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Auletta | Wang

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Quantum Mechanics for Thinkers provides a quick access to quantum mechanics without dealing with a true textbook that demands proper specialized studies in physics and related mathematics.

The book consists of three parts: Basic Issues, in which the basic notions of quantum mechanics are introduced; Formal Issues, in which more advanced topics are discussed; and Ontological Issues, in which the conceptual and interpretational problems are dealt with. The book deals with the most recent developments in quantum information and non-locality. It comprises 70 figures, which are a crucial instrument to acquaint the readers with abstract problems in a representative way, and 30 in-section boxes, which assist the readers to solve even the most difficult mathematical problems. The book has about 130 problems (most of them solved) to help the readers test themselves and verify how well they were able to understand the topics.

One of its kind, this is the only book available in the market that introduces undergraduate students in physics and scholars of adjacent fields (chemistry, mathematics, engineering, information science, biology, and philosophy) to the study of such a difficult field in an easy-to-understand language. It also offers a considerable number of clear and analytical treatments for what are considered the most difficult conceptual problems of the theory. Although a textbook, and therefore not a popular book, it can also prove useful for a very bright spectrum of potential readers.

Gennaro Auletta is a senior researcher at the University of Cassino, Italy. Prof. Auletta has authored and coauthored 17 books, including Quantum Mechanics (with G. Parisi and M. Fortunato; Cambridge University Press, 2009, 2013) and Cognitive Biology: Dealing with Information from Bacteria to Minds (Oxford University Press, 2011), and more than 70 papers. His areas of interest are metaphysics, philosophy of nature, logic, foundations and interpretation of quantum mechanics, quantum information, system biology, cognitive biology,

top-down causation in biology and neurosciences, and mathematical definition of complexity.

Shang-Yung Wang is an associate professor of physics at Tamkang University, Taiwan. His current research interests include elementary particles, cosmology, quantum information, and foundations of quantum mechanics.

Quantum Mechanics for Thinkers

ISBN 978-981-4411-71-4V367

QUANTUM MECHANICS for THINKERS

Gennaro AulettaShang-Yung Wang

QUANTUM MECHANICS for THINKERS

for the WorldWind PowerThe Rise of Modern Wind Energy

Preben MaegaardAnna KrenzWolfgang Palz

editors

Pan Stanford Series on Renewable Energy Volume 2

QUANTUM MECHANICS for THINKERS

Gennaro AulettaShang-Yung Wang

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Published by

Pan Stanford Publishing Pte. Ltd.

Penthouse Level, Suntec Tower 3

8 Temasek Boulevard

Singapore 038988

Email: editorial@panstanford.com

Web: www.panstanford.com

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Quantum Mechanics for Thinkers

Copyright c 2014 Pan Stanford Publishing Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying,recording or any information storage and retrieval system now known or tobe invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying

fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive,

Danvers, MA 01923, USA. In this case permission to photocopy is not

required from the publisher.

ISBN 978-981-4411-71-4 (Hardcover)

ISBN 978-981-4411-72-1 (eBook)

