mechanics. In the EPR thought experiment, Einstein carried the argument to another level with two particles that interact briefly and then separate to some distance, which could be centimeters, kilometers, or even light years. These particles will be denoted as particle A and particle B. The position q A and the momentum p A of the particle A, similarly for the particle B, are complementary variables, hence one cannot be measured without introducing an uncertainty in the other. It was already established in quantum mechanics that the uncertainty relation for the complementary variables like q A and p A followed from their noncommutativity, that is, from the fact that they have non-vanishing commutation relations, [q A , p A ] = q A p A − p A q A = i. Therefore, either q A or p A , similarly for the particle B, could be measured precisely, but not both simultaneously. For commuting variables, on the other hand, both variables could be measured simultaneously precisely. Einstein noticed that the distance between the particles, q A − q B , and the sum of their momenta, p A + p B , are both real and conjugate variables with the all important property that they commute with each other, that is, their commutation relation vanishes: [(q A − q B ), ( p A + p B )] = 0. Now the hearth of the EPR’s argument came. Since these new conjugate variables commute, the distance between the particles and their total momentum could both be measured simultaneously precisely. The essence of the EPR argument was that since particle A and B could be light years apart, we can make our measurements on A without disturbing B in anyway. If we measure the position of A precisely, then by using the separation of the particles, we can simultaneously learn the position of B precisely. According to the EPR criterion of reality, position of B must be a part of physical reality. Similarly, if we measure the momentum of A precisely, then by using the total momentum, we can infer the precise value of the momentum of B simultaneously without performing a measurement on B. Hence, the momentum of B must also be a part of physical reality. Given their definition of physical reality, EPR argued that the position and the momentum of particle B are both elements of physical reality that must exist whether they are observed or not. EPR was striking right at the business end of the Copenhagen interpretation. At this point, it should be emphasized that the EPR experiment was not designed to measure the position and the momentum of B simultaneously precisely. They knew from the uncertainty principle that it cannot be done, but what they were saying was that if we could determine the physical reality of either the position or the momentum of the particle B, which could be light years away from A, then by a measurement we choose to perform on A without disturbing B in anyway, there must exist a spooky action at a distance between the two particles. Since relativity prohibits signals faster than the speed of light, the position and the momentum of B must have been defined all along as a part of physical reality. According to the uncertainty principle, measuring the position of a particle precisely excluded the possibility of measuring its momentum simultaneously and vice

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versa. Now EPR asked the question: Does not the particle have a momentum, or position? Copenhagen interpretation said that unless a measurement was made, momentum, or the position, does not exist. EPR had shown that whether you measure or not the particle has a definite position and a momentum as a part of physical reality. Since there was nothing in the wavefunction that tells how these quantities are defined, EPR’s conclusion was that quantum mechanics is incomplete. To prepare the paper, Podolsky was assigned to write, Rosen did the mathematics, and as Rosen recalled, Einstein “Contributed the general point of view and its implications.”5 The four page paper was published on May 15, 1935, in the American Journal Physical Review. It was entitled Can Quantum Mechanical Description of Physical Reality be Considered Complete. The answer was an unequivocal “No.” Unfortunately, the language and the arguments in the paper lacked the clarity of Einstein’s own papers, a point which Einstein would later regret.6 In a letter to Schrödinger on June 19, 1935, Einstein wrote “For reasons of language this [paper] was written by Podolsky after several discussions. Still, it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by formalism [Gelehrsamkeit].”7 The crux of the paper, which was locality rather than the definition of reality, was obscure in the EPR paper.8 For Einstein, it was unthinkable to have particle, A, effecting particle B, many light years away instantaneously by a measurement performed on A.

9.2 Reactions to EPR As soon as the EPR paper appeared, pioneers of the quantum mechanics in Europe were immediately alarmed. Pauli was furious and wrote to Heisenberg “Einstein has once again made a public statement about quantum mechanics” and continued “if a student in one of his earlier semesters had raised such objections, I would have considered him quite intelligent and promising.”9 Dirac said “Now we have to start all over again, because Einstein proved that it does not work.”10 Unlike Pauli’s condescending reaction, Dirac thought that Einstein had pointed to a fatal flaw in quantum mechanics. At first, Pauli and Heisenberg had considered writing a reply to the EPR paper, but when Bohr learned about the EPR paper from Leon Rosenfeld, who was working with him at Copenhagen, he decided to take the matter into his

5

Jammer (1985), p. 142. Baggott (2016), p. 145. 7 Fine (1986), p. 35. 8 Murdock (1987), p. 173. Letter from Einstein to Schrödinger, June 19, 1935., Baggott (2016), p. 145. 9 BCW, Vol. 7, p. 251. Letter from Pauli to Heisenberg, June 15, 1935. 10 Niels Bohr, AHQP interview, 17 November 1962. 6

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own hands. Rosenfeld later recalled the incident as the EPR “onslaught came down upon us as a bolt from the blue. Its effect on Bohr was remarkable.”11 Bohr immediately put everything on hold and started working on a reply with his colleagues. Something he thought would take him only a few days, extended to a week and then to several weeks. They tried to take the EPR arguments apart to find a weakness. Bohr had realized that this time they had their work cut out for them. It was not going to be easy to refute Einstein’s arguments. Bohr had to forget the disturbance caused by the act of measurement argument that he had used before and had to concentrate on the type of reality. As Bohr was thinking aloud, Rosenfeld remembered that he uttered the words “They do it smartly but what counts is to do it right.”12 This time Einstein’s attack was both ingenious and philosophically deep. Bohr agreed with EPR in being able to predict the results of a possible measurement on B based on the results of an actual measurement on A. On the other hand, Bohr argued that it does not mean that the momentum, or the position, are independent elements of B’s reality. Only when an actual measurement on B is performed, one can talk about B having a momentum, or a position. For Bohr, the presence of a measuring device was essential in defining EPR’s elements of reality. Bohr also accepted that there was no disturbance of any kind on B due to a measurement made on A. He argued that since A and B had interacted briefly in the past before they separated, they were intertwined as a single two-particle system. Therefore, measuring A s momentum, or position, also meant assigning B a definite momentum, or position, instantaneously. But then there appeared a mysterious action at a distance between the particles, which violated locality. Bohr failed to address this challenge, which was the essence of the EPR argument not the definition of reality. For the next six weeks, Bohr worked on nothing but on his response to the EPR paper. He first sent a summary of his response to Nature on June 29, with the title Quantum Mechanics and Physical Reality. His main response appeared as six pages in Physical Review on March 15, with the same title as the EPR paper: Can Quantum-Mechanical Description of Physical Reality be Considered Complete.13 Bohr’s answer was an assertive “Yes,” but he was not able to find an error in the EPR arguments. Instead, he summarized the Copenhagen interpretation and argued that Einstein’s evidence for the quantum mechanics being incomplete was not strong enough to make such a claim.14 Majority of the physicists, including Dirac, who eventually agreed with Bohr, did not care about the philosophical ramifications of the debate and as usual accepted Bohr as the winner and went on with the successful applications of quantum mechanics. Einstein believed that the Copenhagen interpretation of quantum mechanics was incompatible with the objective reality view. On the other hand, according to the Copenhagen interpretation, particles do not have independent reality, they acquire their properties only when they are observed. Bohr said “There is no quantum world. Rosenfeld (1967), p. 128., Rozental (1967), pp. 114 −136. Rosenfeld (1967), p. 129. Also in Wheeler and Zurek (1983), p. 142. 13 Bohr (1935). 14 Born (2005), p. 155. Letter from Einstein to Born, 3 March 1947., Baggott (2016), p. 147. 11 12

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There is only an abstract quantum mechanical description,”15 and John Wheeler summed it: no elementary phenomenon is a real phenomenon until it is an observed phenomenon.16 But what constitutes an observation was never made clear. Does it have to be a conscious observer? Or, are we living in a universe realized by somebody else’s conscious observations. As in the arguments that Bohr had used during the Solvay conferences, the line between the classical and the quantum worlds was never made clear. Sure, statistical physics tells us how atoms manifest themselves as pressure, temperature, etc. in our macroscopic world, but how does information actually propagate, or progress, from the microcosm to the classical realm are still questions that needs to be answered. As Sean Carroll, a Caltech physics professor, said about quantum mechanics in a 2019 New York Times article “What’s surprising is that physicists seem to be O.K. with not understanding the most important theory they have.”

9.3 Schrödinger and Entanglement Shortly after the EPR paper appeared in Physical Review, in a letter dated June 7, 1935, Schrödinger wrote to Einstein “I was very happy that in the paper just published in P.R. you have evidently caught dogmatic q.m. by the coat-tails.”17 He also continued: “My interpretation is that we do not have a q.m. that is consistent with relativity theory, i. e. with a finite transmission speed of all influences. We have only the analogy of the old absolute mechanics… The separation process is not at all encompassed by the orthodox scheme.”18 Schrödinger agreed with EPR about the finite signal speed for all influences, but for the two particles that interacted briefly and then separated, like Bohr, he thought that the two particles form a single two-particle system with an extended wavefunction spreading over large distances. EPR thought the particles after separation are distinct from each other. At the moment measurement is made, EPR thought that the particles are no longer represented by the single twoparticle wavefunction. Particles separating into two locally independent physical entities are sometimes called local reality, or Einstein separability. According to the Copenhagen interpretation until a measurement is performed on one of the particles, say A, and the wavefunction has collapsed, one cannot speak of them as individual particles with physical properties. The Copenhagen interpretation denies that the two particles are separable; hence, they cannot be locally real until a measurement is performed on one of them.

15

Petersen (1985), p. 305. Kumar (2014), p. 312. 17 Moore (1989), p. 304. Letter from Schrödinger to Einstein, June 7, 1935. 18 Moore (1989), p. 304. Letter from Schrödinger to Einstein, June 7, 1935. 16

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9.4 Schrödinger’s Cat During June and August 1935, Einstein and Schrödinger exchanged a series of letters. As they tried to scrutinize the Copenhagen interpretation, Einstein also tried to clear the misunderstandings that the poor language of the EPR paper had caused with additional examples. Eventually, one of these examples inspired Schrödinger his famous cat thought experiment:“A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The wavefunction of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.”19 According to Schrödinger, the cat is either dead or alive, but according to the Copenhagen interpretation, the cat is in a blurred state of both dead and alive and until the box is opened and somebody looked. In other words, it is meaningless to ask is the cat dead or alive? Then, what if nobody looks? Does not the cat know whether it is alive or dead until somebody looks inside. Does the observer has to be a conscious observer? What if another cat looks in through a small window on the box? Einstein saw this paradox as another flagrant evidence for the basic incompleteness of the quantum mechanics. A delighted Einstein wrote to Schrödinger: “We agree completely with respect to the character of the present theory. A wavefunction that contains a living and a dead cat cannot be considered to describe a real state.”20 Schrödinger’s thought experiment showed the difficulty about where to draw the line between the measuring equipment, which is a classical macroscopic object, and the microscopic system that is being observed. On the other hand, the macroscopic measuring device is also made up of microscopic objects like molecules, atoms, protons, electrons, etc., which also obey the laws of quantum mechanics. Besides, should not the observer and consciousness also be included as a part of the measurement. All this shows how complicated the measurement process is. It involves not just one collapse but a sequence of collapses in parts of a rather complicated wavefunction representing the totality of the system observed, the measuring apparatus, and the observer, where the measurement process culminates with the final collapse in the observers consciousness from the mixed state “I don’t see + I see” to a definite “I see” or to “I don’t see.” What is called the measurement problem can be averted if we assume that the measuring device is a classical object and cannot be considered as a part of the wavefunction. But this still leaves the problem of the nature of the collapse of the 19

Schrödinger (1935), p. 157. Mehra and Rechenberg (2001), Vol. 6, Pt. 2, p. 743. Letter from Einstein to Schrödinger, 4 September 1934. 4 September 1935., Baggott (2016), p. 157.

20

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wavefunction open. Schrödinger’s paradox pointed to the difficulties in understanding the collapse of the wavefunction without understanding the measurement process. That is, how information propagates from the microrealm to our classical world and vice versa. Einstein and Schrödinger both believed that the cat-in-a-box paradox is a clear indication of the incompleteness of the Copenhagen interpretation. Bohr simply side stepped the measurement problem by saying that measurement can be made but never elaborated on how. The rest of the physics community deemed such philosophical challenges as unproductive and accepted Bohr’s responses as satisfactory and moved on with the successful and practical applications of quantum mechanics. Schrödinger met Bohr briefly in London in March 1936, where Bohr in his polite way, told him that he finds it appalling and even as high treason that people like Schrödinger and Einstein want to strike a blow against quantum mechanics.21

9.5 Einstein, Bohr Meetings at Princeton Bohr visited Princeton in early February 1937. It was the first time Bohr and Einstein had to talk face to face after the EPR paper. Everybody expected them to continue their heated discussions about the completeness of quantum mechanics. Both men were cordial, but unlike expectations, they talked passed each other with nothing more than repeating their positions.22 In January 1939, Bohr returned to Princeton for four months as a visiting professor. During his entire visit, Einstein was reluctant to talk physics with Bohr. Actually, most of the time they had met only during formal occasions. About Einstein’s unwillingness to talk physics, Rosenfeld, who accompanied Bohr to Princeton, said “Bohr was profoundly unhappy about this.”23 After the war, Bohr was given a permanent non-resident status at Princeton, where he could come and stay whenever he pleased. In 1948, he stayed for almost five months between February and June. This time Einstein was willing to talk physics and the relation between them was quite friendly. In fact, Einstein had allowed Bohr to use his office. However, Einstein was still adamant about the incompleteness of quantum mechanics, which continued to frustrate Bohr as the young Dutch physicist Abraham Pais witnessed.24 indexEinstein-Bohr at Princeton There were other visits, but Bohr never succeeded in convincing Einstein to the completeness of quantum mechanics. Until he died at the age of 76 on April 18, 1955, Einstein continued to work on a theory aimed at unifying the two fundamental interactions: Maxwell’s theory of electromagnetism and his theory of gravitation. Just like Maxwell’s theory united electric and magnetic fields, Einstein was hoping that from such a unified field theory a complete quantum theory would also emerge as a by-product. Eventually, the discovery of strong and weak interactions increased 21

Kumar (2014), p. 318. Brian (1996), p. 305. 23 Fölsing (1997), p. 705. 24 Pais (1967), p. 224. 22

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Fig. 9.2 Bohr’s last blackboard. (AIP Emilio visual archives)

the number of fundamental interactions to four; thus, the problem of unifying all fundamental interactions became even a greater challenge. In 1979, Sheldon Glashow, Abdus Salam, and Steven Weinberg received the Nobel prize for unifying the electromagnetic and the weak interactions as the electroweak theory. The strong interactions joined the unification scheme with what is called the grand unified theories (GUTs). Unification of these three fundamental interactions with the gravitation, which would be called the Theory of Everything, has so far eluded all attempts of unification and is still awaiting to be discovered. Bohr died at the age 77 on November 18, 1962. He was never able to convince Einstein to the completeness of quantum mechanics. Somebody like Einstein not accepting the Copenhagen interpretation of quantum mechanics as a complete theory has always bothered Bohr. Probably, the debate never ended in Bohr’s mind. After all, it was Einstein who was not agreeing with him. In fact, the last drawing on the blackboard of his study was Einstein’s light box (Fig. 9.2).

Chapter 10

Bohm’s Hidden Variables and Bell’s Inequality

David Bohm was a Ph.D. student of Robert Oppenheimer at the University of California at Berkeley. In 1943, after Oppenheimer was appointed as the director of the Manhattan project, Bohm’s communist affiliations made him a security risk, and thus, he was not accepted into the Manhattan project to develop the atomic bomb. Four years later, when Oppenheimer took charge of the Institute of Advanced Study at Princeton, helped Bohm get an assistant professorship at the Princeton University. When it became known that he had joined the American Communist party in 1942, even though he had left after nine months, Princeton University fearing that they might loose wealthy donors did not renew his contract in June 1951. At that time, he had just finished his book, Quantum Theory, which was published in February 1951. Blacklisted by the US government, he was not able to find a faculty position in United States. Oppenheimer recommended his former student to leave the country, and that is what he did. In October 1951, Bohm left for the University of Sao Paulo at Brazil. Even though EPR had claimed that the quantum mechanics is incomplete, they did not tell how to complete it. They said “While we have thus shown that the wavefunction does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible”1 . Bohm met with Einstein at Princeton in spring 1951 and presented a copy of his book. Einstein explained his objections to the Copenhagen interpretation of quantum mechanics and possibly described the EPR paper in more clear terms than it appeared in their paper. Even though in his book Bohm was supportive of the Copenhagen interpretation, this meeting had a profound effect on him. Bohm would later say “Because I then became seriously interested in whether a deterministic and a causal extension of quantum mechanics could be found”.2 Bohm 1 2

Einstein et al. (1935). Bohm (1985), pp. 113–114.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_10

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Fig. 10.1 David Bohm (1917–1992) reading newspaper; after refusing to testify whether or not he was a member of the Communist Party before the House Un-American Activities Committee. (Library of Congress, New York World -Telegram and Sun Collection, courtesy AIP Emilio Segrè visual archives)

was now convinced to find a deterministic and a causal interpretation of quantum mechanics. The simplest way to restore these concepts and to complete quantum mechanics was through hidden variables of some sort. Encouraged by Einstein, Bohm started scrutinizing the Copenhagen interpretation and produced two papers that appeared in Physical Review in January 1952 (Fig. 10.1). In his papers, Bohm outlined a new interpretation of quantum mechanics, which reproduced the predictions of quantum mechanics without giving up a precise, rational, and objective description of individual systems at the quantum level of accuracy.3 Bohm’s theory was basically a more sophisticated version of de Broglie’s pilot wave theory, which de Broglie had decided to abandon after being severely criticized at the fifth Solvay conference in 1927. Whether it was the diffraction or the interference experiment, Bohm noticed that we can only detect whole particles. Patterns appeared only when many such particles were detected. This implied that particles are real entities that follow well-defined paths. Bohm then thought, maybe I could reconsider Schrödinger’s wavefunction (x, t) as describing an objective wave-like field that acts as a guide that real particles 3

Bohm (1952), pp. 369–382.

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follow. He tried to rework the Schrödinger equation into a form that would look like Newton’s second law of motion, which described the motion of real particles with or without forces acting on them.

10.1 Quantum Potential Bohm arrived at his theory by first writing the complex wavefunction (x, t) in terms of two new real variables, R(x, t) and S(x, t), as (x, t) = R(x, t) ei S(x,t)/ , where R is non-negative and x is the three-dimensional position vector. Substituting this into the Schrödinger equation, he obtained a pair of coupled differential equations in terms of the new variables. The first equation was the continuity, or the conservation equation for the probability density ρ(x, t), which was defined as ρ(x, t) = R(x, t)2 = |(x, t)|2 . The second equation was the modified HamiltonJacobi equation for S(x, t), which was actually a statement of the second law of Newton. The second equation differed from the classical Newton’s theory by the presence of an extra term alongside the physical potentials like the harmonic oscillator potential, the Coulomb potential, etc. Bohm called this extra term the quantum potential. Bohm then used this equation just as one uses the classical HamiltonJacobi equation to define the particle trajectories. Bohm assumed that particles are real and follow real trajectories in space. These trajectories were embedded in the wave-like field and guided by the phase function imposed by the guiding equation. Every particle in the field now possessed a definite position and momentum at every point of the trajectory determined by their respective phase function. The equation of motion depended not only on the classical potentials but also on the so called quantum potential. This becomes evident when we put the modified Hamilton-Jacobi equation into a second-order form as.4 d . (m x) = −∇ (V + Q) , dt where V represents classical potentials like the harmonic oscillator potential, Coulomb potential, etc., while Q is the quantum potential. Hence −∇ Q is the quantum force. The quantum potential is intrinsically quantum mechanical and exists even when there are no classical potentials. Take out the quantum potential, what will be left is the usual Newton’s second law. Therefore, free particles, in the classical sense, no longer follow straight lines in Bohm’s theory. One has to keep in mind that Bohm’s theory is not simply classical physics with an additional force term. Since particles

4

Dabin (2009), p. 30.