Printed in the USA

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To our families

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Contents

Foreword xi

Introduction 1

PART I BASIC ISSUES: STATES

1 Classical Mechanics 91.1 Classical-Mechanical Description 9

1.2 Basic Principle of Classical Mechanics 12

1.3 Summary 15

2 Superposition Principle 172.1 Origin and Foundations of Quantum Mechanics 17

2.2 Classical and Quantum Superposition 18

2.3 A Photon in an Interferometer 20

2.4 Probability Amplitudes 24

2.5 Formulation of the Superposition Principle 26

2.6 Transmission, Reflection, and Phase Shift 27

2.7 Action of the Second Beam Splitter 33

2.8 Computing the Detection Probabilities 35

2.9 Summary 37

3 Quantum States as Vectors 393.1 Photon Polarization 39

3.2 Action of the Polarization Filter 40

3.3 Vector Spaces and Bases 42

3.4 Scalar Products and Brackets 44

3.5 Polarization Filters as Projectors 49

3.6 Projectors as Matrices 51

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viii Contents

3.7 Action and Properties of Projectors 55

3.8 Summary 59

4 Bases and Operations 614.1 Corpuscular Nature of Light 61

4.2 Further Experimental Evidences 65

4.3 Quantization Principle 67

4.4 Quantum Observables in General 69

4.5 Different Bases and Superposition 72

4.6 Change of Basis as a Unitary Transformation 75

4.7 Not all Operations Commute 80

4.8 Features vs Properties 84

4.9 Summary 85

5 Complementarity Principle 875.1 Undulatory Nature of Matter 87

5.2 Interferometry with a Blocked Path 88

5.3 Classical and Quantum Probability 90

5.4 Double Slit Experiment 93

5.5 Path Predictability and Interference Visibility 96

5.6 Delayed Choice Experiment 101

5.7 Summary 103

PART II FORMAL ISSUES: OBSERVABLES

6 Position and Momentum 1076.1 Position Operator: Discrete Case 107

6.2 From Summation to Integration 110

6.3 Position Operator: Continuous Case 117

6.4 Derivatives: From Finite to Infinitesimal Quantities 123

6.5 Partial and Total Derivatives 132

6.6 Momentum as Generator of Space Translations 136

6.7 Momentum Representation 143

6.8 Commutation and Uncertainty Relations 147

6.9 Conceptual Aspects of the Uncertainty Relations 153

6.10 Summary 156

7 Energy and Quantum Dynamics 1577.1 Hamiltonian and Classical Dynamics 157

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Contents ix

7.2 Schrodinger Equation 160

7.3 Time Evolution as a Unitary Transformation 162

7.4 Active and Passive Transformations 166

7.5 Schrodinger and Heisenberg Pictures 171

7.6 Free Particle 174

7.7 Harmonic Oscillator 178

7.8 Density Matrix 187

7.9 Composite Systems 192

7.10 Summary 198

8 Angular Momentum and Spin 2018.1 Angular Momentum as Generator of Rotations 202

8.2 Angular Momentum Operator 207

8.3 Quantization of Angular Momentum 210

8.4 Angular Momentum Eigenfunctions 216

8.5 Central Potential and the Hydrogen Atom 223

8.6 Spin Angular Momentum 235

8.7 Addition of Angular Momenta 244

8.8 Identical Particles and Spin 248

8.9 Summary 253

PART III ONTOLOGICAL ISSUES: PROPERTIES

9 Measurement Problem 2579.1 Statement of the Problem 257

9.2 Density Matrix and Projectors 260

9.3 Projection Postulate 262

9.4 Basis Ambiguity 266

9.5 Role of the Environment 269

9.6 Entropy and Information 271

9.7 Reversibility and Irreversibility 280

9.8 Schrodingers Cat 289

9.9 Summary 292

10 Non-Locality and Non-Separability 29310.1 EPR Paper 293

10.2 Bohrs and Schrodingers Criticism of EPR 300

10.3 EPRBohm Experiment 303

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x Contents

10.4 Bell Theorem 306

10.5 Entanglement Swapping 316

10.6 Eberhard Theorem 319

10.7 KochenSpecker Theorem 325

10.8 Summary 331

11 Quantum Information 33311.1 Nature of Information 333

11.2 Information Accessibility 336

11.3 Potential Information 344

11.4 Quantum Computation 348

11.5 Quantum Teleportation 364

11.6 Quantum Cryptography 369

11.7 Mutual Information and Entanglement 374

11.8 Information and Non-Separability 386

11.9 Summary 390

12 Interpretation 39312.1 Information Acquisition 393

12.2 Bounds on Information Acquisition 400

12.3 Operations 408

12.4 Theoretical Entities 410

12.5 Fundamental Information Triad 413

12.6 Summary 416

Bibliography 417Author Index 433Subject Index 437Solutions to Selected Problems 445

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Foreword

The discovery of quantum mechanics and its comprehension are at

the basis of the foundations of modern technology. This fact is not