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are constrained by the guiding equation, unlike in classical mechanics, velocities are no longer independent of position. In Bohm’s theory, position and the momentum of a particle along a trajectory are well defined at all times. However, when we consider large number of particles described by the same wavefunction; in practice, we neither have access nor control of all the initial conditions, and just as in statistical mechanics, we use the classical probability density ρ(x, t) = R(x, t)2 as a practical necessity. This is in contrast to the Copenhagen interpretation, where the wavefunction (x, t) is interpreted as giving the probabilities of various possible outcomes of repeated measurements on a collection of identically prepared systems. Until a measurement is made, the system is in a so-called blurred or a mixed state, and only when a measurement is done, the wavefunction collapses to one of the possibilities, that is, to one of the eigenstates instantaneously. In Bohm’s theory, particle properties are always predetermined, and we resort to probabilities only because we are ignorant of the initial conditions. Since particles move along classical paths determined by the guiding equation, Bohm’s theory reinstated the classical concept of causality and determinism and thus eliminated the need for wavefunction collapse. The theory does not have a measurement problem, since the particles have a definite configuration at all times. Since changing the apparatus changes the guiding wave, particles are forced to adjust instantaneously. The hidden variables, which are the positions of the particles, are in this sense nonlocal; hence, action at a distance is still present in Bohm’s theory. In a quantum experiment, according to Bohm’s theory, the combined system, which is the system observed and the apparatus, we obtain the hidden variables in terms of the randomness of the configuration of the combined system in the usual quantum mechanical way with the distribution given as ρ(x, t) = R(x, t)2 = |(x, t)|2 . The guiding equation for the combined system now translates the initial configuration into the final configuration at the completion of the experiment. In this regard, it is not surprising that both theories make the same predictions. In the double-slit experiment, according to Bohm, particles have well defined trajectories that pass through one or the other slit. However, the guiding wave, which passes through both slits, guides the particles via the quantum potential to their appropriate positions on the screen. Any attempt to determine which slit the electron passed through implies a change in the apparatus, which also changes the guiding wave, hence destroys the interference pattern. In the Copenhagen interpretation, the interference pattern is still formed one point at a time. However, there is no such thing as a particle with a well-defined trajectory. It is the absence of a definite trajectory that is responsible for the interference pattern. It is almost as if the electron passing from one of the slits interferes with itself passing through the other slit to form the interference pattern. Any attempt to find which slit the electron went through would destroy the interference pattern. According to the Copenhagen interpretation, the question: “Which slit did the electron passed through?” does not make sense. Until the electrons are detected, it is as if the electrons did not exist at all. For the double

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slit experiment, Feynman said this experiment “has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mystery and peculiarities of nature one hundred percent”.5

10.2 Reactions to Bohm’s Theory Bohm followed the reactions to his theory from exile in Brazil. In a nutshell, Heisenberg said it was “superfluous, ideological superstructure,”.6,7 Pauli described it as “artificial metaphysics,”.8 Feynman was supportive9 and Oppenheimer said “juvenile deviationism”.10 Einstein who encouraged Bohm at the beginning, called it “too cheap”.11 . Apparently, Einstein was expecting something more. At the time, Einstein was working on his grand unified field theory, and he was probably hoping that a quantum theory to his liking would naturally emerge from such a theory. Bohm’s papers were largely ignored. For one thing, in rewriting Schrödinger’s equation in terms of two new real variables, R(x, t) and S(x, t), Bohm had paid a heavy price. Unlike Schrödinger’s theory, which was simple and linear, the HamiltonJacobi equation in Bohm’s theory was highly nonlinear. Besides, he needed R(x, t) for its closure. The quantum potential was even to Bohm “rather strange and arbitrary”.12 . But most importantly, twenty years before Bohm’s papers, in 1932, the legendary mathematician von Neumann had written a book, Mathematical Foundations of Quantum Mechanics, which was considered as the bible of quantum mechanics. In his book, von Neumann offered a mathematical proof that banned all hidden variable theories. Even though von Neumann had cautioned that despite its experimental success, there could still exist a small chance that quantum mechanics might be wrong, everybody interpreted his proof as no hidden variable theories can be right.13 Considering von Neumann’s word of caution and Einstein’s initial encouragement, Bohm had gone ahead to work on his theory. However, even though he was not able to point it out, Bohm was probably aware of a gap in von Neumann’s proof. This would later be spelled out by John Bell, who noticed that one of von Neumann’s assumptions was unwarranted (Fig.10.2).

5

Feynman (1967), p. 130. Bell (1987), p. 160. 7 Heisenberg (1958), p. 133. 8 Pauli (1953), pp. 33–42. 9 Baggott (2016), p. 305. 10 Baggott (2016), p. 305. 11 Born (2005), p. 192. Letter from Einstein to Born, 12 May 1952. 12 Bohm (1980), p. 80. 13 Von Neumann (1955), pp. 327–328. 6

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Fig. 10.2 John Steward Bell and his wife Mary Bell in Amherst, MA in the summer of 1990. (Photograph by Kurt Gottfried, courtesy AIP Emilio Segrè visual archives)

10.3 Bell’s Inequality Bell first encountered von Neumann’s proof in Max Born’s book, Natural Philosophy of Cause and Chance, when he was still a student. He later recalled, “I was very impressed that somebody—von Neumann—had actually proved that you couldn’t interpret quantum mechanics as some sort of statistical mechanics”.14 Von Neumann was one of the legendary mathematicians of his time. A proof with his name naturally was immediately accepted by most physicists without bothering to check. Almost everybody took it as no hidden variable theory could reproduce the same experimental predictions as the quantum mechanics. Since von Neumann’s book was in German, Bell had to rely on Born’s book as his only reference. In his book, Born had also praised von Neumann’s work for establishing the axiomatic framework of quantum mechanics. He also reaffirmed von Neumann’s theorem as “no concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one”.15 He also argued that if someday a deterministic theory could be constructed, then it must be totally different, not a modification, or a reinterpretation, of the present one. Born’s message was loud and clear: Quantum mechanics is complete; therefore, it cannot be modified. It was 1955 when von Neumann’s book appeared in English, but Bell had already read Bohm’s theory on hidden variables from his two papers. Bell’s first reaction 14 15

Bernstein (1991), p. 64. Bell (1987), p. 159.

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was that “he saw the impossible done”.16 He immediately realized that von Neumann must have been wrong.17 Like most people, he was under the impression that Bohm’s alternative to the Copenhagen interpretation of quantum Mechanics was ruled out by von Neumann’s theorem. He was appalled that Bohm’s theory was not taught in schools and ignored in textbooks. For 25 years, hidden variable theories had been outlawed by the decree of von Neumann. If a theory that would remove Einstein’s objections and restore causality and determinism, and make the same predictions as quantum mechanics, then there would be no reason left for the scientific community to ignore it. Bohm’s theory had offered an alternative, but by that time as the only correct and complete theory, the Copenhagen interpretation of quantum mechanics was so well fortified that it was hard to budge. Besides, Bohm’s theory had its own problems. Schrödinger’s theory was simple and linear, while Bohm’s theory was highly nonlinear and difficult to apply to many particle systems. In 1964, during his year-long sabbatical from CERN, Bell decided to dwell into Bohm’s theory and the EPR thought experiment. He first concentrated on the nonlocality of Bohm’s theory and wondered whether it would be possible to construct a local hidden variable theory. After many attempts, he eventually admitted “Everything I tried didn’t work”.18 Each time Bell had tried to eliminate Einstein’s spooky action at a distance, he failed. Thinking that may be it cannot be done, he decided to concentrate on another version of the EPR experiment that Bohm had introduced in 1951, which was simpler to understand. Bohm considered a zero spin particle that disintegrated into two entangled electrons, A and B, and then flied off into opposite directions until no physical interaction between them is possible. Instead of their relative separation and the total momentum of the particles, Bohm used their spins. Naturally, these electrons would have opposite spins. If one is spin up, the other would be spin down. In three dimensions, spin of the electrons could be measured along any one of the Cartesian axes-(x, y, z). When the spin of the electron A is measured along in any one of these axes, say z, it would be either spin up or down with an equal probability of 1/2. When a simultaneous spin measurement along the same axis is performed on B, there will be a perfect correlation with the spin of A. When the spin of A is up, the spin of B will be down and vice versa. According to the Copenhagen interpretation, until a measurement is performed on A, or B, neither electron has a preexisting spin in any direction. It is almost as if the electrons existed in a blurred state, a superposition, of both spin up and spin down states at the same time. When the spin of A is measured, its entangled partner B, which could be light years away, instantaneously assumed a spin opposite to that of A. This implied that the Copenhagen interpretation is non-local. Since preexisting spin states cannot be accommodated in quantum mechanics, this was what led Einstein to think that quantum mechanics must be incomplete. Einstein believed in local reality that particles cannot be influenced instantaneously by what happens elsewhere. 16

Bell (1987), p. 160. Bernstein (1991), p. 65. 18 Bernstein (1991), p. 73. 17

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A compromise between Einstein and Bohr was reached when it became clear that one cannot use this effect to communicate superluminally between two points. Unfortunately, Bohm’s clever reworking of the EPR experiment still was not enough to cast a deciding vote between Bohr’s and Einstein’s views. Both were able to account for the result of such an experiment. For many such entangled electron pairs, Bell noticed that when the detectors measuring the A and B electrons are aligned, that is, when their magnetic fields are parallel, then there is a 100 percent correlation between the spins of electrons; when one of the detectors measures spin up, then the other one measures spin down for sure. When the detectors are not perfectly aligned, the correlation is less than 100 percent. When A is measured as spin up, then some of the B electrons are spin down, but some are also spin up. In particular, when the detectors are oriented at 90◦ C to each other, then when A electrons are measured as spin up, then only half of the B electrons will be spin down. When the detectors are oriented at 180◦ C to each other, then the spins of the A and the B electrons will be completely anti-correlated. That is, when the A electrons are spin up, then all the B electrons will also be spin up. At this point, Bell had a stroke of genius. Even though the Bohm modified EPR experiment is a thought experiment, he realized that it might be possible to decide experimentally between the quantum mechanics and any local hidden variable theory by measuring the correlations of pairs of entangled electrons for a given orientation of the detectors and then repeating the measurements for different orientations. Quantum mechanics made it possible to calculate the precise value of the spin correlation for any orientation of the detectors. Independent of the nature of the local hidden variable theory, since the outcome of one detector, A, cannot affect the outcome of the other detector, B, Bell was able to calculate the limits on the degree of spin correlation between entangled electron pairs to conclude that it is possible to find measurement configurations that are compatible with any hidden variable theory but incompatible with quantum theory. Bell later described his theorem as “If the [hidden variable] extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics, it will not be local. This is what the theorem says”.19 Independent of the local hidden variable theory, Bell had shown that spin correlations generated numbers called the correlation coefficients between −2 and 2. However, for certain configurations of the detectors, quantum mechanics produced correlation coefficients outside this range, which is called the Bell’s inequality. In his 1964 paper, On the Einstein Podolsky Rosen Paradox, Bell concluded with “In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal must propagate instantaneously, so that such a theory could not be [consistent with special relativity]”.20 Based on the experimental success of quantum mechanics, where not single experiment existed that conflicted with it, and von Neumann’s theorem, the vast majority of 19 20

Bell (1975), pp. 2–6. Bell (1964), pp. 195–200.

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the physicists looked at the debate between Einstein and Bohr like Pauli, who wrote to Born in 1954 “One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about ancient question of how many angels are able to sit on the point of a needle”.21 Bell’s brilliant theorem had changed all that. He not only proved that von Neumann’s theorem was wrong, but also gave experimentalists a tool that they can use to settle this decades old debate with concrete experimental results. If Bells’s inequality is satisfied, then Einstein’s claim that quantum mechanics is incomplete would be right. However, if the inequality is violated, then Bohr would be victorious.

10.4 Bell’s Challenge and Clauser’s Acceptance A friendly rivalry has always existed between experimentalists and theorists. Rutherford, who had a condescending attitude toward theorists, has never missed a chance to mock them. He once told a friend “They play games with their symbols, but we turn out the real solid facts of nature”.22 On another occasion, when he was asked to deliver a lecture on atomic physics, he said “I can’t give a paper on that, it would only take two minutes. All I could say would be that the theoretical physicists have got their tails up and it is time that we experimentalists pulled them down again”.23 Of course, theorists also have their ways of mocking experimentalists. Friction between the theorists and the experimentalists in interdepartmental politics is not uncommon. In Munich, it almost costed Heisenberg his doctorate. During his thesis defense, when he failed to answer simple experimental physics questions like how a storage battery works, Wien the head of experimental physics wanted to fail him. After a heated argument between his thesis advisor Sommerfeld and Wien, a compromise was reached and a humiliated Heisenberg was allowed to pass by the skin of his teeth with the lowest of the three passing grades. Wien had always complained about theorists taking their laboratory responsibilities lightly. In principle, theorists should always keep an open eye on the new results in experimental science and also be aware of developments in experimental techniques. Similarly, experimentalists should have at least a basic understanding of the theory and its limitations. Sometimes, as John Bell challenged experimentalists to check his inequality, where in 1964 he wrote, “it requires little imagination to envisage the measurements involved actually being made”, 24 theorists take the initiative and recommend new experiments. At other times, as in the case of Einstein’s discovery of the general theory of relativity, minute discrepancies, like the one in the perihelion of Mercury, from the predictions of the prevailing theory that refuse to go away, inspire theorists to think outside the box and make major discoveries. 21

Born (2005), p. 218. Letter from Pauli to Born, 31 March (1954). Andrade (1964), p. 210. 23 Andrade (1964), p. 209, note 3. 24 Bell (1964), p. 199. 22

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Of course, it is one thing for a theorist to think of a thought experiment and another thing for an experimental physicist to actually do it. In 1967, a young PhD student, John Clauser, at the University of Columbia at New York, went to his Professor to inquire about the possibility of designing an experiment to test Bell’s inequality. The response he got was no different than “Are you crazy.” It was in keeping with the status of the Copenhagen interpretation of quantum mechanics. Any young physicist who dared to challenge it either faced risking not being able to find a job or risked being treated as a cuckoo. Even Einstein was not immune from this type of treatment.25 Three years later, Clauser had now settled at Berkeley at the University of California, in doing radio astronomy. When he told his boss about his interest in designing an experiment to test Bell’s inequality, he was allowed to allocate half of his time to this project. After five years, in 1969, Bell had finally attracted an experimental physicist, who was willing to devote his time to checking his inequality. By 1972, with the help of a graduate student, Stuart Freedman, Clauser was finally ready to test Bell’s inequality. They used entangled photons instead of electrons. Photon pairs were easier to produce and their optics was simpler, where their polarization played the role of spin. Photon pairs were created when excited Calcium atoms dropped to their ground state in two steps by emitting a pair of entangled photons, one green, and the other blue. The photons were then sent in opposite directions until two detectors measured their polarizations simultaneously. For the first set of experiments, the detectors were oriented at 22.5 degrees to each other. They then changed the orientation of the detectors to 67. 5 degrees and obtained a second set of data. After 200 h of measurement, they found that the level of photon correlation they measured violated Bell’s inequality.

10.5 Aspect’s Experiment Of course, for such an important result to be trusted, it had to be repeated by other teams. From 1972 to 1977, other groups conducted nine separate experiments. Only in seven of them, Bell’s inequality was violated.26 In these experiments, among other loopholes, there were concerns regarding the use of single-channel detectors that measured polarizations only in one direction. To obtain photon pairs Clauser had heated calcium atoms to raise them to an excited state, which was not a very efficient method. Obviously, more experiments were needed before a final verdict could be given. While Clauser and others were trying to improve their experiments, a French physics graduate, Alain Aspect, was fascinated by the EPR thought experiment and Bell’s inequality. At the time, Aspect was in Africa doing volunteer work and reading up on quantum mechanics. When he returned from Africa in 1974, Aspect was a 2725

Smith and Wiener (1980), p. 190. Letter from Robert Oppenheimer to Frank Oppenheimer, 11 January 1935. 26 Redhead (1987), p. 108, table 1.

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year-old graduate student. He was trying to make his dream come true in a basement laboratory at the Institute for Theoretical and Applied Optics at the University of Paris at Orsay, France. Even though he had a permanent position at the institute, he was still a graduate student trying to earn his doctorate.27 When Aspect met Bell in Geneva in 1975 and told him about his experiment, Bell’s response was “You must be a very courageous graduate student”.28,29 Bell was worried that Aspect might be damaging his career as a young physicist by working on such a difficult and risky project. However, he also added that the experiment that Aspect was proposing would be a substantial improvement on the existing results and encouraged him to pursue his goal. In the early 1980s, progress in modern optics, computers, and laser physics allowed Aspect and his collaborators to conduct the second generation of experiments on the Bell’s inequality. Like Clauser, Aspect also measured the correlation of the polarization of the photons moving in opposite directions. Their experiments were based on a highly efficient source of entangled photon pairs, produced by nonlinear laser excitations from calcium atoms. Their experiments, which used two-channel polarizers, gave an unambiguous violation of the Bell’s inequality by tens of standard deviations.30,31 When Aspect received his doctorate in 1983, Bell was one of his examiners. Bell expressed his concern over the possibility, however small, that the two detectors might be communicating with each other. To avoid this, he recommended changing the settings of the detectors during the flight of the photons so that no signal exchange would be possible.

10.6 Third Generation Experiments The third generation of tests was conducted in late 1980s at Maryland and Rochester. They used nonlinear splitting of ultraviolet photons to produce entangled photon pairs. A remarkable property of such photon sources was that two narrow beams of entangled photons can be sent through two optical fibers, thus allowing them to be measured at great distances from each other (1988).32,33 The first test was over four kilometers at Malvern (1998) and the other was over tens of kilometers in Geneva (1998).34,35,36

27

Aczel (2003), p. 186. Aczel (2003), p. 186. 29 Aczel (2003), p. 186. 30 Aspect (1999). 31 Aspect et al. (1982). 32 Shih et al. (1988). 33 Ou et al. (1988). 34 Baggott (2016), p. 325. 35 Tapster et al. (1994). 36 Tittel et al. (1998). 28

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Motivated by Bell’s comment, experimentalists at Innsbruck in 1998 decided to eliminate the loophole in Aspect’s experiment. They used the same method as the Maryland and Rochester with a separation of 400 m between their detectors, and with 1.3 μs of random settings of the polarizer. They obtained an unquestionable violation of the Bell’s inequality.37 Over these years, various loopholes have been noticed. Some were based on the imperfections of the equipment used, others like the locality loophole were because the polarization detectors were set before the entangled photon pairs were emitted. Another loophole called the efficiency loophole was because only a small fraction of the photon pairs produced were observed at the detectors. All these loopholes have been identified and closed by various groups. Even though it is not possible to claim that all loopholes have been closed, most physicists now accept that Bell’s inequality is violated.38 Bell’s inequality was based on two assumptions. The first one was that there exists an observer independent reality. This implied particles have well-defined properties like spin polarization, etc. before the measurement. Second, locality is preserved. In other words, there is no action at a distance. Aspect’s and finally the Innsbruck results indicated that one of these assumptions cannot be true. Bell was willing to give up locality. He said, “One wants to be able to take realistic view of the world, to talk about the world as if it is really there, even when it is not being observed”.39 When Bell died at the age of 62 in 1990, he was convinced that “quantum theory is only a temporary expedient that would eventually be replaced by a better theory”.40 Einstein would not have objected to the existence of observer independent reality, but the violation of Bell’s inequality had shown that Einstein’s local reality is not true. In 1985, Bell commented on Aspect’s experiment as “It is a very important experiment, and perhaps it marks the point where one should stop and think for a time, but I certainly hope it is not the end. I think that the probing of what quantum mechanics means must continue, and in fact it will continue, whether we agree or not that it is worth while, because many people are sufficiently fascinated and perturbed by this that it will go on”.41

10.7 Aftermath of Bell’s Theorem The detection of the violation of Bell’s inequality showed that no local hidden variable theory with the assumption of objective reality can reproduce the predictions of quantum mechanics. In 2003, Anthony Leggett published a new inequality that if violated excluded certain types of non-local hidden variable theories. In April 2007, Markus Aspelmeyer and Anton Zeilinger announced their detection of the 37

Weihs et al. (1998) Hensen et al. (2015). 39 Davies and Brown (1986), p. 50. 40 Davies and Brown (1986), p. 51. 41 Baggott (2016), p. 327. Also, John Bell in Davies and Brown (eds.) (1986), p. 52. 38

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Fig. 10.3 Richard Feynman, 1918–1988. 1965 Nobel Prize in Physics won for fundamental work in quantum electrodynamics with deep-plowing consequences for the physics of elementary particles. (Harvey of Pasadena, courtesy AIP Emilio Segrè Visual Archives)

violation of Leggett’s inequality in Nature, which implied that objective reality and certain types of non-local hidden variable theories are not compatible with quantum mechanics. But it still did not rule out all possible non-local hidden variable theories. Alain Apect, John F. Clauser, and Anton Zeilinger were awarded the 2022 Nobel Prize in Physics for their work on demonstrating the violation of Bell’s inequality, which proved to be at the foundation of quantum information science. Einstein had never proposed a hidden variable theory, even though he had encouraged such attempts up to 1940s. By the beginning of 1950s, he was no longer heartening attempts to develop hidden variable theories. In a letter dated November 10, 1954, he wrote to Aron Kupperman “it is not possible to get rid of the statistical character of the present quantum theory by merely adding something to the latter, without changing the fundamental concepts about the whole structure”.42 Einstein was obviously aiming for something fundamentally different. He was hoping that the theory he had devoted the last 25 years of his life, the grand unified field theory, would not only unite general theory of relativity and Maxwell’s electromagnetism but would also yield the quantum mechanics that would be complete. However, the discovery of weak and strong interactions had already broadened the scope of such a unified theory dramatically. In the meantime, Bohr’s effect on the new generation was growing, while Einstein’s was weakening. Einstein was viewed by Bohr’s followers as an old man who could not keep up with the new developments. Backed by the enor42

Fine (1986), p. 57. Letter from Einstein to Aron Kupperman 10 November 1954.

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mous successes of quantum mechanics, which kept accumulating, most physicists left such philosophical concerns behind as fruitless endeavors and concentrated on the applications of quantum mechanics that could be tested in laboratory. However, ten years after Einstein’s death, Feynman said “I think I can safely say that nobody understands quantum mechanics (Fig. 10.3)”.43 Even though this is still true, majority of the physicists are OK with it as long as they know how to calculate stuff that can be checked in laboratory. In David Mermin’s words they have learned to “shut up and calculate”.44 The difficulty in understanding quantum mechanics is not that it possesses plenty of counterintuitive results or that it uses probabilities—classical physics also has plenty of counterintuitive examples—but that it goes against our firm belief that the world exists independent of the presence of an observer. Bell’s inequality has dragged this argument from the realm of endless philosophical bickering into the laboratory. When Bell said “I think that the probing of what quantum mechanics means must continue, and in fact, it will continue, whether we agree or not that it is worthwhile, because many people are sufficiently fascinated and perturbed by this that it will go on”,45 he was right. Einstein’s EPR thought experiment had not only revived the interest in problems like the completeness of quantum mechanics and the nature of reality, but it had also proven that to dwell on such problems is not a waste of time. Einstein was not able to find his grand unified theory, but the EPR paradox had ushered in the era of quantum information with ground breaking applications to quantum computers, spy-proof communication, teleportation, etc. Einstein’s persistence in searching for the true meaning of quantum mechanics had started all this. Not bad for an old man! It is now clear that discovering the theory that would replace quantum mechanics is much more difficult than the discovery of quantum mechanics itself.46 As Murray Gell-Mann said, part of the problem is “Niels Bohr brain-washed a whole generation of physicists into believing that the problem had been solved”.47 Unlike the prevalent opinion that made Bohr an icon,48 even in Bohr’s mind these arguments were probably not settled entirely. After all, it was Einstein that he failed to persuade. Bohr probably convinced himself, just like he had convinced everybody else that such questions are not to be asked. When Bohr died in 1962, they found Einstein’s light box drawn on his blackboard. Apparently, the debate in the sixth Solvay Conference was still ongoing in his mind. The Copenhagen interpretation requires an outside observer, which brings a host of questions to mind—Does the observer have to be conscious? In the Schrödinger’s cat thought experiment, will the poor cat remain in the ghostly state of dead and alive at the same time until somebody observes it? Who observed the observer? Or, does not the moon exist? if you are not looking at it. One can go on and on. It is the 43

Feynman (1965), p. 129. For Mermin’s comment see Ball (2013). Feynman (1965), p. 129. For Mermin’s comment see Ball (2013). 45 Baggott (2016), p. 327. Also, John Bell in Davies and Brown (eds.) (1986), p. 52. 46 Pais (2000), p. 55. 47 Gell-Mann (1979), p. 29. 48 Blaedel (1988), p. 11. 44

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so-called measurement problem. It is the fact that Bohr has never clearly drawn the line where the classical world stops and where the quantum world begins that allows us to ask confusing questions like these. The unsolved measurement problem and Bohr’s failure to draw the line where the quantum realm starts and where the classical world ends have also induced others to think about these issues. Nobel Prize laureate Gerard ’t Hooft has said that “A theory that yields may be as an answer, should be recognized as an inaccurate theory”.49 When the British physicist and mathematician Roger Penrose was asked the question “could Einstein be profoundly wrong as Bohr and his followers proclaimed?” His answer was “I do not believe so, I would, myself, side strongly with Einstein in his belief in a submicroscopic reality, and with his conviction that present-day quantum mechanics is fundamentally incomplete”.50 A leading experimentalist, Nicolas Gisin, working on entanglement said “[I] have no problem thinking that quantum theory is incomplete”.51 Bell who died in October 1990 at the age of 62 was convinced that “quantum theory is only a temporary expedient that would eventually be replaced by a better theory”.52 It is at least clear that the better theory will not be just a reinterpretation of the present theory, but it will be fundamentally different from any of the existing theories. It would not be surprising if gravity still played the central role.

10.8 Everett and Many-Worlds Interpretation In 1957, Hugh Everett III, a graduate student of John Wheeler at Princeton, in his thesis: On the Foundations of Quantum Mechanics, offered an alternative, where he demonstrated that it is possible for each and every possible outcome of an experiment to be realized in a different universe. According to Everett in the Schrödinger’s cat experiment two real universes exist. The one in which the cat is alive and the other, in which the cat is dead. This eliminated the need for an observer and the collapse of the wavefunction in the Copenhagen interpretation. Everett’s many-worlds interpretation is different from the multiverse models, where multiple universes are created by separate big bangs. In Everett’s interpretation, every time the universe encounters a point where there are multiple possibilities for the outcome, the universe splits into multiple universes to accommodate all possibilities. Everett’s view implied that the entire universe is described by a huge wavefunction representing a superposition of all the constituent particles. He called this in his thesis the universal wavefunction. As this wavefunction evolves continuously, there is no collapse, each term in the superposition corresponds to a possible reality that continues to branch out at each observation point. In this scenario, all possibilities 49

Buchanan (2007), p. 37. Stachel (1998), p. xiii. 51 Buchanan (2007), p. 38. 52 Davies and Brown (1986), p. 51. 50

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are accommodated in different (parallel) universes. Of course, an individual observer lives only in one of the possibilities (universes). Everett wrote “Thus with each succeeding observation (or interaction), the observer state branches into a number of different states. Each branch represents a different outcome of the measurement and the corresponding eigenstate for the [superposition]. All branches exist simultaneously in the superposition after a given sequence of observation”.53 Even though Everett himself used the word splitting, It is not that new universes are created from one at each measurement, but they are just separated. According to Everett, at each point of interaction, or measurement, or decision, the universe splits into separate universes, where in each universe one of the possibilities is realized. I used the term decision on purpose. Whether it being the electron’s spin that turns out to be up or down in an experiment, or you choosing the right or the left road when you arrive at a junction, where the road bifurcates. In Everett’s interpretation, in one universe, the electron is spin up, and in the other, it is spin down. Similarly, in one universe, you follow the road on the left, while an identical you in another universe follows the road on the right. Before the fork, there is only you, but after the fork, you and your replica begin to have different lives in different universes. In the other universe, not just you but everything else, people, stars, galaxies, etc. will have their replicas. You cannot interfere or interact with your other self in the other universe. No matter how trivial the situation is and no matter how inconsequential you think the decision you have made is, you will have different futures with your other self who chose the other alternative. In terms of the Schrödinger’s cat, the cat is never both dead and alive simultaneously. It is always dead in one universe and alive in another. This view gets really extravagant if we take any interaction between any quantum entity anywhere in the universe and any act of decision as measurement, then “Every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies.”54 said Bryce deWitt. Of course, there are also problems with the many-worlds interpretation.55 One of them is how the split actually occurs. For this the theory of decoherence is used, where the split does not occur instantaneously. It simply evolves through decoherence and continues till the decoherence is complete when all possibility of interference between the universes is gone. Decoherence is basically the problem of microscopic quantum events giving rise to a macroscopic classical behavior through interactions with the environment. Every minute change caused by quantum measurement causing a proliferation of worlds was another source of objection. Of course, a serious objection was the notion of self. In what sense does my replicas in the other universes represent me? Wheeler initially supported Everett’s interpretation. In fact, in May 1956, he traveled to Copenhagen to get at least a partial support, but he failed.56 Even-

53

Everett III (1957). Ball (2019), p. 292. 55 Ball (2018), p. 367. 56 Olival (2005). 54

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tually, Wheeler rejected it arguing that it carried too much “metaphysical baggage”.57 Many others dismissed it on similar grounds. Everett’s interpretation also attracted interest among science fiction writers like Jorge Luis Borges in his book The Garden of Forking Paths and movies like the Sliding Doors. Since Everett’s theory required no changes in the mathematical structure of quantum mechanics, in the observer’s universe it predicted the same experimental outcomes. In this sense, its prediction of other worlds is not the type of prediction that scientific theories are expected to make. It is a deduction from the assumption that other possibilities are real too. As Max Tegmark poetically said “I feel a strong kinship with parallel Maxes, even though I never get to meet them. They share my values, my feelings, my memories—they’re closer to me than brothers”.58 At first, it may be appealing to think that our exact replica is experiencing things that you are missing in this universe, but before feeling kinship with your replicas, keep in mind that some of those experiences will not only be horrible, but also you would be doing some terrible things in those universes. In 1984, Robert Griffiths introduced another interpretation of quantum mechanics based on consistent histories. In this approach, there is no right way to look at a quantum event. Instead, many possible consistent histories are equally valid. One simply chooses the histories that are consistent with the experiment performed. In this regard, some of the possible histories are particle histories, yielding which-way paths, while others are wave histories yielding interference effects. In 1990, GellMann and Hartle introduced decoherent histories, which aside from minor differences same as the Griffiths’ consistent histories approach. In 1994, Gell-Mann wrote “We believe Everett’s work to be useful and important, but we believe that there is much more to be done. In some cases too, his choice of vocabulary and that of subsequent commentators on his work have created confusion. For example, his interpretation is often described in terms of many-worlds, whereas we believe that many alternative histories of the universe is what is really meant.” He also continued as “the manyworlds are described as being all equally real, whereas we believe it is less confusing to speak of many histories, all treated alike by the theory except for their different probabilities”.59 Everett’s many-worlds interpretation, as well as Griffiths’ consistent histories or Gell-Mann and Hartle’s decoherent histories are basically the standard quantum mechanics presented in a coherent way. In this regard, they are not the fundamentally different theory that Einstein was hoping.

57

Baggott (2016), p. 401. Ball (2019), p. 297. 59 Gell-Mann (1994), p. 138. 58

Chapter 11

The Gist of Quantum Mechanics

The upshot of the Copenhagen interpretation is that the quantum reality is dictated by what the experiment measures, and there is no other reality beyond that. In the famous Bohr-Einstein debates, even though Bohr was declared as the winner by the majority of physicists, there were also question marks about Bohr’s arguments. He had used macroscopic measuring devices with springs, scales, and pointers that obeyed classical laws, while the system being measured obeyed the laws of quantum mechanics. Even though the equipment is also made up of atoms, Bohr was criticized by some for never making the line between the classical and the quantum worlds clear. According to Bohr, the physical properties of a quantum system do not exist until they are observed and the wavefunction has collapsed. Prior to observation, the system exists in a ghost-like state as the superposition of all possible outcomes with amplitudes ci ,where the probability of the wavefunction collapsing to the ith state is given as |ci |2 . Until the wavefunction collapses, nobody can certainly predict to which state the wavefunction will collapse. One could only predict probabilities. After Born’s probabilistic interpretation of the wavefunction, in 1926, Einstein responded to a letter from Born as “God does not play dice.” This was usually interpreted as Einstein being after classical determinism. In 1954, Pauli spent two months at Princeton, where he had a better chance of understanding what actually Einstein’s objection to the Copenhagen interpretation was. Pauli realized that Einstein’s objections went far deeper than quantum mechanics being expressed in terms of probabilities and wrote to Born “Einstein does not consider the concept of determinism to be as fundamental as it is frequently held to be.”1 . Pauli continued: “Einstein’s point of departure is realistic rather than deterministic.”2 What Pauli meant by “realistic” was that Einstein believed that particles have preexisting properties prior to any act of measurement. In other words, Einstein believed in the existence of an observer1 2

Born (2005), p. 216. Letter from Pauli to Born, 31 March 1954. Born (2005), p. 216. Letter from Pauli to Born, 31 March 1954.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_11

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independent reality. However, for Einstein, reality also had to be local with no room for action at a distance. Bell’s theorem could not decide about the completeness of the quantum theory, but the violation of Bell’s inequality indicated that if we stick with the observer-independent reality, then we cannot have local hidden variable theories that reproduce the same results with the quantum mechanics. That is, any hidden variable theory that reproduces the predictions of quantum mechanics had to be non-local. Einstein would have agreed with the observer-independent reality part, but for someone who discovered relativity, which had passed all experimental tests, it would be difficult to accept non-locality. However, Einstein might have accepted the partial solution that the no signaling theorem offered. Schrödinger saw that the two particles forming a joint wavefunction, where he used the term entanglement, is not unique to the EPR case. In a subsequent paper published later that year, he said “Any entanglement of predictions that takes place can obviously only go back to the fact that the two bodies at some earlier time formed in a true sense one system, that is were interacting, and have left behind traces on each other.”3 Again, in the same paper he continued; “If two separated bodies, each by itself known maximally, enter a situation in which they influence each other, and separate again, then there occurs regularly that which I have just called entanglement of our knowledge of the two bodies.”4 Schrödinger described entanglement as the gist of quantum mechanics, displaying its difference from classical physics in the most unequivocal way. The two entangled particles have to be seen as a whole. If one concentrates on each particle separately, as EPR did, one looses the significance of entanglement. For almost fifty years, the EPR paper and entanglement were viewed as a strange effect that Einstein used to demonstrate his philosophical concerns over the completeness of quantum mechanics. However, starting with the last decade of twentieth century, EPR and entanglement ushered in the new era of quantum information that extended practical applications of quantum mechanics to areas like spy-proof communications, teleportation, quantum dense coding, and quantum computers, etc.5

11.1 Entanglement and No Signaling Theorem Let us now consider a pair of entangled electrons produced in a common process and then sent in opposite directions to observers A and B. Electrons are produced such that if the spin of one of the electrons is up, ↑, then the other must be down, ↓, and vice versa. Electrons are also produced such that the probability of an electron’s spin being up or down is 1/2. A measurement by one of the observers, say A, collapses the wavefunction to either spin up or down with the probability 1/2. This removes all uncertainty in any measurement that B will make on the second electron. Note 3

Schrödinger (1935), p. 161. Schrödinger (1935), p. 161. 5 Bayin (2018), Chap. 19. 4

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that separation of A and B could be anything. This immediately reminds us action at a distance and the possibility of superluminal communication via quantum systems. As soon as A (usually called Alice) makes a measurement on her particle, spin of the electron at the location of B (usually called Bob) adjusts itself instantaneously. It would be a blatant violation of special relativity if Alice could communicate with Bob by manipulating the spin of her particle. This would also be an open violation of locality−the main objection of the EPR paper. However, if we notice that Bob has to make an independent measurement on his particle, and also the fact that he still has to wait for Alice to send him the spin of her electron, which can only be done by classical means at subluminal speeds, we realize that even though the wavefunction collapses at an instant, it cannot be used to send superluminal signals. We now scrutinize this problem further. Alice and Bob share an entangled pair of electrons.6 Alice performs a measurement on her electron. She has a 50/50 chance of finding its spin up or down. Let us assume that she found spin up. Instantaneously, Bob’s electron assumes the spin down state. However, Bob does not know this until he has performed an independent measurement on his electron. Bob has 50/50 chance of seeing either spin, but Alice knows that the wavefunction has collapsed and that for sure Bob will see spin down. Bob conducts his measurement and indeed sees spin down. But to him this is normal, he has just seen one of the possibilities. Now Alice calls Bob and tells him that he must have seen spin down. She could even call Bob before he makes his measurement and tell him that he will see spin down. In either case, it would be hard for Alice to impress Bob. He would think that after all, Alice has 50/50 chance of guessing the right answer anyway. They now share a collection of identically prepared entangled electrons. Alice measures her electrons one by one and calls Bob and tells him that she observed the sequence ↑ ↓ ↓ ↑ ↓ ↓ ↑ ··· and that he should observe the following sequence, where the direction of the spins is reversed: ↓ ↑ ↑ ↓ ↑ ↑ ↓ ···. When Bob measures his electrons, he is now impressed by the astounding accuracy of Alice’s prediction. Even if this experiment is repeated with Alice calling Bob before he had conducted his measurements, Alice would still be able to predict Bob’s results with stunning accuracy. Quantum mechanics says that the wavefunction collapses instantaneously no matter how far apart Alice and Bob are. However, this still cannot be used to communicate superluminally. First of all, in order to communicate they have to agree on a code. Since Alice does not know what sequence of spins she will get until she performs measurements on her set of electrons, they cannot do this beforehand. Once she does measure her set of particles, she is certain of what Bob will observe. She can now embed the message into the sequence that she has observed by some kind of mapping. For Bob to be able to read the Alice’s message, Alice has 6

Bayin (2018), Chap. 19.

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to send him that mapping, which can only be done through classical channels. Even if somebody intercepts Alice’s message, it will be useless without the sequence that Bob has. Hence, Alice and Bob can establish secure, spy-proof, communication through entangled states. One of the main technical challenges in quantum information is decoherence, which is the destruction of the entangled states by interactions with the environment. It is for this reason that internal states like spin or stable energy states of atoms are preferred to construct entangled pairs of particles, which are less susceptible to external influences by gravitational and electromagnetic interactions. In the mathematical theory of information, we use bit as the unit of information, where 1 bit is the information content of the answer to a binary question like yes or no, right or left, up or down, etc.?7

11.2 Single-Particle Systems and Quantum Information To observe the profound differences between the classical and the quantum systems, we start with the following experimental setup, where the light source S emits a coherent monochromatic light beam with the intensity I Fig. (11.1): The beam impinges on a beam splitter B, where it separates into two beams as transmitted, T, and reflected, R, with equal intensities I /2. Aside from a decrease in intensity, the transmitted beam on the left goes through the beam splitter unaffected. However, the reflected beam on the right suffers a phase change of π/2 with respect to the transmitted beam. The detector on the left, D1 , receives the transmitted, while the detector on the right, D0 , receives the reflected beam. We now reduce the intensity of the beam so that we are sending one photon at a time to the beam splitter, which diverts these photons to the right or the left detectors with the equal probability of 1/2. The detectors respond equally to each photon they receive. For an experiment repeated many times, half of the time D1 and the other Fig. 11.1 Quantum pinball machine. T stands for the transmitted and R for the reflected beam

7

Bayin (2018), Chap. 19.

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half of the time D0 will respond. So far, everything is like a classical pinball machine, where we have balls instead of photons, and instead of the beam splitter, we have a pin that diverts the balls with equal probability into the right or the left bins. However, the differences begin to appear when we ask the question: Which way did the photon, or the ball, go? In the case of the ball, it has two possibilities; it will either go through the left or the right paths with equal probability. The source of randomness is the pin, which diverts the ball into the right or the left bin with equal probability. Once the ball clears the pin, it has a definite path and ends up either in the left or the right bin. The observer still does not know it yet. But whether the observer checks the bins or not has no effect on the result. For the photon, we write |0  and |1  as the eigenstates of the which-way operator, where |0  corresponds to the photon following the right path and |1  corresponds to the photon following the left path. When the detectors are turned off, or when there are no detectors, the photon is in the mixed, superposed state: | = c0 |0  + c1 |1  ,

(11.1)

where the amplitudes c0 and c1 satisfy the normalization condition |c0 |2 + |c1 |2 = 1. In this state, the photon is following neither the right path, |0  , nor the left path, |1  , but it is in both simultaneously. To find which way the photon goes through, we turn on the detectors. Now, one of the detectors responds and the wavefunction | collapses to either |0  or |1  , with the probabilities |c0 |2 or |c1 |2 , respectively. Once the wavefunction has collapsed, the photon is in a pure state: |0  or |1 . During this process, we have gained 1b of information about the system. But the price we have paid is that we have destroyed the initial state function Eq. (11.1) irreversibly. That is, we can no longer construct the initial state from the collapsed state | = |0  or | = |1  . By repeating the experiment many times under identical conditions, all we can gain is the statistical information about the initial wavefunction. That is, the probabilities of the initial wavefunction collapsing to either |0  or |1 , which are given as p0 = |c0 |2 and p1 = |c1 |2 .

(11.2)

For a symmetric beam splitter, the probabilities are equal, p0 = p1 = 1/2, hence, the wavefunction can be written in any one of the following forms: | =

1 1 [|0  + |1 ] or | = [|0  − |1 ] . 2 2

(11.3)

In the case of a pinball machine, the ball is always in a pure state. That is, it is either in state |0  or in |1  . The ball is following either the left or the right path. It has nothing to do with the presence of an observer or a measuring device, or somebody checking or not checking the bins. Observation, or measurement, in

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classical physics never has the same dramatic effect on the state of a system as it has in quantum mechanics. In the next section, we show how all this can be demonstrated in laboratory by the Mach-Zehnder interferometer.8

11.3 Mach-Zehnder Interferometer In the Mach-Zehnder interferometer Fig. (11.2), after going through the first beam splitter B1 , the transmitted and the reflected beams are reflected at the mirrors M L and M R , respectively. They then go through a second beam splitter B2 , and they eventually get picked up by the detectors D1 and D0 . The transmitted and the reflected beams of the first beam splitter, each with intensity I /2, are referred to as the left, L, and the right, R, beams, respectively. The light source S produces a coherent monochromatic beam with intensity I . The transmitted waves suffer no phase difference, while the reflected waves lead the incident wave by a phase difference of π/2. The transmitted wave at B1 follows the left path and after getting reflected at M L , splits again at the second beam splitter B2 into its reflected and transmitted parts. These are joined by the parts of the wave that follows the right path, which gets reflected at B1 and gets reflected again at M R . Finally, the left beam reflected at B2 meets the right beam transmitted at B2 . Since both waves suffered two reflections, they are in phase and interfere constructively to shine on D1 with the intensity I. The part of the left beam transmitted at B2 is joined by the part of the right beam that is reflected at B2 . Since the beam that followed the right path has suffered three reflections, while the beam that followed the left path has suffered only one reflection, they are out of phase by π, hence interfere destructively to produce zero intensity at D0 . In conclusion, D1 will get the full original beam with the intensity I , while D0 will get nothing. We are assuming that the legs of the interferometer have equal optical length. This result is completely consistent with the macroscopic result that all photons are detected by D1 , while D0 detects no-photons. As we have seen, this can be understood easily in terms of the interference of electromagnetic waves. We now dial down the intensity so that we are sending one photon at a time to B1 . However, in terms of individual photons, we find ourselves forced to accept the view that the photon has followed both trajectories to interfere with itself to produce a null response at D0 and a sure response at D1 . As far as the information theory is concerned, we already know the answer. The detector D1 responds for sure, but we know nothing about which path the photon has taken to get there. It is not like the photon has split into two. No one has seen a split photon. It is not as if half of the photon went through the left and the other half went through the right path. To learn which path the photon has taken, we remove the second beam splitter B2 . Now, one of the detectors, either D0 or D1 , responds and we learn which path the photon has taken. We gain 1b of information in finding that out. In the Mach-Zehnder interferometer, we know exactly which detector responds, that is, D1 , but we have no 8

Bayin (2018), Chap. 19.

11.4 Mach-Zehnder Interferometer with a Channel-Blocker

167

Fig. 11.2 Mach-Zender interferometer

knowledge of how the photon got there. There is a region of irremovable uncertainty about where the photon is in our device. There is no classical analog of this. If we do the same experiment with the pinball machine, we see that half of the time the ball is in bin 1, and the other half of the time it is in bin 2 Fig. (11.3, left). This follows from the fact that the events corresponding to the ball following the left path and the right path to reach bin 2 are mutually exclusive with their respective probability of 1/4 Fig. (11.3, right). If one occurs, the other one does not. In this regard, their joint probability is the sum: 41 + 14 = 21 . A similar argument can be given for the ball reaching bin 1. No matter how many times the experiment is repeated, we will never see a case where the ball that followed the left path interfered or in superposition with itself that followed the right path. This is the strange position that we always find ourselves in when we try to understand quantum phenomena in terms of our intuition shaped by our everyday experiences.

11.4 Mach-Zehnder Interferometer with a Channel-Blocker We now modify our thought experiment and introduce a channel-blocker to the right path of the Mach-Zehnder interferometer that could be turned on or off Fig. (11.4). When the blocker is on, if the photon gets reflected at the beam splitter B1 and goes to the right path, then it will be absorbed by the channel-blocker and no-photon will

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Fig. 11.3 Mach-Zehnder experiment with the pinball machine. On the right we show the two mutually exclusive paths for the balls seen in bin 2

be counted in either D1 or D0 . If the photon gets transmitted at B1 and goes to the left path, then either D1 or D0 will register a photon, where D1 will get the reflected photon at B2 and D0 will register the one transmitted at B2 . If the channel-blocker is off, then we have interference, and only D1 registers a photon and no-photons will be registered by D0 . But this time, we have no idea about which path the photon has taken. Now, let us assume that we have a channel-blocker that works sporadically Fig. (11.4). Can we tell when the channel-blocker was on or off? First of all, if no photon is registered in either D0 or D1 , we can be sure that it was on. Since the photon that followed the right path would be absorbed before reaching B2 and the photon that followed the left path would have registered in one of the detectors. If a photon is registered in D0 , then we can conclude that the channel-blocker was on. Because, if the channel-blocker was off, then the photon that followed the right path would show wave-like behavior and interfere destructively with itself that followed the left path to produce a no response at D0 . How on earth then did the photon that followed the left path, the one that got transmitted at B1 , knew that the right path was blocked without ever coming into contact with it and decided to behave like a particle and followed the left path straight to D0 ? Finally, if D1 responds, then we cannot tell whether the channel-blocker was on or off, since it could either be the photon that took the left path and got reflected at B2 , or the channel-blocker was off, and the photon that followed the right path interfered constructively with itself that followed the left path and got reflected at B2 to register in D1 .

11.4 Mach-Zehnder Interferometer with a Channel-Blocker

169

Fig. 11.4 Mach-Zehnder interferometer with a channel-blocker. The glass plate is inserted to equilize the optical paths

Note that when we do not know which path the photon is taking, it behaves like a wave to interfere with itself, and when we know the path it takes, it behaves like a particle. How it behaves depends on the experimental setup. It is never possible to construct an experiment that displays the wave and the particle behavior simultaneously. This is described by Bohr in terms of the complementarity principle, which is similar to the uncertainty principle that makes simultaneous measurement of the position and momentum of a particle impossible. In addition to all this, there is another mysterious effect of quantum mechanics. To confuse photons about whether they should behave like particles or waves, let us activate the channel-blocker after the photon has passed the first beam splitter B1 . By making the legs of the interferometer sufficiently large, we can introduce a significant delay to the choice that the photon has to make about behaving like a particle or a wave. The experiment still works. Not only the photon going through the left path knows that there is a channel-blocker active on the right side, but it also appears to know even before it is activated. Nowadays, such delayed choice experiments can be done. But wait! So far we have talked about photons, and particles, as if they have a mind and as if they know something. A photon does not have a brain to think and to act accordingly. Also, a photon does not split into two: One going one way and the other going the other way, and then interfere with each other to produce a sure response in D1 and a no response in D0 . It is when we try to understand quantum phenomena in terms of our classical ways of thinking and intuition that produces these strange paradoxical interpretations. We now go back a little, where we shine light with intensity I onto the first beam splitter B1 . In this case, there are zillions of photons in the beam. The fact that the

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Fig. 11.5 Feynman’s double slit experiment: When the light source is turned on, we know exactly which slit the electron passed through, hence there is no interference pattern (B). When we turn off the light source, the interference pattern appears (A)

result, no response in D0 and a sure response in D1 , is 100 percent the same with the case when we reduced the intensity to a level where only one photon is sent at a time, is a flagrant demonstration of the continuity of physical behavior as we transit from the classical macroscopic world to the quantum domain. This is incorporated into quantum mechanics in terms of the correspondence principle.

11.5 Feynman’s Double Slit Thought Experiment In his 1964 Messenger lectures at Cornell,9 Feynman introduced a double slit thought experiment that depicts the mystery of quantum mechanics, where he used electrons instead of photons. To be able to see which slit the electrons pass through one needs to shine light on the electrons Fig. (11.5). So Feynman also put a light source behind the slits. He argued that each time a flash of light appears behind one of the slits, it would reveal the slit that the electron went through. When the light source is turned off, one cannot tell which slit the electrons are going through; hence, electrons behave like waves and produce an interference pattern on the second screen (A). Of course, in this case, many electrons are passing through the first screen to produce the interference pattern. Next, Feynman arranged the electron gun so that it sends one electron at a time. We still do not know which slit the electron passed through, but it leaves a single point on the second screen−not an interference pattern, as one may expect from a wave. It is not like half of the electron goes through slit 1 and the other half goes through slit 2 to interfere with itself to produce an interference pattern. Electrons behave as a whole. However, as more electrons are sent, one sees that their distribution on the second screen slowly begin to produce an interference pattern (A), rather than the distribution in (B) expected from particles. 9

Feynman (1967), p. 130.

11.6 Realization of Feynman’s Thought Experiment

171

To see which slit the electrons are passing, we turn on the light source. Similar to Bohr’s arguments in the fifth Solvay Conference, Feynman argued that the very act of finding out which slit the electrons are passing through is what destroys the interference pattern. To recover the interference pattern, one may try to be gentle on the electrons, thus reduce the intensity of the light source, which means sending fewer photons. This time, some of the electrons will manage to get through without being seen. These electrons will produce an interference pattern on the second screen (A). The remaining electrons, the ones that we know which slit they passed through, will behave like particles and produce two humps centered around the points corresponding to the location of the slits on the first screen (B). Unlike in classical physics, where the energy of radiation is proportional to its intensity, in quantum mechanics energy of the photons is proportional to their frequency with the proportionality constant being the Planck constant. A single photon scattering from an electron will reveal the slit it passed through, but it will also have a dramatic effect on its behavior, which will be enough to destroy the interference pattern. To minimize the disturbance of the electrons, we may try using a light source with a lower frequency, thus hoping that it will allow us to learn the slit that it passed through without destroying the interference pattern. However, the frequency of the photon, ν, is inversely proportional to its wavelength, λ = c/ν. Therefore, as we try to be gentle on the electrons, we realize that there is a limit to how gentle we can be. As we use photons with lower energy, their wavelength becomes larger, and as their wavelength approaches the separation between the two slits, we can no longer resolve which slit the electrons are going through; hence, the interference pattern is recovered. In other words, as we try to corner nature to observe the wave and particle properties of the electrons simultaneously, as always does, it manages to slip through our fingers. If we know which slit the electron went through, there is no interference pattern, and if we do not know which slit it went through, the interference pattern appears. Bohr summarized this with the complementarity principle.

11.6 Realization of Feynman’s Thought Experiment Feynman probably did not think that his thought experiment would ever be realized, at least in some foreseeable future. But in mid-2000s, Stefano Frabboni and collaborators in Italy demonstrated interference pattern with electrons passing through 83 nm width slits.10,11 They were able to reduce the current of the electron beam to such a level that only one electron was present between the source and the detector at any given time. However, due to the limitations of their detectors, they were not able to observe interference from single electrons. It was not until 2013 that double slit interference from single electrons was finally convincingly observed. Working at the University of Nebraska-Lincoln in USA, Roger Bach and his team fired electrons 10 11

Frabboni et al. (2007) and (2008). Bach (2013).

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with a beam energy of just 0,6 kev through 62 nm width slits.12 Compared to the beam energy of 200 kev used by Frabboni and coworkers, this allowed them to work with electrons with higher de Broglie wavelengths, which not only produced interference pattern with wider separation but also allowed them to count single electrons via a channel plate detector. Bach and his team was also physically able the mask across the slits so that they could perform experiments where only one or both slits were open. To greater than 99.9999% probability, their experiment assured that only one electron is present between the source and the detector at any given time. Initially, as expected, individual electrons appeared to arrive at random points on the screen. But as more electrons were sent, an interference pattern with dark and bright fringes began to appear.13 Since each electron was detected before the next electron was emitted, they could not have influenced each other. Therefore, as Feynman so eloquently described, we have to accept that each electron has wave-like nature when both slits are open and behaves like a particle when only one slit is open.

11.7 Scully-Drühl Version of the Double Slit Experiment Bohr initially used clumsiness in the act of measurement in his defense of the uncertainty principle. But after the EPR paper, he had to switch his defense to the complementarity of the wave-particle nature of the quantum particles, which prevents simultaneous observation of the wave-like and the particle-like behavior. From Bohr’s new position, we can infer that it is the complementarity that prohibits us from observing electrons in Feynman’s experiment behaving as particles and waves simultaneously−not the clumsiness that was initially associated with the uncertainty principle that causes uncontrollable momentum transfer to the particle−Feynman’s experiment has demonstrated that avoiding clumsiness in beating the uncertainty principle is impossible. But, Scully and Drühl thought that they had found a way. In April 1982, Marlan Scully and Kai Drühl at the Max Planck Institute for Quantum Optics near Münich and the Institute for Modern Optics at the University of New Mexico proposed a new thought experiment.14 They considered the interference of photons emitted by atoms located at points 1 and 2 in Fig. 11.6 i. Similar to Young’s classical double slit experiment and Feynman’s quantum version of the double slit experiment, these atoms act as distinct light sources. These atoms are assumed to have two levels, which are pumped from the ground state c to an excited state a by a laser L Fig. (11.6 ii). Both atoms subsequently emit a photon γ by dropping to their ground state. Since we cannot tell which photon came from which atom, Scully and Drühl showed that this indeed produces an interference pattern as expected. Next, they assumed that there is also an intermediate level b with energy lower than a but higher than the ground state c Fig. (11.6 ii and iii). Arranging the experiment 12

Bach (2013). Bach (2013). 14 Scully and Drühl (1982). 13

11.7 Scully-Drühl Version of the Double Slit Experiment

173

Fig. 11.6 Quantum eraser and the Skully and Drühl thought Experiment

such that only the photons corresponding to the transition from the excited state a to the intermediate state b are detected, i.e., transitions from a to c are ignored. One may expect to see the interference pattern again. However, there is a subtle difference between the two cases. In the second case, one could determine which atom the photon came from by checking which atom is in the ground state and which atom is in the intermediate state after the emission has taken place. For example, if we find atom 1 in the intermediate state and atom 2 in the ground state, then we can immediately conclude that the photon came from atom 1 and therefore learn its trajectory. This allows us to find which path the photon has taken without in anyway disturbing it after it has been emitted. This is equivalent to finding which slit the electron went through in Feynman’s double slit thought experiment without disturbing the electron. It appeared that Scully and Drühl had found a way of observing the particlelike and the wave-like behavior of the photons simultaneously, which the Copenhagen interpretation categorically denies that it can be done. Ten years later in 1991, commenting on a different but equivalent thought experiment, Scully, et at. argued that even though the photons are left to travel toward the detector unaffected by any act of measurement, they are entangled inseparably with the atoms they left behind.15 It is this entanglement that destroys the interference pattern. In other words, what safeguards the system is that the entire system has to be considered from the perspective of quantum mechanics. By introducing the final states of the atoms left behind, Scully et al. mathematically demonstrated that the terms responsible for the interference pattern disappear. That is, one cannot observe wave-like and particle-like behavior together. This has nothing to do with the act of measurement being gentle or clumsy. If one could discover which atom the photon has come from−even in principle−the interference term is destroyed. If we are pre-

15

Scully et al. (1991).

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vented from finding out which atom the photon has come from, then the interference terms come back. Complementarity is generally associated with complementary variables like position and momentum, and their uncertainty relation. The more precisely we measure one of the complementary variables, the less precise the other becomes. The famous Bohr-Einstein debate during the fifth Solvay Conference was about the double slit thought experiment, where Einstein tried to beat the uncertainty principle. The debate was concluded with Bohr using the uncertainty principle and the idea of uncontrollable momentum transfer to the screen with the slits. On the other hand, in Feynman’s double slit thought experiment, Feynman’s argument was still based on the uncertainty principle and the amount of uncontrollable momentum transferred to the electron by the light source, which changed the electron’s behavior after it had cleared the slits. What is special about the Scully and Drühl thought experiment is that as Scully et al. said “... this disappearance [of the interference pattern] originates in correlations between the measuring apparatus and the systems being observed. The principle of complementarity is manifest although the position-momentum uncertainty relation plays no role.”16 In the Scully and Drühl thought experiment, if we force the photon to reveal which atom it came from, or which path it followed, quantum theory denies the possibility of observing its wave and particle-like nature simultaneously by destroying the interference pattern. If the wave-particle duality is at the core of the mystery of quantum mechanics, then complementarity−not the uncertainty principle−is the way it works.

11.8 Quantum Eraser and the Delayed Choice Thought Experiment Scully and Drühl advanced their thought experiment in 1982 further to answer the question: What if we set up the experiment so that it allows us the opportunity to find the path that the photons are taking but we choose not to look.17 Will the interference pattern come back? Furthermore, what if we wait until the photons have passed through the equipment and got detected and then choose to look, or not to look, to see which way the photons have taken? Is it possible to turn the interference pattern on and off by simply not looking and looking after the photons have been detected? According to their original analysis, they found that indeed the interference pattern comes back if one chooses not to look. In the previous case, we have seen that when the atoms have an intermediate level, it is possible to learn from which atoms the photons are coming from by looking at the state of the atoms after the emission of the photon. Now, instead of looking at the final state of the atoms, what if we hit them with another laser pulse which excites any atoms in the intermediate level to 16 17

Scully et al. (1991). Scully and Drühl (1982).

11.8 Quantum Eraser and the Delayed Choice Thought Experiment

175

Fig. 11.7 In Scully and Drühl thought experiment s stands for shutter, d for the common detector wall

another higher energy level which quickly decays to the ground state by emitting a photon, which they called the φ−photon Fig. (11.6iv). This erased the whichway information from the system. Their analysis showed that the application of the second laser pulse L  did not bring the interference pattern. To see the interference pattern, they had to look at the φ−photons carefully. Their arguments were subtle, but they showed that indeed the interference pattern comes back when one chooses not to look. Now, imagine we have two cavities in front of the screen with the double slit capable of trapping the φ−photons (Fig. 11.7). These cavities are separated by a common detector wall at the center, so that the top slit is on the top cavity and the bottom slit is on the bottom cavity. The cavity material is such that it allows the laser pulses and the interference photons to pass unaffected. The common detector wall made up of thin-film semiconductor that absorbs the φ−photons and acts as a photodetector. There are also (electro-optic) shutters on both sides of the common detector wall that can be opened and shut selectively. The box that contains the φ−photon indicates which slit, hence which atom the interference photon detected on the screen came from. The first laser pulse excites both atoms, where one of them drops to the intermediate state and emits a photon that goes to be detected on the screen. A second laser pulse erases the information about the final states by leaving a φ−photon in one of the cavities. Until the φ−photon is detected, it is in a superposition of 50/50 being in either cavity. We wait until the interference photon is detected hence left a point on the screen, and then open the shutters. The φ−photon is now absorbed by the common detector wall, which assures that the erasure has been completed. Scully and Drühl calculated that this will lead to a count, or a no-count, on the detector with 50/50 probability. The question is; will this bring the interference pattern back? The interference pattern does indeed come back, but to see it, one has to correlate the interference photon with the detection, or no-detection, of the φ−photon. Let us now assume that we could somehow distinguish each spot on the screen with respect to its correlation with the detection or no-detection of the φ−photon as a yes-, or a no-photon, respectively. When we plot the distribution of the interference

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photons on the screen separately, we begin to see an interference pattern for both types of photons. However, there is a difference; the interference pattern for the yes-photons is shifted with respect to the no-photons. The interference fringes corresponding to the no-photons are called anti-fringes, since the peaks of the yes-photons correspond to the troughs of the no-photons. If we do not look at the photons separately, then the result is their sum comparable to a scatter distribution that is associated with a which-way detection leading to a smearing of the fringe pattern. As Scully and Drühl said “If on the other hand we choose not to read our φ− photon counter and keep all scattering events, no interference pattern will be found in the complete ensemble of all γ −photon counts. Thus in our (thought) experiment the total ensemble of scattering events is decomposed into two subassemblies showing interference fringes and antifringes, respectively.”18 Finally, Scully and Drühl19 proposed a delayed choice thought experiment, where one closes the upper shutter in the top cavity so that the φ−photons from atom 1 can never reach the common detector wall. Now, any count in the φ−detector arises from atom 2 and provides a which-way information. In other words, for every γ −photon detected, there will either be a count in the φ−detector, thus signaling that the γ −photon came from atom 2, or a no-count indicating that the γ −photon came from atom 1. In this mode of the delayed choice thought experiment, the φ−detector provides the which-way information; hence, no interference pattern is expected. In the previous case, where both shutters are open, the apparatus is sensitive to the wave nature of the photons; hence, interference is expected. Scully and Drühl concluded their paper as “Hence, we are potentially able to display either the particle (path) or wave (interference) nature of the scattered radiation even though we delay this choice until long after the γ − photons have been emitted.”20

11.9 Complementarity and the Quantum Eraser Experiment The thought experiments proposed by Scully and Drühl in 1982 were fascinating.21 Even though quantum mechanics allows us to calculate the results, they are extremely difficult and tricky experiments to perform. However tricky and difficult these experiments may be, in 1995, Thomas Herzog and collaborators at Insbruck, Austria, accepted the challenge and presented their results in Physical Review Letters.22 In July 1995, Herzog et al. produced a vertically polarized photon pair by using type I spontaneous parametric down conversion of a UV photon.23 They directed the 18

Scully and Drühl (1982). Scully and Drühl (1982). 20 Scully and Drühl (1982). 21 Yam (1996). 22 Herzog et al. (1995). 23 Herzog et al. (1995). 19

11.9 Complementarity and the Quantum Eraser Experiment

177

Fig. 11.8 The basic setup of the Herzog et al. interferometer. A nonlinear crystal is pumped by a laser P. Reflecting the pump beam back into the crystal gives a second pair, which makes them indistinguishable from the pair created in the first pass, thus interference occurs

351-nm beam of a single-mode Ar+ −ion laser, called the pump, onto a nonlinear crystal (LiO3 ) to produce a photon pair, called the “signal” and the “idler” with frequencies in the red (633 nm) and in the near infrared (789 nm) end of the spectrum, respectively (Fig. 11.8). When the pump beam was reflected back into the crystal, a second signal-idler pair with identical wavelengths was created. Using two more mirrors, the pair created in the first pass of the pump beam was reflected back into the crystal so that they overlap with the pair created in the second pass, which made them indistinguishable from the pair created by the reflected beam. They called the first pair created by the first pass of the pump beam “reflected” and the pair created by the reflected pump beam “direct.”24 The two photon pairs, reflected and direct, effectively act like the two slits in the classical interference experiment. Since the reflected and the direct photons are indistinguishable, as long as no attempt is made to find out which pair they belong to. Herzog and coworkers observed an interference pattern in the signal and the idler detectors as functions of the differences in the paths they have taken. Similar to the double slit experiment, as long as no attempt was made to identify which pair the photons belong, reflected or direct, an interference pattern was observed. On the other hand, a which-way information could be obtained by tagging one of the photons. For this, they changed the polarization of the idler photons from vertical to horizontal and put a polarization analyzer in front of the detector. Since tagging consists of a rotation of the polarization of the photons, which does not affect their momentum, it helped physicists to find what kind of photon they were: reflected or direct. Obtaining which-way information for the idler photons, also implied equivalent which-way information for the signal photons. Thus, interference in both the signal and the idler photon rates was destroyed. They were also able to erase the tag, hence the which-way information, by simply rotating the analyzer to an angle of 45 degrees. This prevented the possibility of finding the polarization of the idler photons, hence the opportunity to learn which type, reflected or direct, they belong to. This was enough to restore the interference in the idler photon detection rate and in the rate of the signal and idler photon coincidences, but not enough to restore the interference in the signal photon detection rate. The 24

Herzog et al. (1995).

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Fig. 11.9 Strange things happen in microcosmos

Innsbruck experiments used photon pairs with different types of quantum markers to perform which-way and quantum eraser operations. Their markers and erasers allowed continuous variation of the degree of obtainable which way information, thus causing continuous loss of interference. They ended their paper with “The use of mutually exclusive settings of the experimental apparatus implies the complementarity between complete path information and the occurrence of interference. In conclusion, our results corroborate Bohr’s view that the whole experimental setup determines the possible experimental predictions.”25

11.10 Finally the Delayed Choice Experiment Done Despite the interesting results of the Innsbruck experiment, they had not quite reproduced all the aspects of the original thought experiment. In particular, they have not demonstrated the delayed choice quantum eraser experiment, where the possibility of delaying the choice between observing the particle (which-way) and the wave (interference) nature until the measurement is completed. Such an experiment was performed by Yoon-Ho Kim and collaborators in 1999, where they reported their results in a Physical Review Letters article.26 Their experiment confirmed the bizarre nature of the quantum realm by showing that the interference pattern can indeed be turned “on” and “off” by choosing not to look and to look at the whichway information after the experiment is over. Additional experiments to this effect were performed and reviewed by Xiao-song Ma and collaborators.27 There is a direct connection between complementarity and non-locality in quantum mechanics. Interference is a direct consequence of wave-like behavior and a manifestation of non-locality. In the mathematical structure of quantum mechanics, these effects appear as the entanglement of the states responsible for the interference and the states that register which-way information. These states cannot be disentan25

Herzog et al. (1995). Kim et al. (2000). 27 Ma et al. (2016). 26

11.11 Quantum Tunneling and More Quantum Weirdness

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gled to reveal the wave-like and the particle-like behavior simultaneously. This has nothing to do with our ability or ingenuity to conceive an experiment that is smart enough to reveal both particle-like and wave-like behavior at the same time. It is complementarity that says it just cannot be done. This is the crux of complementarity that lies at the core of quantum weirdness.

11.11 Quantum Tunneling and More Quantum Weirdness Another important example of the quantum weirdness with practical significance is the quantum tunneling effect. In the quantum realm, an electron traveling toward a cliff, that is, a potential drop of V, with the Energy E (Fig. 11.9), has a finite probability of reflecting at the edge. In our classical world, this is analogous to a tennis ball thrown out of a window bouncing back at you. In a classical roller coaster, conservation of energy forbids a particle that starts from the point a with zero velocity from overshooting the point d. In quantum mechanics, it is possible for an electron to sneak through the forbidden region between d and e, to emerge on the other side of the hill at e. Around 1920, Eddington speculated that the energy source of stars is nuclear fusion. At that time, the source of stellar energy was a mystery. When critiques objected that the center of a star could not be hot enough to allow fusion, Eddington’s historic response was go and find a hotter place.28 In classical physics, inside the sun there is almost no chance for protons to come close enough to fuse by overcoming each other’s Coulomb repulsion. It was George Gamow who first applied quantum mechanics to this problem and showed that due to tunneling there is a significant chance for the protons to fuse.29 After attending a seminar by Gamow, Born realized that the tunneling effect is not just restricted to nuclear physics, but it is one of the rudimentary features of quantum mechanics with wide range of applications.30 In 1957, in the study of semiconductors, tunneling of electrons in solids played an important role in the development of diodes and transistors. Leo Esaki, Ivar Giaever and Brian Josephson received the Nobel Prize in physics in 1973 for predicting the tunneling of superconducting Cooper pairs. Quantum tunneling has many interesting practical applications like the scanning tunneling microscope.31

28

Stanley (2005). Gamow (1928). 30 Razavy (2003). 31 Razavy (2003). 29

Chapter 12

Can We Ever Hope to Understand Quantum Mechanics?

Wheeler once said that if we really understood quantum mechanics, then we ought to be able to state it in one simple sentence. Of course, if this means that in future quantum theory will be stripped from all of its counterintuitive features, then we could safely say that the new theory will neither be a modification nor a reinterpretation of quantum mechanics. It will be a profoundly different theory. Some may argue that it is the language of mathematics that should be used−not words. As Feynman knew, for the true meaning of reality we cannot hide behind equations. What has always amazed me is that in mathematics, such as in number theory, some of the deep and challenging problems like Fermat’s last theorem can be stated in simple everyday language that everybody can understand, yet their solution was possible only after centuries and only after the development of new branches of mathematics. Some still await a satisfactory solution like the three-body problem, which has no general closed-form solution. Some like the four color problems were solved only with the help of modern computers. So far, including Bohm’s hidden variables theory and Everett’s many-worlds interpretation of quantum mechanics, none of the alternate interpretations of quantum mechanics, can make its weirdness go away.

12.1 How to Define the State of a System Even though the concept of quanta has played a key role in its discovery, quantum mechanics is not just about quanta. Nature could very well be granular and yet classical at small scales. A more general description of quantum mechanics would be that it is about the state of a system and how that state evolves and what changes are brought upon it by observation.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_12

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How do you define the state of a system? In classical systems, it is defined by telling what the system “is” at a given time. For example, a golf ball is localized with a definite position and velocity. Additional characteristics like color, mass, and shape could also be given to complete the picture. Given the forces acting on a body, theory allows us to find its state precisely at a later time. The state of a system in classical physics is always directly accessible to observation, and hence, it is a part of physical reality. Sure, even in classical physics observation induces some changes in the state of a system, but these changes can either be minimized or could be calculated precisely to make the necessary corrections to the results. In quantum mechanics, is an electron a wave or a particle? These are mutually exclusive properties of the electron. A wave is spread out in space. When you shout, everybody around you can hear. Sound waves spread out like the ripples on a lake. On the other hand, a particle is localized. It can only be present at a single point at a given time. According to quantum mechanics, an electron is neither a wave nor a particle until it is detected. Depending on your experimental setup, it will reveal either its particle-like or wave-like nature, but not both at the same time. In the double-slit experiment, if your experiment is designed to find out which slit it went through, the electron will behave like a particle. Otherwise, it will behave like a wave and you will get an interference pattern. The state of a quantum system is defined, not in terms of what the system “is” at a given time, but as a superposition of all the things that it can be. It is described by a complex wavefunction, also called the state function, that is not available to direct observation. Until an observation or a measurement is made, the system is in a state like suspended animation, as if it is in all the possible states that it could be in at the same time.

12.2 Weird or Just Counterintuitive Until Born came up with the probabilistic interpretation of the wavefunction, Schrödinger thought that the wavefunction represented physical waves like the sound waves. Born argued that the coefficients of the individual states in the superposition represented the probabilities of the individual states coming up when a measurement is performed. Until a measurement is performed, the wavefunction evolved continuously with the system existing in the ghostly state of superposition of all the possible states that the system could be in. It is now Schrödinger’s equation that describes how the wavefunction evolves continuously. Coefficients of the individual states in the superposition changes, but the total probability is conserved. In other words, the total information about the system remains fixed. We say the wavefunction’s evolution is unitary. However, measurement does something else. It collapses the wavefunction to one of the states in the superposition with its respective probability. For example, consider a system with three states, A, B, and C, with their respective probabilities as 20%, 30%, and 50%. With time, these probabilities will change but unitarity

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demands that their sum always be 100%. A measurement on this system will cause the wavefunction to collapse to one of A, B, or C, with its respective probability, say B with 30%. What happens to A and C? They are gone. The system is now in state B 100% and no matter how many times it is observed, it will be in state B for sure. There is no way to get back to the original superposition unitarily from B. Of course, one can always prepare a new system under the same conditions with the same superposition, but once the wavefunction has collapsed, it won’t happen on its own. What is the source of this abrupt change that takes place at an instant everywhere? There is nothing in Schrödinger’s equation that describes it. You can start with the superposition of the states A, B, and C, and end up with any one of them, but there is no way you can go back to the original superposition you started. The standard machinery of quantum mechanics is unitary, which describes how the wavefunction evolves continuously. Yet every observation/measurement causes the wavefunction to collapse, which is non-unitary. This is why the measurement problem is so enigmatic in quantum mechanics. Measurements are classical by definition. They require classical equipment that humans can interact with. Wavefunction collapse is how quantum states are turned into classical observed phenomena. We cannot predict the outcome of the collapse with certainty, but quantum mechanics allow us to calculate the probabilities of specific outcomes. In a single experiment, our chance of getting A, B, or C is 20, 30, and 50%, respectively. If we conduct our measurements on 100 identically prepared systems, then 20 times A, 30 times B, and 50 times C will be observed. It was the Hungarian mathematical physicist von Neumann who first made the wavefunction collapse an essential part of quantum mechanics.1 He argued that collapse is due to the act measurement, and hence, it must also have something to do with the presence of an observer. This induced Nobel Prize winning physicist Eugene Wigner to hypothesize in 1961 that collapse results from conscious intervention in the quantum system.2 He demonstrated his idea with a lesser-known thought experiment, where he conducts an experiment on a superposition of some quantum system with two states. When the measurement is done, a flash is registered (or not) indicating that one of the possibilities has been realized, thus signaling that the wavefunction has collapsed. Now, suppose that the measurement was conducted after Wigner has left the lab. Wigner cannot know that the wavefunction has collapsed until his friend in the lab calls and tells him the result. In this view, Wigner’s friend is also in a superposition until Wigner collapses the wavefunction by extracting the information from his friend. Then, Wigner himself is in a superposition until he shares the result with his other friend awaiting the news in the office. Naturally, this could go on forever. This is the kind of awkward situation we find ourselves in when we try to introduce consciousness into the measurement process. Does a dog trained to react to a flash or to some other stimulus from an instrument count as a conscious observer and 1 2

Neumann (1955). Wigner (1961).

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can the dog collapse the wavefunction. Seeing or not seeing, understanding or not understanding may cause a partial collapse in the combined wavefunction of the observed phenomena including the apparatus, and the observer’s mind. However, like Einstein can we push this as far as saying that the moon does not exist until you or somebody looks at it. The corresponding part(s) of the wavefunction of the moon and the observer has collapsed and “decohered” long time ago, thus establishing the moon as a classical object obeying Newton’s laws for everybody. However, still there has to be a difference in the parts of the of the total wavefunction of the observed system and the observer, corresponding to the observer’s state of mind between knowing and not knowing, understanding or not understanding, being aware or not aware of something. Until we have a theory of body, mind, and consciousness, thought-induced collapse will remain a highly controversial area.

12.3 Wigner’s Thought Experiment Realized in Laboratory In 2019, physicists noticed that it can be possible to reproduce Wigner’s thought experiment in a real experiment. In other words, it can be possible to create different realities at the same time and compare them in the lab to see if they can be reconciled. Massimiliano Proietti at the Heriot-Watt University in Edinburgh and colleagues claimed that they have just done such an experiment.3 Their conclusion was that Wigner is right−the two realities, Wigner’s and Wigner’s friend’s, coexist and are irreconcilable. In other words, it is impossible to agree on objective reality. They used a state-of-the-art six photon setup and a system of four entangled observers to show that while one of the components of the system generated a (measurement) result, the other remained in a superposition showing that a measurement had not yet been made. Physicists were able to measure both realities at once. According to Proietti and co-workers, this supports the claim made by quantum theories that take observer dependence into account, where two irreconcilable, and hence mutually exclusive, realities might exist simultaneously, at least in the quantum realm. For the observers to be able to reconcile their results demand three conditions4 : (I) Existence of universal facts that observer can agree on, that is, existence of observer-independent facts. (II) Observers are free to make their decision about whatever they want to observe. In other words, free choice exists. (III) Choices made by one observer do not affect the choices that the other observer makes. This is also called locality. If there is an objective reality, then all these conditions must hold. Proietti and co-worker’s results indicate that objective reality does not exist. Their experiment indicates that at least one or more of these conditions must not be true. Of course, 3 4

Proietti et al. (2019). Proietti et al. (2019).

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in coming years, one still needs additional experiments that create alternate realities with their possibly overlooked loopholes investigated and closed.

12.4 Non-locality and the EPR Thought Experiment Aside from the mysterious wavefunction collapse, the EPR paradox demonstrated another feature of quantum mechanics that challenged our common sense about locality. In other words, properties of particles are localized on themselves and what happens here cannot affect what happens elsewhere instantaneously. This looks so natural that it hardly qualifies as an assumption. Yet it is exactly this core concept of reality that entanglement undermined. However, the two particles in the EPR thought experiment are no longer separate entities; they are parts of a single object even though they are separated in space. Properties of one of the particles are not located solely on that particle alone. In quantum mechanics, properties can be non-local. In a superposition, we write the wavefunction of a quantum system in terms of all the possible outcomes of a measurement with their respective probabilities. Entanglement is the same idea, but applied to two or more particles. For example, in a two-particle system, when one of the particles has spin down, the other has spin up and vice versa. Even though the particles could be miles or even light years apart, they are represented by a single wavefunction. We cannot untangle the wavefunction into some combination of the single-particle wavefunctions. Because non-locality is so counterintuitive that it has been tested rigorously by various groups, which we have discussed in conjunction with Bell’s inequality. Weirdness is when we try to understand entanglement in terms of a measurement on one of the particles, say its spin, which affects the spin of the other particle instantaneously and vice versa. A compromise with relativity was reached when it was realized that one cannot use this to send signals faster than the speed of light. The quantum mechanical way is to understand this in terms of non-locality. Counterintuitive results are present even in classical physics. When you ask; ignoring friction, which one falls faster, a hammer or a feather? You would be surprised how many people would answer hammer. Galileo is usually credited for demonstrating from the leaning tower of Pisa that they fall at the same time−a consequence of the equality of inertial and gravitational mass. Our intuition is shaped by our everyday interactions with our environment and our expectations and projections based on them. Even in such counterintuitive examples, the truth or reality is always directly accessible to us. When we try to understand quantum phenomena by our intuition, it looks down right weird. Just because quantum mechanics is a consistent theory and allows us to calculate things that can be tested in laboratory, does not make it any less weird.

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12.5 Does Quantum Mechanics Need Imaginary Numbers? One of the mysterious aspects of quantum mechanics is the appearance of the imaginary i in the formulation of the theory. Imaginary numbers are written in units of √ i, which is the square root of minus one, −1, or i 2 = −1. Imaginary numbers do not correspond to physical quantities, but this does not mean they are useless in science. Complex numbers are composite numbers composed of two real numbers (a, b), which are also written as z = (a + ib). In electromagnetism and various other branches of science and engineering, even though the problem is real, complex numbers are usually used as a means of calculational convenience. At the end, we use either the real or the imaginary part of the solution as the physical solution. For example, in the study of oscillations and waves, the real part of the solution in the complex plane gives the solution in terms of the cosine function, while the imaginary part gives the result in terms of the sine function. In all such cases, the physical phenomenon is real and can be formulated entirely in terms of real numbers. In quantum mechanics, even though the probabilities and the other observables are by necessity real, for the first time in science, imaginary i has appeared as an essential part in the formulation of the theory. Do we really need the imaginary i? Could it be just an artifact of the way the theory was constructed by its discoverers? Could the quantum theory be reconstructed entirely in terms of real numbers? And will the weirdness go away, if somehow we could reconstruct quantum mechanics entirely in terms of real numbers? Despite the successes of the quantum mechanics that continue to pile up, such questions continued to lurk in physicist’s and philosopher’s minds−at least in the minds of the ones who did not listen to “shut up an calculate” advice. Thanks to the work of John Bell, it was possible to design and perform experiments, in which local hidden variable theories and the quantum theory predict different results. It was the experiments of Alain Apect, John F. Clauser, and Anton Zeilinger, who were awarded the 2022 Nobel Prize in Physics, that demonstrated unequivocally the violation of Bell’s inequality. As far as the local hidden variable theories, the verdict was in favor of quantum mechanics. In a 2021 Nature article, Miguel Navascués and colleagues of the Institute for Quantum Optics and Quantum Information in Vienna shed some light on the possibility of reconstructing the quantum mechanics entirely in terms of real numbers.5,6 They found that subject to four basic postulates that the quantum mechanics is based on, no real-valued version of quantum mechanics can duplicate all the predictions of the usual complex-valued formulation. They also designed an experimentally feasible test capable of eliminating out real-valued quantum theories. Since their paper appeared in January 2012, two groups carried out the experiment and both found results that supported the standard complex-valued quantum mechanics.7,8 5

Renou et al. (2021). Miller (2022). 7 Li et al. (2022). 8 Chen et al. (2022). 6

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The necessity of complex numbers in quantum mechanics has a lot to do with Heisenberg’s uncertainty relation, where the position and the momentum operators are represented by matrices that do not commute Eq. (5.5). However, it is more subtle than this. There exist some reformulations of quantum mechanics based on real vectors and real operators that play the role of physical states and physical measurements.9,10 But they begin to differ from the standard quantum mechanics when one considers multiparticle states, where the combination of two states is represented by tensor (matrix) products. The mathematical proof of the impossibility of real-valued quantum mechanics was an ambitious goal. Navascués and colleagues didn’t just have to prove that the most obvious real-valued formulations of quantum mechanics do not work, but they also had to prove that none of them work. It was an audacious task. Using entanglement swapping experiments, Navascués and colleagues succeeded in finding a function of measurement correlation that could be 8.49 in standard quantum theory but could never exceed 7.66 in real-valued formulations. This does not leave much room for error, but Jian-Wei Pan and colleagues at the University of Science and Technology of China in Hefei carried out experiments via superconducting qubits and found the value 8.09.11,12 Like the early attempts to test Bell’s inequality, these experiments still have quite a few loopholes. It will certainly take some time to close all the loopholes, but at the moment, the results are in favor of the standard complex-valued quantum mechanics.

9

Stueckelberg (1960). Aleksandrova et al. (2013). 11 Li et al. (2022). 12 Chen et al. (2022). 10

Chapter 13

Navigating Between the Classical and the Quantum Worlds

Schrödinger’s cat metaphor was successful in challenging Bohr’s Copenhagen interpretation of the quantum mechanics. Like Einstein, Schrödinger had also approached Copenhagen interpretation with a great deal of skepticism. It was OK for Bohr to impose strict separation between the quantum and the classical worlds and to make a measurement/observation via a process where they are distinguished. But, what about before an observation was made. Schrödinger showed that when taken literally superposition of macroscopic states leads to bizarre consequences like objects being in different places at once, or as in the cat metaphor, the poor cat being both alive and dead until somebody observes it. Schrödinger’s cat metaphor demonstrated the paradox that one encounters when the world is separated into classical and quantum realms without defining where the boundary is. The quantum realm is probabilistic defined by the probability waves obeying the Schrödinger’s wave equation, which allows superposition and entanglement. On the other hand, the classical realm is orderly and deterministic described by Newton’s equations. Yet we need a way that allows us to navigate between the two worlds. At the core of the problem lies the wave nature of quantum systems that allow superposition between possible outcomes of an observation. Until an observation is made, the cat is neither dead nor alive but exists in a mixed state of both dead and alive. This thought experiment could be repeated in many ways where the cat is replaced by any system with two states. Such as a coin tossed by a mechanical device in a closed box. Until the box is opened and an observation is made, the coin will be in the mixed state of both heads and tails. We could also consider an electron in the box that could assume either the spin up or down state. Again, the electron will be in a superposition of the spin up and down states with equal probability. This thought experiment could be repeated at different scales with various quantum systems that obey Schrödinger’s equation, where the results could be understood within the context of quantum mechanics in terms of superposition. But with the coin and the cat that obey Newton’s laws, superposition looks downright weird. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_13

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13.1 Coherence Is What Determines “Quantumness” Where does the “quantumness” begin or end? Can we navigate from the quantum realm into the classical domain and vice versa? At the core of this distinction lies the fact that quantum objects have wave nature that allows superposition and entanglement. Even though these waves are neither physical waves like the sound waves nor their probabilistic interpretation is the same as in classical probability theory, they still have wave characteristics like interference and superposition. Another key concept borrowed from ordinary waves is coherence. When the crests and the troughs of two waves coincide we say they are coherent, or in phase. Light from an ordinary tungsten light bulb is incoherent, since photons emitted from different parts of the filament have different phases. On the other hand, light from a laser is coherent, since all the photons emitted are in phase. Coherence is the key concept in defining the quantumness of an object. In the double-slit thought experiment, coherence of the waves emanating from the two slits is essential in obtaining the interference pattern on the screen. It is this interference that verifies the wave-like behavior of the electrons. Any attempt to find out which slit the electron went through destroys the coherence, hence the interference pattern. In the case of the wavefunction of a system with two states, if the two states are not coherent, then there will neither be an interference nor the superposition can be maintained. The two states will pretty much behave as distinct classical systems. Macroscopic objects do not show interference or maintain superposition because their wavefunctions have lost their coherence, or they have decohered. A loss of coherence essentially destroys the quantumness of the system.

13.2 Do All Objects Have a Wavefunction? Notice that we are talking as if all objects have wavefunctions. After all, macroscopic objects are made up of quantum entities with their separate wavefunctions. In principle, we expect the wavefunction of a macroscopic object to be a combination of the wavefunctions of the quantum entities that it is composed of. Yet, we don’t see wave-like behavior among macroscopic objects like superposition, interference, collapse of the wavefunction, and entanglement. Consider a case where you used your lucky vintage pen as a placeholder in one of the many books on your terribly disorganized desk and you forgot which book it was in. You begin to look frantically for your pen. It is certainly in one of the N books on your desk. The probability of your pen being in any one of the books on your desk is given by the classical probability 1/N . Of course, the “wavefunction” of the pen will contain information not just about the many internal energy states of the atoms and the molecules that it is composed of, but it will also have information about the positions of these atoms and molecules with respect to the desk. This also means information about in which book the pen is in. However, the finite number of

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states corresponding to which book the pen is in are mutually exclusive states that cannot be superposed. In other words, it is not like the pen exists in the ghostly state of superposition, where the pen is in all the books at once until you find your pen. The pen is always in one of the books, whether you find it or not. The probability of you finding your pen in one of the books is given by the classical probability 1/N . It is also not possible for the pen being entangled with other pens on your desk. We now further our example as a thought experiment and consider miniaturizing our vintage pen. We are not worried about whether our miniaturized pen works or not. We are not miniaturizing the atoms. They are still the same, but our miniature pen is made up of fewer and fewer atoms. We are interested in finding out at what point our miniaturized pen begins to show quantumness. We would also be interested in seeing whether we could navigate between the classic and the quantum realms. Technologically we are almost at a point where we could explore these ideas in laboratory with systems at mesoscales, roughly at the order of nanometers. But first, we will discuss the mechanism that objects lose or gain their quantumness.

13.3 Where Does the Weirdness Begin or End? What happens? At what point does the pen lose or gain its “quantumness”. The pen is always composed of quantum entities−atoms and molecules−that are governed by Schrödinger’s wave equation. According to the Copenhagen interpretation, the wavefunction of a system, which is also called the state function, gives the complete description of that system. It includes everything that can be known about that system. There are no hidden variables. Therefore, the wavefunction of the individual quantum entities possesses information not only about their internal energy states, but also about their locations within the pen and where the pen is on the desk. Assuming that the conditions outside the pen, temperature, pressure, etc., remain constant, in which book the pen is in has no bearing on the internal states of the quantum entities that the pen is composed of. Therefore, in principle, we should be able to separate the part of the wavefunction that carries information about the location of the pen among the N books on the desk. Then, why don’t we see interference between the wavefunctions of the pen corresponding to its presence in different books? What was the cause for these states to exist as mutually exclusive classical states governed by classical Newton’s laws?

13.4 The Essence of Quantum Mechanics The key to quantumness is coherence. It was the coherence of the two waves coming from the two slits that produced the interference pattern. Any attempt to find which slit the electron went through destroys the coherence, hence the interference pattern. If the wavefunctions of two states are not coherent, then they cannot maintain superposition

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and therefore display interference. A loss of coherence is called decoherence, which destroys the quantumness of an object. It is still not clear why objects no matter how big they are cannot maintain coherence−provided there is no observation or measurement made on them. It was known in Bohr and Einstein’s time that the act of measurement destroys coherence and hence causes decoherence, which collapses the wavefunction. A single particle in quantum mechanics is described by a wavefunction that spreads out over a large region, but it has never been detected in two (or more) places. Einstein never believed that the non-local collapse of a wavefunction could be real. His first argument in the fifth Solvay Conference was about a single particle wavefunction diffracted through a tiny hole, which dispersed it over a large hemispherical region that eventually encountered a screen covered with a photographic film. Since the film registered only a single particle, quantum mechanics implied that the wavefunction has instantaneously collapsed to a single point on the screen and to nothing everywhere else. To Einstein, this meant spooky action at a distance, which violated his special theory of relativity. In 2015, by splitting a single photon between two laboratories, researchers from the Griffith University, Center of Quantum Dynamics in Australia, and the University of Tokyo in Japan have successfully demonstrated the non-local collapse of the wavefunction.1 Using homodyne detectors, which allowed them to measure wave-like properties, one of the groups were able to make different measurements, while the other group used quantum tomography to test the effects of the first group’s choices. Through these different measurements, they were able to observe wavefunction collapse in different ways, thus proving non-local collapse of a particle’s wavefunction. Unlike the usual quantum entanglement, which involves two particles, this is yet the strongest proof of the entanglement of a single particle, which is being explored for its applications to quantum communication and computation. Superposition and entanglement are usually viewed as delicate and fragile states that are easily destroyed when put in a noisy environment, which cause them to collapse. But why should quantum states be fragile? After all environment is also made up of atoms and molecules that are governed by Schrödinger’s equation. In fact, quantum states are not fragile. On the contrary, they are highly contagious. They tend to spread out very quickly. And that is what destroys them. When a quantum system interacts with another one, they become linked into a composite system called entanglement. Their interaction has turned them into a single quantum entity. Similarly, when a particle hits an air molecule, the two will be in an entangled state. When that entangled air molecule hits another air molecule, the second molecule will also be captured in an entangled state too. This way the initial quantum entity will be entangled more and more with its environment. Eventually, the system and the environment will merge into a single system−a single superposition.

1

Fuwa et al. (2015).

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13.5 Defining Decoherence In other words, the quantumness of the original objects is not destroyed but spreads and leaks into the environment. There is nothing that can stop this. It is this spreading of quantumness of the original object into the environment that makes it discernible in the original object. Superposition is now a shared property with the environment. The original system has lost its integrity and exists in a shared state with all the particles in the environment. One can no longer observe superposition by just looking at a small part of the combined system. Decoherence is basically not a loss of quantumness but it’s a loss of our ability to see it in the original system. Decoherence is a real physical process that happens in time. For example, how long does it take for a dust grain floating in air at room temperature to decohere. A dust grain is roughly one-hundredth of a millimeter. Ignoring photons in the room and considering only the collisions with the air molecules, quantum mechanics allows us to calculate the decoherence time for a dust particle as ∼ 10−31 s. This is roughly equal to the time that light takes to travel across a proton. If we evacuate the room, there will still be photons due to the finite temperature of the walls. Now the decoherence time becomes ∼ 10−18 s. This is roughly the travel time of light across a gold atom.2 Can we ever hope to catch decoherence in the process of happening? For macroscopic objects under ordinary conditions, decoherence is almost at an instant. However, for objects at mesoscales at the order of nanometers (millionth of a millimeter), where nanotechnology and biology operate, we may be able to navigate between the classical and the quantum realms. For a large molecule roughly the size of a protein floating in air at room temperature, decoherence happens within 10−19 seconds. But in perfect vacuum at room temperature, coherence may survive for almost a week. It is decoherence that stops us from seeing macroscopic superpositions. This includes Schrödinger’s cat and also answers Einstein’s question about whether the moon exists or not until somebody looks at it. Yes the cat is either dead or alive regardless of you looking into the box and the moon exists independent of the existence of somebody looking at it. Wavefunction of the cat has decohered due to the billions of collision with the air molecules and the photons within the box. Similarly, the wavefunction of the moon has decohered billions of years ago due to bombardment by the photons from the sun. In none of these examples, we need a conscious observer or a measurement made for the wavefunction to collapse. Environment does the job of spreading the quantum coherence, where size matters. The larger the object is, the greater the number of interactions with the environment, and hence the faster the decoherence is. In other words, what we have called measurement before, at least most of it, could be attributed to decoherence. However, we can never completely isolate a system from the environment. We are all inseparable parts of the same grand structure. The system is a part of the environment and the environment is also a part of the system. In the quantum world, there are no sharp boundaries. In this regard, even if we consider an object in space, 2

Ball (2019), p. 210.

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there will still be decoherence due to the collisions with the photons from the cosmic microwave background radiation and with other cosmic particles.

13.6 Decoherence as a Theory Decoherence was known in Bohr and Einstein’s time as the cause of the wavefunction collapse. Its development and introduction as a formal theory had to wait the work of German physicist H. Dieter Zeh. In 1970s, Zeh laid down the foundations of the decoherence theory.3,4 At the beginning, he was largely ignored, but with the two important papers by Wojciech Zurek in 1981 and 1982, a former student of John Wheeler, interest in decoherence theory was awakened when its use in navigating between the classical and the quantum realms was recognized. The first experiment that allowed controlled decoherence came in 1999 from Zeilinger, Arndt, and colleagues in Vienna.5 They were able to change the amount of decoherence so that one could compare the theory with the experiment. The groundwork for the Vienna experiment was prepared in 1985 and 1990, where researchers devised techniques for producing coherent molecular beams, hence matter waves, that could be used in the double-slit interference experiments.6,7 Zeilinger and colleagues observed interference of molecules called “fullerenes” consisting of 60 or 70 carbon atoms (C60 , C70 ) joined into closed cages, each almost millionth of a millimeter across. At that time, this was the most massive and complex object that displayed quantumness. In particular, with their many internal states and with their many possible couplings to the environment, C60 was almost like a classical body that acted like our hypothetical nano-pen. When Zeilinger and coworkers sent a beam of fullerene molecules through an array of slits, they obtained an interference pattern due to their coherent wave-like nature. In their paper, they also said “we emphasize that for calculating the de Broglie wavelength, λ = h/Mv, we have to use the complete mass M of the object. Thus, each C60 molecule acts as a whole undivided particle during its center-of-mass propagation”.8 Zeilinger and colleagues were also able to control the rate of decoherence in the molecular beams by changing the pressure of the gas in the chamber.9 The presence of more gas molecules implied more collisions with the fullerene molecules, hence more opportunity for them to decohere. As more methane gas molecules entered the chamber, interference fringes became (weaker) fainter, which reflected the fact that “quantumness” is lost to decoherence. Zeilinger and coworkers said “In quantum 3

Joos et al. (2003). Papers by Zeh can be found in his website: www.zeh-hd.de. 5 Arndt et al. (1999). Also Hornberger (2003). 6 Kroto et al. (1985). 7 Kratschmer et al. (1990). 8 Arndt et al. (1999). Also Hornberger (2003). 9 Arndt et al. (1999). Also Hornberger (2003). 4

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interference experiments, coherent superposition only arises if no information whatsoever can be obtained, even in principle, about which path the interfering particle took. Interaction with the environment could therefore lead to decoherence”.10 In their analysis, quantum mechanics allowed Zeilinger and coworkers to determine how strongly the environment suppresses the interference. They found that their predictions agreed remarkably well with the experimental results. Eleven years later in 2011, Markus Arndt, Stefan Gerlich, and coworkers used large and massive tailor-made organic molecules in new high-contrast quantum experiments to show the quantum wave nature and delocalization of components composed of 430 atoms.11 They argued that complex systems with more than 1000 internal degrees of freedom can be prepared and isolated sufficiently well from the environment to show quantumness. The compound they used had a maximal size of 6 millionth of a millimeter (6 nm), comparable to the size of a small protein found in living organisms.

13.7 Pointer States and Einselection Quantum mechanics is extremely successful in practical applications. So far there is not a shred of experimental evidence that is in conflict with its predictions. Yet over half a century after its inception the debate about the relation between the quantum theory and the physical world we perceive continues. States of quantum systems are defined by a state function, also called the wavefunction, that evolves according to the deterministic linear Schrödinger’s equation. Linearity implies that any superposition of states is also a valid state that satisfies Schrödinger’s equation. In this regard, the state of a quantum system is a superposition of all the possible states that it could be in. This seems to conflict with our everyday experience of reality. Classical objects are either here or there, either in this or that state, but never seen in more than one place or state at the same time. The second problem with the quantum correspondence is the delicateness of the quantum states. Upon measurement, a general quantum state collapses suddenly into one of the possibilities in the superposition. Even though the deterministic evolution of the state function has been verified in many carefully controlled experiments, Schrödinger’s equation fails to describe the collapse. Measurement problem has a fascinating history. The Copenhagen interpretation proposed by Bohr has offered the widely accepted solution to the problem of how a simple outcome emerges from many possibilities.12,13 Bohr insisted that a classical apparatus is necessary for the measurement process to be completed, and thus quantum mechanics cannot be universal. At the core of the Copenhagen interpretation lies the dividing line between the classical and the quantum worlds. This line 10

Arndt et al. (1999). Also Hornberger (2003). Gerlich et al. (2011). 12 Wheeler et al. (1983). 13 Bohr (1928). 11

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separated the quantum and the classical worlds, where each world is governed by its own laws. According to Bohr, the line could not be fixed so that even the human brain and the nervous system could be used as a part of the quantum system provided that a classical device is available to carry out the measurement. This was an ingenious move on Bohr’s part. Physicists could now investigate the quantum world without paying attention to the interpretational or philosophical issues. Even though the thought experiments like Schrödinger’s cat and the problem of wavefunction collapse continued to baffle some, Bohr’s rule of thumb was sufficient for majority of the physicists. The theory of decoherence is at the core of modern understanding of the quantumclassical interface. Macroscopic quantum systems can never be isolated from their environment. Therefore, they should not be expected to follow Schrödinger’s equation, which is applicable only to closed systems. In this regard, systems usually considered as classical suffer from the natural loss of quantumness (quantum coherence), which leaks out into the environment.14,15 The resulting decoherence cannot be evaded when we consider the problem of wavefunction collapse. In quantum mechanics, there is nothing in the theory that selects particular states from all possible states and says that only these correspond to the allowed outcomes of a measurement. Decoherence theory changes all that, where one seeks answers in terms of the combined system-apparatus-environment wavefunction, and to what states it collapses to. Decoherence tends to mix things up. Given a quantum state prepared in a superposition, decoherence mixes and dilutes it until it is not recognizable in the initial quantum state. But if decoherence does this in a flash to every quantum state, then we would never be able to find anything about the quantum system. The fact that we are able to make reliable measurements is due to the robustness of certain quantum states that survive despite the toxic effect of decoherence. In other words, some states are special, even when inserted into an environment. Zurek calls these pointer states, because they correspond to the possible positions of a pointer on a dial of a measuring instrument.16 It is the existence of these well-defined and stable pointer states that make classical behavior of macroscopic quantum systems possible. In other words, environment, hence decoherence, does not destroy quantumness randomly: It selects certain states and throws away the others. Zurec calls this process environment-induced superselection or einselection. Survivors are the pointer states. Note that superposition of pointer states does not have this stability. Surviving decoherence doesn’t immediately mean that we will be able to measure them. Survival essentially means that the state is in principle measurable. We still have to get that information to detect the state. In other words, we have to find out how that information gets to the experimenter. A measuring equipment must always have a macroscopic classical element that the experimenter can interact with. This could be a pointer or a display large enough to 14

Zeh (1970). Zurek (1981), (1982). 16 Zurek (1981). 15

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be seen. The measuring equipment also acts as a part of the environment that interacts with the system being observed, and hence it also causes decoherence. As we said, this interaction is not completely destructive. Decoherence versus entanglement with the environment is actually how the information from the quantum system passes onto the environment. It is this what makes the pointer move and what makes the information accessible to the observer. Einselection is how the information gets filtered and reaches the observer through a process where only pointer states survive. An object’s, such as a dust grain, decoherence-inducing collisions with the air molecules carry away information about it, such as its position. By looking at the air molecules bouncing of the dust grain, one can deduce the position of the dust grain. In fact, the trajectories of the rebounding air molecules encode a kind of replica of the dust grain in the environment. In principle, whenever we determine the position or any other property of anything, we read it from its replica imprinted on the environment. It is important to emphasize that the information carried away from the object into the environment fundamentally changes the quantum state of the object. This change is not necessarily due to any transfer of momentum or energy between the object and the environment. Hence, it does not have anything to do with the Heisenberg uncertainty principle. Sure, due to the unevenness of the collisions from different directions some disturbance will be imparted onto the object. but decoherence does not depend on it. It results from a transfer of quantum information; when the object is entangled with the environment, where information about each object is no longer confined to the object itself. The role of decoherence in measurement is to create a replica or an imprint of the object in the environment that eventually produces a reading in our classical apparatus. One can consider an object’s properties to be measured to the degree that those properties are entangled with the environment and thus decohered. It is not important that the information imprinted on the environment is actually read by some observer or not. What matters is that information is available on the environment and could be read in principle. Pointer states are not always imprinted on the environment in the same robust and easily accessible way. As in the collisions with the air molecules, some environments are quite good at inducing decoherence of a quantum system, but not maintaining a sharply defined replica of it. Sure, we can reconstruct the position of an object from the air molecules bouncing of it, but that information is quickly degraded by the collisions with the other air molecules. Photons, on the other hand, are much better at retaining an image, since they generally do not collide with one another after they scattered from the object, thus information does not get destroyed very quickly. That is why most living things rely on vision. Some use smell, or sound, when vision won’t work.

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13.8 Quantum Darwinism The efficiency of imprinting images on the environment also depends on how strongly the system and the environment are coupled and on how a measurement is done. In some cases, using quantum mechanics it is possible to calculate how efficient this process is. Some quantum states are apparently better at generating replicas. In other words, they leave a more robust footprint on the environment. These are the states that we tend to measure, and they are the ones that eventually produce a unique classical picture. We could say that only the fittest states survive the measurement process, since they are the best ones in replicating images in the environment that a measuring device can detect. Zurek calls this the quantum Darwinism.17 In 2010, Jess Riedel and Wojciech Zurek showed how collisional decoherence, a common event in everyday life, leads to the spread of information about objects into the environment at extremely short time scales. A grain of dust 1μm across after illuminated by the sun for just 1μ s will have its location imprinted about 100 × 106 times in the scattered photons. Such explosive proliferation of replicas on the environment allows multiple observers to determine the position of the dust grain independently by examining the environment, thus establishing the fact that the object has an objective classical location. In general, one does not make a measurement by exhausting all the available information in the environment. One uses only a small fraction of the photons that scattered from the object. When we measure (observe) a property of a quantum system by looking at its replica, we essentially entangle it with the measuring apparatus, which destroys the replica. Is it then possible to drain all available information (copies) from the environment by repeated measurements? Sure, too much measurement, in principle, will eventually make the initial quantum state disappear. This is not something you should worry, if you are frequently looking at your vintage pen or classical painting. However, when you are looking at something small enough like a single quantum spin, take a peek and you will have used up all the available information. Another measurement will then may reveal another result. Measurement consumes some of the information that the environment holds about what is measured. In this new view of measurement, a quantum state can never imprint all that can be known about it on the environment, and that can be extracted in a single experiment. In classical physics, this is no problem, where different properties like mass, position, etc. could be measured separately. In quantum systems, this is no longer possible. Obtaining each piece of information introduces new entanglements that significantly alter some or all the rest of the system. Since we cannot get all the information about a system at once, we cannot duplicate it exactly either. In other words, cloning is not possible. To suppress decoherence, one naturally needs high vacuum and low temperatures. For large molecules, roughly the size of proteins, decoherence happens in room conditions almost immediately within 10−19 s. However, for perfect vacuum and at 17

Zurek (2009).

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the same temperature, one could maintain quantumness (coherence) for more than a week. While researchers continue to push the limits of seeing quantumness in mid-sized objects, they are also aware of the difficulties in suppressing environmentinduced decoherence as the physical size increases. Even though the decoherence can in principle be suppressed, In 2007, Johannes ˇ Kofler at the Max Planck Institute for Quantum Optics and Caslav Brunker at the University of Vienna thinks that we may still not be able to see quantum behavior in large objects due to the limited precession of the measurements, in which there is always a margin of finite error.18 In other words, it is not that the quantumness does not exist in macroscopic objects, but it is the fact that as the size increases, the discrete energy states get so close that eventually they blend into the continuum of the classical energies that we perceive. They also showed that for macroscopic objects classical physics emerges from quantum mechanics, not just because measurement is always imprecise to distinguish between successive quantum states, but also in those circumstances the specific laws that emerge from quantum mechanics are exactly the classical laws. That is, classical world is how the quantum world is perceived if you are macroscopic. Despite the technical difficulties in suppressing decoherence, on two occasions, we could get a glimpse of what macroscopic quantum mechanics looks like. Superconductivity is where we can conduct electricity with no electrical resistance−a property that some materials have at very low temperatures. When a material superconducts, a heavy magnet can be levitated, like magic, on top of it to display quantum mechanics visibly in action. Including the large hadron collider (LHC) at CERN all large-scale hadron colliders are built using superconducting magnets. Superfluidity is another quantum effect that allows ultracold liquid helium to creep up the sides of a container and out of the top. These events are visible by the naked eye, but they are not quantum in the sense that we have been talking. They are simply large-scale manifestations of the underlying principles of quantum mechanics. They are not superpositions, that is, two states at once.

18

Kofler et al. (2007).

Chapter 14

Mathematics, Physics, and Nature

During the first quarter of the twentieth century, new concepts like the wave-particle duality and the principle of uncertainty, which eventually led to the discovery of quantum mechanics, shook the foundations of classical physics. Similarly, the Galilean relativity gave way to Einstein’s special theory of relativity, where Einstein showed that there is a universal upper limit to how fast one can travel and that energy and mass are equivalent. Newton’s theory of gravitation yielded to Einstein’s theory of gravitation, when Einstein discovered that gravity is nothing but a manifestation of space-time curvature, thus leaving Euclidean space only as an approximation in the limit of small masses and slowly moving objects. Development in science does not always leave the theories of the past on the dusty shelves of libraries. Yes, the wave-particle duality, the principle of uncertainty, and a new type of determinism are all among the core concepts of quantum mechanics that are new to Newton’s theory. However, in the classical limit, where the Planck constant h goes to zero, quantum mechanics still reduces to Newton’s theory. In the realm of macroscopic objects, Newton’s theory is still the practical theory to use. In Einstein’s special theory of relativity, the speed of light is an upper limit to how fast an object can move. Compared to the speeds we encounter in our daily lives, the speed of light is so large that Galilean relativity still shapes our intuition. Even though Einstein’s theory of gravitation has replaced Newton’s theory, for low masses, where the curvature of space-time is negligible, Newton’s theory of gravitation remains an excellent working theory. We have been rather successful in describing physical processes in terms of mathematical models. However, mathematics is the world of numbers, and if we want to understand nature by mathematics, we have to represent nature and its processes in terms of numbers. But aside from integers, all the other numbers are constructs of the free human mind. Besides, mathematics has a certain logical structure to it, thus implying a closed or a complete system. Even though our universe is not chaotic, our knowledge of it will always be far too limited to be understood by logic. One naturally wonders why mathematics is so successful as a language in describing nature? © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1_14

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What could be the mystery behind this intriguing connection between physics and mathematics? In 1920, David Hilbert suggested that mathematics be formulated on a solid and complete logical foundation such that all mathematics be derived from a finite and a consistent system of axioms. This philosophy of mathematics is usually known as formalism. In 1931, Kurt Gödel shattered the foundations of the formal approach to mathematics by his famous incompleteness theorem. This theorem not only showed that Hilbert’s goal is unachievable but also proved to be the first in a series of deep and counterintuitive statements about the rigor and the provability of mathematics. Gödel’s theorem simply states that if you start with a set of axioms and prove theorems only by using these axioms, then there is no guarantee that eventually you will not run into a theorem that will conflict with your axioms. Extending your axiom set will not save you from this fate either. Could Gödel’s incompleteness theorem be the source of this mysterious link between mathematics and nature? It is true that certain mathematical models have been rather successful in expressing the law and order in the universe. However, this does not mean that all possible mathematical models and concepts will somehow and someday will find a place in physics. If it was possible to extend our understanding of nature by logical extensions of our existing theories, then physics would have been rather easy, and I might add−rather dull. An approach that has worked beautifully once, may not work another time. As Einstein once said to young Heisenberg, “A good trick should not be tried twice.” Sometimes physicists are so mesmerized by the sophistication and the beauty of their theories that they begin to lose contact with nature. We should not get upset and insist, if it turns out that nature has not chosen our way.

14.1 Laws of Nature At first, only the dynamical theory of Newton and his theory of gravitation existed. Then followed Maxwell’s theory of electromagnetism. After the discovery of quantum mechanics in the early twentieth century, there was a brief period when some scientists thought that everything in nature could in principle be explained in terms of four particles: electron, proton, neutron, and photon and the electromagnetic and gravitational interactions among them. But not for long, the discovery of strong and weak interactions along with a proliferation of new particles complicated the picture. Introduction of quarks as the new elementary constituents of particles did not help to simplify the picture either. Today, string theorists are trying to build a theory of everything in which all known interactions are unified. What is a true law of nature? We have been rather successful in expressing laws of nature in terms of differential equations, which are composed of the derivatives of the unknowns in our theories. However, for a complete description of physical phenomena we also need the boundary or the initial conditions. In our expressions of the natural laws, there are also some universal constants like the constant of gravity G, the speed of light c, and the Planck constant h, which are also called the fundamental

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constants of nature. From the earliest moments of the universe to its far reaches, as far as we know, the value of these constants has not changed. However, their values are so sensitive that slight changes in them renders the universe unsuitable for intelligent life as we know it. Understanding the inner workings of nature also allows us the ability to predict and control our environment. This helps us to develop technologies that make the universe a safer and a more predictable place to live. Our technologies are essentially based on our ability to manipulate the initial conditions. One naturally wonders whether intelligent life in the universe can ever develop enough to discover higher generations of theories where the natural constants appear as initial conditions. In such a case, could it ever become possible for intelligent life to develop technologies based on manipulations of these constants, at least locally in space-time? Developments in technology also bring new developments in science by making new and more accurate experiments possible, thus forcing the boundaries of our understanding of the universe. Aside from the natural constants like h, G, and c, there are also the curious transcendental numbers like π and e. A transcendental number is a number that is not algebraic−that is, it cannot be the root of a nonzero polynomial of finite degree with rational coefficients, where rational numbers are the numbers that can be expressed as the ratio of two integers. They appear in all sorts of unexpected and unrelated areas in mathematics and science. The mysterious number π not only appears in its relation to circles and spheres but also appears in places like the normal distribution, which among its many uses, is encountered in natural and social sciences to represent random processes. Normal distribution also gives the velocity distribution of atoms in gases. Among numerous places in physics, π appears in expressions like the quantum vacuum energy and the black hole temperature. It is possible that eventually genuine laws of nature, if they exist, will turn out to be relatively simple and in general that could be expressed as inequalities like the uncertainty principle: xp ≥ h/4π (14.1) and the second law of entropy: δS(total entropy of the universe) ≥ 0.

(14.2)

Others expressed in terms of equalities are most likely to be theories or models that are based on certain assumptions, and hence they will be the ones that are most prone to change in time. Sometimes, assumptions in science are made to make complex problems tractable. Confining ourselves to small velocities, low masses, uniform distributions, etc., are to name a few. However, the most dangerous assumptions are the ones that sneak into our theories without us noticing them. They look so natural that they don’t even appear as assumptions. Absolute space, universal time, and the principle of locality in classical physics are a few to name. Such assumptions may take decades or even centuries to recognize and when they are recognized and replaced with the correct

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concepts, dramatic changes take place in our basic understanding of the universe. As in the discovery of special relativity and quantum mechanics, sometimes the development of experimental techniques, like the developments in interferometry and spectroscopy, brings out contradictions and unexplained discrepancies from the predictions of the existing theories that refuse to go away, force us to think outside the box. An exception to this is the general theory of relativity. At the time, there was no immediate need for such a theory. Sure, there was the problem of the advance of the perihelion of Mercury’s orbit that did not behave as predicted by Newton’s theory of gravitation. As the orbit of Mercury followed an ellipse, its point of closest approach to sun, called the perihelion, advanced slowly due to the influences of the other planets. When all the effects were taken into account, there still remained an unexplained advance of the perihelion of Mercury by 43 s of arc per century. However, the general opinion was that there is no need for a new theory and that in due time Newton’s theory will eventually explain the discrepancy in terms of some yet unaccounted effect of other planets and their moons. It was only Einstein who considered this minute discrepancy to be a serious problem and discovered the general theory of relativity, which besides the correct amount of the perihelion of Mercury, predicted blackholes, expansion of the universe, etc. It was Einstein’s pure genius that brought us the general theory of relativity several decades ahead of its time. Today, measuring the anisotropies in the cosmic microwave background radiation− the relic radiation left from the Big Bang−per ESA Planck Satellite data indicate that our universe is 13.8 ∓ 0.02 billion years old and the Hubble constant H0 is 67.4 ∓ 0.5 km/sec/Mpc. However, measurements pertaining to the recent universe via the Hubble Telescope give a larger value for H0 as 74.2 ∓ 3.0 km/sec/Mpc.1 This discrepancy continues to baffle astronomers. While some of the astronomers think that as more data is gathered the discrepancy will eventually disappear, independent and accurate measurements continue to firmly establish its reality. Some astronomers even think that we may indeed need some new physics for its explanation.2 On the quantum side, due to recent technological developments, some of the experiments proposed as thought experiments became possible to do in laboratory. Including Bell’s inequality tests, Feynman’s double-slit thought experiment, Wheeler’s delayed choice experiment, and several other incredible experiments became possible. In fact, the 2022 Nobel Prize in physics was awarded to Alain Aspect, John F. Clauser, and Anton Zeilinger “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science.” So far there is not a shred of evidence against quantum mechanics. Or, may be there is, and we keep misinterpreting it. Development in science and our understanding of the universe usually come with surprises.

1 2

Riess (2020). Riess (2020).

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14.2 Could the New Physics Be Hiding in Living Matter? What separates life from inanimate matter is one of the mysteries of the universe. Sure, living matter is also made up of atoms and molecules that obey the laws of quantum mechanics. It is not just the incredible complexity of living matter that separates ordinary matter from living matter. Could the quantum weirdness be a part of what makes life special and miraculous? When you also add consciousness and mind into the equation, the problem becomes even more insurmountable. In 1930s, many of the prominent physicists like Bohr, Heisenberg, and Wigner believed that there is something different−some new physics−involved in living matter. Schrödinger said “One must be prepared to find a new kind of physical law prevailing in it (living matter)”.3 Even though Schrödinger didn’t make clear what he meant from new physics. The new physics reminds us of vitalism, a school of thought that dates back to Aristotle, which basically attempts to explain life in terms of a new kind of force or energy that pertains to living matter. Vitalism eventually lost credibility as the evidence for the chemical and the physical nature of the living matter and its processes accumulated. Astrobiologists are looking for life beyond earth. They are either looking for suitable environments for known life, or they look for the remains of once living organisms. They also search for signs of some other forms of alien life. However, whatever we are looking for as the evidence of life beyond earth, we still lack a general definition of living matter, and also a set of universal principles that would manifest detectable signatures of life or even pre-life.4 Whatever the definition of life may be, scientists are amazed by the abundance of life in all kinds of hostile environments. Such as in acidic environments, in deep ocean, where there is no light, and in space, where scientists thought life would never exist before. With the help of decoherence theory, we are now able to construct experiments that allow us to study the boundary where quantumness begin and end. Similarly, we also need experiments that would allow us to navigate between the living and the non-living matter.

14.3 Physics Versus Biology The separation between physics and biology is more than a matter of complexity; there is also a fundamental difference in conceptual framework and language. Physicists use concepts like energy, mass, entropy, temperature, force, and reaction rates. Biology, on the other hand, use terms like signals, codes, transcription, and translation, which is the language of information science. Genome editing technologies, including the amazing new CRISPR technology, allow scientists to edit the codebook of life.5 It appears that life is making full use of information storage and processing 3

Schrödinger (1944). Spencer (2011), p. 38. 5 Palermo et al. (2019), p. 30. 4

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techniques at all levels, not just in DNA. At the cellular level, physical mechanisms allow signaling between cells to lead to cooperative behavior. Similarly, ants and bees use complex information exchange mechanisms to show collective decision making.6,7 The human brain processes a staggering amount of information with incredible efficiency and uses only the energy equivalent of a dim light bulb, while it has the power of a megawatt supercomputer.8 From another standpoint, physics is also based on observation and perception, which is about how information flows from the source to the observer. Analyzing each such flow, we could derive the physical laws at the source that give rise to that flow. We should also emphasize that biology is not just about genes. Yes, genes play an important role in our health and determine our certain physical characteristics, but so do our behaviors and environment, such as what we eat and the social environment that we live in. Epigenetics is the branch of science that studies how our environment and behaviors can affect the way our genes work. Unlike genetic changes, epigenetic changes are reversible. They do not change our DNA sequence, but affect how our body reads a DNA sequence. Information storage is about reorganizing matter with an effective code in an environment where it is stable and retrievable, and hence it is basically governed by the laws of physics, which are normally immutable. But inflexible laws are not suitable for biological systems, whose evolution depends on a variety of ever-changing complex networks of local and global conditions.9,10 However, biological evolution is neither random nor chaotic. A more realistic description of changes in biological systems would be in terms of changes in their states. Information processing is like a software that can adapt easily and quickly to the changing state of the system and to the changing environment. Then, where do we find state-based description in physics? In fact, we have already discussed that the state-dependent description and dynamics is at the core of quantum mechanics. At what point then, could the transition from the classical ball-and-stick description of biology to the quantum mechanics make life happen? Could the quantum weirdness embodied in phenomena like superposition, entanglement, and tunneling be the secret behind life? The field of quantum biology and non-equilibrium quantum statistics, even though approached with some skepticism, is also under heavy scrutiny.11

14.4 Consciousness and Quantum Mechanics The controversial idea that quantum effects in the brain can explain consciousness passes a critical test as presented by Jack Tuszynski and co-workers from the Uni6

Peleg (2019), p. 66. Davies (2020), p. 34. 8 Loewenstein (1999). 9 Davies et al. (2019). 10 Goldenfeld et al. (2011). 11 McFadden et al. (2014). 7

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versity of Alberta in Canada at the Science of Consciousness Conference in Tucson Arizona on April 18, 2022. In 1990s, Physicist Roger Penrose and anesthesiologist Stuart Hameroff proposed the orchestrated objective reality (Orch OR) theory. They postulated that brain microtubules are the place where gravitational instabilities in the structure of space-time break the delicate quantum superposition between particles, which in turn give rise to consciousness. Due to the lack of experimental evidence, this theory remained in the fringes of the consciousness science. Tuszynski and coworkers have shown that anaesthetic drugs shorten the time it takes for microtubules to re-emit trapped light. Microtubules are hollow tubes that are made up of tubulin protein, which form the part of the skeletons of the plant and animal cells. When researchers shone blue light on microtubules and tubulin proteins for over several minutes, they observed that the light was caught in an energy trap inside the molecules and then re-emitted in a process called delayed luminescence. Tuszynski and colleagues suspect that this has quantum origin. They say it took hundreds of milliseconds for tubulin units to emit half of the light, and more than a second for full microtubules, which is comparable to the time scales that the human brain takes to process information. They also observed that the presence of anaesthetics quenches the delayed luminescence, thus decreasing the time it takes by about a fifth. They concluded that if the brain works at the brink of the classical and the quantum realms, then even a small amount of quenching could turn on or off consciousness. Similar results were obtained by Gregory Scholes and Aarat Kalra at the Princeton University. Critiques say that the phenomena seen in these experiments can also be explained by classical physics rather than quantum mechanics. However, they also add that the successes of the classical neuroscience do not preclude quantum mechanics playing a key role either. While these results support the idea that microtubules control consciousness at the level of brain cells, further research is needed before we could conclude about the role played by quantum mechanics in consciousness. https://blog. scientiststudy.com/2022/04/quantum-experiments-add-weight-to.html

14.5 Mathematics and Mind Almost everywhere mathematics is very useful and a powerful tool or language in expressing the law and order in the universe. However, mathematics is also a world of ideas, and these ideas occur as a result of some physical processes at the cellular and molecular levels in our brain. Today, not just our physical properties such as eye and hair color, but also the human psyche is thought to be linked to our genes. We have taken important strides in identifying parts of our genes that are responsible for certain properties. Research is ongoing in developing technologies that will allow to us to remove or replace parts of our genes that may represent potential hazard to our health. Scientists are working on mechanisms to turn off or silence toxic genes in a cell. Such efforts will eventually lead to the development of new medicines

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for protecting cells from harmful genes and treating diseases. Already smart drugs that specifically target diseased cells are quite effective on certain types of cancer. Even though we still have a long way to go, we have covered significant distance in understanding and controlling our genetic code. To understand and codify ideas in terms of some basic physical processes naturally requires a significantly deeper level of understanding of our brain and its processes. If ideas could be linked to certain physical processes at the molecular and cellular level that produce certain patterns of neural networks, then there could also exist a large but a finite upper limit to the number of ideas that we could ever come up with, no matter how absurd they may be. This limit basically implies that one’s brain has a finite phase space that allows only a finite number of configurations corresponding to ideas. Of course, most of these configurations will correspond to nonsense, but a few will correspond to great ideas. This also means that there is an upper limit to all the mathematical statements, theorems, concepts, etc., that we could ever come up with. We simply cannot think of anything that requires a process that either violates some of the working protocols or codes of the brain, which are naturally going to be based on the fundamental laws of nature, or requires a brain with a larger phase space. In the true sense of the words, it may not be possible to think outside the box. Some of the unsolved problems of mathematics may indeed be related to our brain’s finite capacity. Among the most difficult problems of mathematics, we could name Fermat’s last theorem, which was scribbled along the margin of a copy of an ancient book Arithmetica by Diophantus. Fermat’s comments in Latin (1637) can be translated as “It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.” This proposition remained unproven for almost four centuries and eventually proven by Andrew Wiles and Richard Taylor in 1995. Its proof came along with a number of discoveries like the algebraic number theory and the proof of the modularity theorem, tools none of which were available to Fermat. Even though the odds are against it, the situation begs the speculation that Fermat may still have possessed a simpler proof. Even though Fermat had continued to work on the special cases for n = 3 and 4 during the last thirty years of his life, he never published on this problem. It is most likely that he himself saw the flaws in what he called a “truly marvelous demonstration.” Among the exceptionally difficult problems of mathematics, the four-color problem, first conjectured by Guthrae in 1878, only got partially solved with the aid of computers in 1977. There are still other exceptionally difficult problems of mathematics. Some solved after many years, some still awaiting to be solved, and some yet to be discovered. Even for the solved ones, one cannot dismiss the existence of a better and more elegant proof. What is amazing is that in mathematics sometimes problems with extraordinary level of difficulty can be stated in terms of simple everyday language that everybody can understand. Maybe, this is what inspired Wheeler to say “if we really understood quantum mechanics, then we ought to be able to state it in one simple sentence.”

14.6 Cracking the Brain’s Memory Code

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A quick way to increase the phase space of the brain is to have a bigger brain. In fact, to some extent nature has already utilized this alternative. It is evidenced in fossils that as humans evolved, brain size increased dramatically. The average brain size of Homo habilis, who lived approximately 2 million years ago, was approximately 750 cc. Homo erectus, who lived 1.7−1 million years ago, averaged 900 cc in brain size. The modern human skull holds a brain of around 1400 cc. However, brain size and intelligence are only correlated loosely. A much more stringent limit to our mental capacity naturally comes from the inner efficiency of our brain and the number of internal states it accommodates. Research on subjects like brain stimulators, hard wiring of our brain, and mind reading machines are all aiming at a faster and much more efficient use of our brain.

14.6 Cracking the Brain’s Memory Code A better understanding of our brain’s inner workings may also bring a more efficient way of using our creativity, much needed at times of crisis or impasses, the workings of which are now left to chance. The possibility of tracing ideas to their origins in terms of physical processes at the molecular and cellular level and also the possibility of codifying them with respect to some finite, probably small, number of key processes implies that the relation between mathematics and nature may actually work both ways. One of the mysteries of the brain is how information is encoded. Despite a century of research, this problem continues to elude scientists. Neural synaptic connection strengths are expected to play a crucial role in the way information is encoded in the brain. However, synaptic components are short lived, while memories can last lifetimes. This suggests that synaptic information is encoded and hard-wired at a deeper level within the post-synaptic neuron. In the March 8, 2012, issue of the journal PloS Computational Biology, physicist Travis Craddock and Jack Tuszynski of the University of Alberta, and anesthesiologist Stuart Hameroff of the University of Arizona presented a plausible model for encoding synaptic memory in microtubules, which are the major components of the structural cytoskeleton within neurons. 12 Microtubules are cylindrical hexagonal lattice polymers of the protein tubulin. They constitute 15% of the total brain protein. These microtubules characterize the neuronal architecture and regulate synapses and are thought to process information by the interactive bit-like states of tubulin. But until Craddock et al. any resemblance to a common code connecting microtubules to synaptic activity was missing. Craddock and co-workers give a feasible and robust model for encoding synaptic information into structural and energetic changes of the microtubule lattices via calcium-activated calcium/calmodulin-dependent protein kinase kinase 2 (CaMKII) phosphorylation, which transforms the electrical information that a neuron receives 12

Craddock et al. (2012).

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for the building blocks of the cell. They demonstrate microtubule-associated protein logic gates and show how patterns of phosphorylated tubulins in microtubules can control neuronal functions by triggering axonal firings, regulating synapses, and traversing scale.

14.7 Is Mathematics the only Language for Nature? We have been extremely successful with mathematics in understanding and expressing the law and order in the universe. However, can there be other languages? Can the universe itself serve as its own language? It is known that intrinsically different phenomena occasionally satisfy similar mathematical equations. For example, in two-dimensional electrostatic problems, potential (x, y) satisfies the Poisson equation: ρ(x, y) − →2 , (14.3) ∇ (x, y) = − ε0 where ρ(x, y) is the charge density, and ε0 is the constant permittivity of vacuum. Now consider an elastic sheet stretched over a cylindrical frame like a drum head with uniform tension τ0 . If we push this sheet by small amounts at various points by rods, its displacement, u(x, y), from its equilibrium position will satisfy another Poisson equation: f (x, y) − →2 , (14.4) ∇ u(x, y) = − τ0 where f (x, y) is the force applied to the rods. If we now replace u(x, y) with (x, y) and f (x, y)/τ0 with ρ(x, y)/ε0 , then all the electrostatics problems with infinite charged sheets, long parallel wires, or charged cylinders will have a representation in terms of a stretched membrane. In fact, this method has been used to solve complex electrical problems. By pushing rods at various points to various heights against a membrane corresponding to the potentials of a set of electrodes, we can obtain the electric potential by simply reading the displacement of the membrane. The analogy can even be pushed further. If we put little balls on the membrane, their motion is approximately the corresponding motion of electrons in the corresponding electric field. This method has actually been used to obtain the complicated geometry of many photomultipliers.13 The limitation of this method is that Equation (14.4) is valid only for small displacements of the membrane. Also, the difficulty in preparing a membrane with uniform tension and density restricts the accuracy. However, the beauty of the method is that we can still find the solution to a complex boundary value problem without actually solving a partial differential equation. Note that even though we have not solved the boundary value problem explicitly, we still have used mathematics to link the two phenomena that are physically different. 13

Feynman et al. (1964)

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Recently, scientists have been intrigued by the uncanny similarity between the propagation of light in curved space-time and the propagation of sound in uneven flow. Scientists are now trying to exploit these similarities to gain insight into the microscopic structure of space-time. Even black holes have acoustic counterparts. Acoustic analogs of the Casimir effect, which is usually introduced as a purely quantum mechanical phenomenon, are now being investigated with technological applications in mind. The development of fast computers has slowed the progress of this approach. However, one could still drive important hints and information about the inner workings of this universe by using it as its own language. After all, quantum mechanics has taught us that everything is connected.

14.8 Nature and Mankind Let us now consider an imaginary civilization whose world is the entire collection of novels ever written on earth. Their members are mesmerized by the characters and their lives depicted in these novels. They wonder about the reason and purpose behind all the drama and the intricate relations among the characters. One day, one of their scientists comes up with a model and claims that all these novels are composed of a finite number of words and prepares a dictionary. They are all excited and begin to search every page, every paragraph, and every sentence they could find. In time, a few additions and subtractions come to this dictionary but what does not change is that their universe is composed of a finite number of words. Soon, a new scientist comes along and claims that all these words in the dictionary and the novels themselves are made up of a small number of letters, numbers, and punctuation marks. After intense testing, this theory also gains enormous support. As the quality of their observations increases, they begin to discover the grammar rules, which appear to hold in all the novels that they could check. These rules of grammar are the laws of nature in their universe. We all know that grammar rules alone cannot tell us why a novel is written and why it is great, but it is not possible to understand a novel properly without knowing the grammar rules either. Everybody has wondered why this universe exists and what our place is in this delicate yet elegant universe. Delicate, since the slightest tempering with the values of the natural constants, renders it unsuitable for intelligent life as we know it. Elegant, since everything so gracefully fits together. Even though no simple answers exist, it is incredible that almost everybody has somehow come to a peaceful coexistence with such questions. As Einstein once said, “The most incomprehensible thing about this universe is that it is comprehensible.”

Appendix

The Collected Papers of Albert Einstein (CPAE), published by Princeton University Press: Volume 5—The Swiss Years: Correspondence, 1902–1914. Edited by M.J. Klein, A.J. Kox, and R. Schulmann (1994). Niels Bohr Collected Works (BCW), published by North-Holland, Amsterdam: Volume 2—Work on Atomic Physics, 1912–1917. Edited by U. Hoyer, general editor L. Rosenfeld (1981). Volume 4—The Periodic System, 1920–1923. Edited by J.R. Nielsen (1977). Volume 6—Foundations of Quantum Physics I, 1926–1932. Edited by J. Kalckar, general editor F. E. Rüdinger (1985). Volume 7—Foundations of Quantum Physics II, 1933–1958. Edited by J. Kalckar, general editors F. Aaserud and E. Rüdinger (1996).

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Index

A Addition of velocities, 42 Alpha particles, 55 Angström, Anders, 62 Annus Mirabilis Miracle Year, 35 Anomalous Zeeman effect, 84 Arndt, Fritz Georg, 131 Arndt, Markus, 195 Aspect, Alain, 204 Bell’s inequality, 152 Aspelmeyer, Marcus, 155 Atoms and kinetic theory, 21

B Bach, Roger, 172 Balmer, Johann Jacob, 62 Balmer series, 62 Becquerel, Henri, 55 Bell, John, 186 Bell, John Steward Bohm’s theory, 148 Bell’s challenge, 151 Bell’s inequality, 150 aftermath, 155 Aspect, Alain, 152 Clauser, John, 152 loopholes, 154 third generation experiments, 154 Benjamin, Thomson, Sir, 19 Besso, Michele, 35 Beta particles, 55 Big bang, 204 Bit, 164 Black body, 24 Black body radiation, 26

Bloch, Felix, 96 Bohm, David Princeton years, 143 Bohm’s theory, 144 first reactions, 147 guiding equation, 145 Hamilton-Jacobi equation, 145 quantum potential, 145 Bohr at Cambridge, 57 Bohr at Manchester, 58, 59 Bohr atom first reactions, 64 Bohr atom model, 61 Bohr festspiele, 68 Bohr is stunned, 128 Bohr, Niels, 57, 89 festspiele, 68 triology, 64 Bohr’s festival, 87 Bohr-Sommerfeld model, 65 Bohr’s triology, 64 Boltzmann distribution, 29 gases, 30 Boltzmann, Ludwig, 21 Boltzmann’s death, 28 Born, Max, 77, 89, 90, 101, 104, 148 Born rebellious, 33 Bose-Einstein condensation, 32 Bose-Einstein distribution, 31 Bose-Einstein statistics, 29 Bragg, William L., 120 Brain’s memory code, 209 Brownian motion, 36, 39 Brown, Robert, 39 Brunker, Caslav , 199

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. S. ¸ Bayın, The Pursuit of Reality, https://doi.org/10.1007/978-981-99-1031-1

223

224 C Calculus, 7 Carnot, Sadi, 18 Cartesian coordinates, 40 Cavendish, Henry, 8 Chandrasekhar, Subrahmanyan, 81 Classical determinism, 107 Clauser, John, 152, 204 Clausius, Rudolf, 19, 20 Cloud chamber, 109 Cloud chamber tracks, 110 Coherence quantumness, 190 Complementarity, 116 Completely cuckoo, 132 Complimentarity, 109 Compton, Arthur, 71 Compton effect, 114 Compton scattering, 113 Compton’s discovery, 72 Conservation laws, 8 Consistent histories, 159 Cooper pairs, 179 Copenhagen interpretation, 117, 157, 161 Corpuscular theory, 12 Correlation coefficients, 150 Correspondence principle, 68, 91 Covariance, 42 Crab nebula, 84 Craddock, Travis, 209 CRISPR, 205 Curie, Marie, 55

D Darwin, Charles Galton, 59 Davisson, Clinton, 72 Davisson-Germer experiment, 72 De Broglie, Louis, 69 De Broglie theory, 71 De Broglie wavelength, 69 Deby, Pieter, 95 Decartes, Rene, 41 Decoherence, 164, 190, 192, 193 dust grain, 197 Decoherence theory Zeh, H. Diether, 194 Decoherent histories, 159 Delayed choice, 204 Delayed choice experiment, 175, 178 Demon in quantum, 67 Dirac, Paul, 94, 125 Double slit

Index Einstein’s version, 122 Feynman’s, 170 Scully and Drühl, 172 Drühl, Kai, 172 Dust grain decoherence, 197

E Eddington, Arthur, 82, 179 Eddington-Chandrasekhar contraversy, 82 Ehrenfest, Paul, 85 Eigenvalue equation, 98 Einselection, 197 Einstein, Albert, 33, 101, 108 difficult times, 34 patent office, 35 Einstein and quantum mechanics, 133 Einstein at Princeton, 133 Einstein-Bohr at Princeton, 140 Einstein-Bohr debate double slit, 122 Einstein-Podolsky-Rosen (EPR), 134 argument, 134 onslauth, 137 reactions, 137 Einstein’s double slit, 123 Einstein separability, 138 Electron ptychography, 40 Electron waves, 72 Element 72, 69 Energy and mass equivalence, 52 Energy-mass equivalence, 36 Entanglement, 162 Schrödinger, 138 Entropy, 23 Eötvos, 6 Ether, 43 Everett III, Hugh many-worlds interpretation, 157 Exclusion principle, 77, 80 Existence of atoms, 39

F Faraday, Michael, 58 Fermat’s theorem, 208 Fermi–Dirac distribution, 80 Feynman, Richard, 108 quantum mechanics, 156 Feynman’s double slit, 170, 171 Fictitious forces, 9 Fifth Solvay Conference, 119

Index FitzGerald, George, 43 Formalism, 202 Four-color problem, 208 Fourth quantum number, 80 Fowler, Ralph, 94 Frabboni, Stefano, 171 Franck-Hertz experiment, 65 Franck, James, 65 Frank, Alfred Eric, 131 Freedman, Stuart, 152 Fresnel, Augustine, 14 Freud, Sigmund , 131 Fullerene, 194 Fundamental constants, 203

G Galilean transformations, 42, 49 Gamow, George, 179 Geiger, Hans, 55 Gell-Mann, Murray many-worlds interpretation, 159 quantum mechanics, 156 General theory of relativity, 204 Genetic code, 208 Germer, Lester, 72, 73 Gibbs, Williard, 35 Gödel, Kurt, 202 Göttingen, 131 Goudsmit, Samuel, 85 Griffiths, Robert consistent histories, 159 Grossmann, Marcell, 33, 34

H Hameroff, Stuart, 207, 209 Hansen, Hans, 62 Heisenberg’s microscope, 112 Heisenberg’s quantum theory matrix algebra, 91 Heisenberg’s theory, 90 Heisenberg’s uncertainty, 187 Heisenberg, the magician, 90 Heisenberg, Werner, 87 complimentarity, 109 Helgoland, 90 Hertz, Gustav, 65 Hertz, Heinrich, 35 Herzog, Thomas, J. quantum eraser experiment, 176 Hevesy, Georg von, 58 Hidden variables, 145

225 von Neumann, 132 Hilbert, David, 131, 202 Homedyne detectors, 192 Hooft, Gerard’t quantum mechanics, 157 Hubble constant, 204 Huygens, Christien, 12 I Iconic formula, 52 Imaginary i wavefunction, 105 Imaginary numbers, 186 Incompleteness theorem, 202 Information era, 162 Interferometer Mach-Zehnder, 166 Invariance, 42 J Jordan, Pasqual, 92 Joules, James, 18 K Kalra, Aarat, 207 Kelvin, Lord, 19 Kim, Yoon-Ho, 178 Kinetic theory atoms, 21 Kirchhoff, Gustav, 25 Kofler, Johannes, 199 Kramers, Hendrik, 90 Kronig, Ralph, 84 L Landé, Arthur, 84 Langevin, Paul, 71 Laplace technique, 99 Large hadron collider, 44 Larmor, Joseph, 43 Lasers, 67 Laue, Max, 76 Leanard, Phillip, 38 Leggett’s inequality, 155 Leibniz, Gottfried, 6 Light-quanta, 37 Locality, 185 Local reality, 138, 149 Lorentz-FitzgGerald contraction, 49 Lorentz, Hendrik, 43, 121 Lorentz transformations, 43, 49

226 M Mach, Ernst, 11, 21 Mach’s principle, 11 Mach-Zehnder interferometer, 166 Mach-Zehnder interferometer, 166 with a channel blocker, 168 Many-worlds interpretation, 157 Maric, Mileva, 33 Mathematics as a language, 210 Matrices Heisenberg’s theory, 91 Ma, Xiao-song, 178 Maxwell, James Clerk, 15 Maxwell’s theory, 15 Meaning of Psi, 104 Measurement problem, 118 Mermin, David quantum mechanics, 156 Michelson, Morley, 42 Microscope Satellite, 6 Microtubules, 209 Millikan, Robert, 38 Minkowski, Hermann, 46, 48 Miracle Year, 35 Morley, Edward, 42 Moving frames, 41 Mutually exclusive events, 167

N Nano scales, 193 Navascués, Miguel, 186 Nernst, Walther, 19 Neumann, John von hidden variables, 147 wavefunction collapse, 183 Neutrinos, 9 Neutron stars, 80 Newton’s bucket, 10 Newton’s first law, 3 Newton’s gravitation, 5 first test, 8 Newton, Isaac, 6, 125 Newton’s second law, 3 Newton’s third law, 3 Nicholson, William, 61 Noncommutative algebra, 91 Nonlocal collapse wavefunction, 192 Nonlocality and EPR, 185 No signaling theorem, 163

Index O Objective reality, 134, 184 Oppenheimer, Robert, 132, 133 Bohm, David, 143 Orchestrated objective reality, 207 Ostwald, Wilhelm, 21, 34, 39

P Pais, Abraham, 36 Particle-wave duality, 69 Patent office, 34 Pauli at Princeton, 161 Pauli, Wolfgang, 76, 92 exclusion principle, 77 Penrose, Roger, 207 quantum mechanics, 157 Perpetual motion machine, 18 Perrin, Jean Baptiste, 40 Photoelectric effect, 36, 37 Photon, 28 Pilot wave theory de Broglie, 144 Planck-Einstein formula, 63 Planck, Max, 21, 24, 101 black body, 26 Planck’s radiation formula, 28 Planck’s son Ervin, 28 Planck’s student Frank, James, 28 Podolsky, Boris, 133 Poincare, Henri, 43 Pointer states, 196 Probabilistic interpretation of Psi, 106 Probability and causality, 107 Proietti, Massimiliano Wigner’s thought experiment, 184 Pulsars, 84

Q Quantum and complex numbers Navascués, Miguel, 186 Quantum coherence, 196 Quantum concept, 28 Quantum Darwinism Zurek, Wojciech, 198 Quantum eraser, 175 Quantum information, 164 Quantum mechanics footsteps, 76 imaginary numbers, 186

Index Quantum mechanics and consciousness, 207 Quantumness coherence, 190 Quantum potential, 145 Quantum reality, 136 Quantum theory footsteps, 87 Quantum tunneling, 179 Quarks, 202

R Radioactivity, 55 Radio waves, 35 Ramanujan, Sirinivasa, 82 Rayleigh-Jeans law, 25 Real valued quantum mechanics Pan, Jian-Wei, 187 Reichenbach, Hans, 46 Riedel, Jess, 198 Röntgen, Wilhelm, 55 Rosefeld, Leon, 137 Rosen, Nathan, 133 Rubens, Heinreich, 25 Rutherford, Ernest, 56 Rutherford model, 57 Rutherford’s atom, 57 Rydberg constant, 63

S Scholes, Gregory, 207 Schrödinger at Copenhagen, 101 Schrödinger’s cat, 139, 189 Schrödinger’s equation, 98, 100 Schrödinger Erwin, 138 living matter, 205 wave mechanics, 95 Schrödinger’s hydrogen atom, 98 Scully-Drühl double slit, 172 Scully, Marlan, 172 Second law of thermodynamics, 203 Shell model, 68, 78 Sixth Solvay Conference, 127 Bohr could not believe his eyes, 129 Bohr is stunned, 128 Bohr turns the tables, 130 Einstein accepts, 129 Einstein’s lightbox, 128 Smart drugs, 208 Soddy, Frederick, 59 Solvay, Ernest, 119 Sommerfeld, Arnold, 65, 76, 88, 101

227 Space and time, 45 Space-time, 46 regions, 50 Special relativity, 36, 40 postulates, 44 Speed of light, 44 Spherical harmonics, 98 Spin, 85 Spin discovery, 86 Spin-orbit effect, 84 Spontaneous emission, 67 Stark effect, 65, 94 State function wavefunction, 190 State of a system, 182 Stephen-Boltzmann law, 27 Stimulated emission, 67 Stoner, Edmund, 79 Superconductivity, 32, 199 Superfluidity, 32, 199 Superposition, 182, 189 Synaptic information, 209

T Taylor, Richard Fermat’s theorem, 208 Tegmark, Max, 159 Thermodynamic equilibrium, 23 Thermodynamics laws, 19 Thomson, George Paget, 73 Thomson, J.J., 56, 57 Time dilation, 50 Toxic genes, 208 Tumultuous years in Europe, 131 Tuszynski, Jack, 209 consciousness theory, 207

U Uhlenbeck, George, 85, 101 Uncertainty principle, 203 cloud chamber tracks, 112 Uncertainty relation matrices, 93 Unitary, 182 Universal wavefunction, 157 University of Istanbul, 131 Uranic rays, 55

V von Neumann, John, 132

228 von Neumann’s proof, 147 W Watt, James, 18 Wavefunction, 107 state function, 190 Wavefunction (Psi) wavefunction, 104 Wavefunction collapse, 106 Wigner, Eugene, 183 Wave mechanics, 95 first reactions, 101 Wave-particle duality, 69, 116 uncertainty, 114 Weber, Heinrich, 33 Weinberg, Steven, 92 Weyl, Herman, 95 Wheeler, John, 194, 204 many-worlds interpretation, 157 White dwarfs, 80 Wien-Sommerfeld argument, 88

Index Wien, Wilhelm, 25, 88 Wigner, Eugene, 184 thought experiment, 183 Wigner’s thought experiment done, 184 Wiles, Andrew Fermat’s theorem, 208 Wilson, C.T.R., 109 Worldline, 51

Y Young’s double slit, 13 Young, Thomas, 12

Z Zeeman effect, 65, 77 Zeeman, Pieter, 65 Zeh, H. Diether, 194 Zeilinger, Anton, 155, 194, 204 Zurek, Wojciech, 198