Quantum Mechanics II. Advanced Topics

433 Pages·2023·17.381 MB·English

by  S. Rajasekar, R. Velusamy

Quantum Mechanics II
Quantum Mechanics II: Advanced Topics offers a comprehensive exploration of the state-of-the-
art in various advanced topics of current research interest. A follow-up to the authors’ introductory
book Quantum Mechanics I: The Fundamentals, this book expounds basic principles, theoretical
treatment, case studies, worked-out examples and applications of advanced topics including quan-
tum technologies.
A thoroughly revised and updated this unique volume presents an in-depth and up-to-date
progress on the growing topics including latest achievements on quantum technology. In the
second edition six new chapters are included and the other ten chapters are extensively revised.
Features
• Covers classical and quantum field theories, path integral formalism and supersym-
metric quantum mechanics.
• Highlights coherent and squeezed states, Berry’s phase, Aharonov—Bohm effect and
Wigner function.
• Explores salient features of quantum entanglement and quantum cryptography.
• Presents basic concepts of quantum computers and the features of no-cloning theorem
and quantum cloning machines.
• Describes the theory and techniques of quantum tomography, quantum simulation
and quantum error correction.
• Introduces other novel topics including quantum versions of theory of gravity, cosmol-
ogy, Zeno effect, teleportation, games, chaos and steering.
• Outlines the quantum technologies of ghost imaging, detection of weak amplitudes
and displacements, lithography, metrology, teleportation of optical images, sensors,
batteries and internet.
• Contains several worked-out problems and exercises in each chapter.
Quantum Mechanics II: Advanced Topics addresses various currently emerging exciting
topics of quantum mechanics. It emphasizes the fundamentals behind the latest cutting-edge
developments to help explain the motivation for deeper exploration. The book is a valuable
resource for graduate students in physics and engineering wishing to pursue research in quan-
tum mechanics.

Quantum Mechanics II
Advanced Topics
Second Edition
S. Rajasekar and R. Velusamy
Second edition published 2023
by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
and by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
© 2023 S. Rajasekar and R. Velusamy
First edition published by CRC Press 2014
CRC Press is an imprint of Informa UK Limited
The right of S. Rajasekar and R. Velusamy to be identified as authors of this work has been asserted in accordance with sections 77 and 78
of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging‑in‑Publication Data
Names: Rajasekar, S. (Shanmuganathan), 1963- author. | Velusamy, R., 1952-
author.
Title: Quantum mechanics / S. Rajasekar, R. Velusamy.
Description: Second edition. | Boca Raton : CRC Press, 2022. | Includes
bibliographical references and index. | Contents: v. 1. The fundamentals
-- v. 2. Advanced topics. | Summary: “Quantum Mechanics I: The
Fundamentals provides a graduate-level account of the behavior of matter
and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It
covers basic concepts, mathematical formalism, and applications to
physically important systems. This fully updated new edition addresses
many topics not typically found in books at this level, including: Bound
state solutions of quantum pendulum Morse oscillator Solutions of
classical counterpart of quantum mechanical systems A criterion for
bound state Scattering from a locally periodic potential and
reflection-less potential Modified Heisenberg relation Wave packet
revival and its dynamics An asymptotic method for slowly varying
potentials Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and
Bell’s theorem Delayed-choice experiments Fractional quantum mechanics
Numerical methods for quantum systems A collection of problems at the
end of each chapter develops students’ understanding of both basic
concepts and the application of theory to various physically important
systems. This book, along with the authors’ follow-up Quantum Mechanics
II: Advanced Topics, provides students with a broad, up-to-date
introduction to quantum mechanics. Print Versions of this book also
include access to the ebook version”-- Provided by publisher.
Identifiers: LCCN 2022021033 | ISBN 9780367769987 (v. 1 ; hardback) | ISBN
9780367776367 (v. 1 ; paperback) | ISBN 9781003172178 (v. 1 ; ebook) |
ISBN 9780367770006 (v. 2 ; hardback) | ISBN 9780367776428 (v. 2 ;
paperback) | ISBN 9781003172192 (v. 2 ; ebook)
Subjects: LCSH: Quantum theory.
Classification: LCC QC174.12 .R348 2022 | DDC 530.12--dc23/eng20220518
LC record available at https://lccn.loc.gov/2022021033
ISBN: 978-0-367-77000-6 (hbk)
ISBN: 978-0-367-77642-8 (pbk)
ISBN: 978-1-003-17219-2 (ebk)
DOI: 10.1201/9781003172192
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
To our teachers.

Contents
Preface xiii
About the Authors xvii
1 Quantum Field Theory 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why Quantum Field Theory? . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 What is a Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Quantum Equations for Fields . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Quantization of Nonrelativistic Wave Equation . . . . . . . . . . . . . . . 8
1.7 Electromagnetic Field in Vacuum . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Interaction of Charged Particles with Electromagnetic Field . . . . . . . . 17
1.9 Quantization of Klein–Gordon Equation . . . . . . . . . . . . . . . . . . . 20
1.10 Quantization of Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Gauge Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.12 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.13 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Path Integral Formulation 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Time Evolution of Wave Function and Propagator . . . . . . . . . . . . . 34
2.3 Path Integral Representation of Propagator . . . . . . . . . . . . . . . . . 35
2.4 Connection Between Propagator and Classical Action . . . . . . . . . . . 36
2.5 Schr¨odinger Equation From Path Integral Formulation . . . . . . . . . . . 39
2.6 Transition Amplitude of a Free Particle . . . . . . . . . . . . . . . . . . . 40
2.7 Systems with Quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . 42
2.8 Path Integral Version of Ehrenfest’s Theorem . . . . . . . . . . . . . . . . 48
2.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Supersymmetric Quantum Mechanics 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Supersymmetric Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Relations Between the Eigenstates of Two Supersymmetric Hamiltonians 58
3.4 Hierarchy of Supersymmetric Hamiltonians . . . . . . . . . . . . . . . . . 61
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Generation of Complex Potentials with Real Eigenvalues . . . . . . . . . 66
3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
vii
viii Contents
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Coherent and Squeezed States 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 The Uncertainty Product of Harmonic Oscillator . . . . . . . . . . . . . . 76
4.3 Coherent States: Definition, Uncertainty Product and
Physical Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Generation and Properties of Coherent States . . . . . . . . . . . . . . . . 80
4.5 Spin Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Coherent States of Position-Dependent Mass Systems . . . . . . . . . . . 87
4.7 Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.8 Deformed Oscillators and Nonlinear Coherent States . . . . . . . . . . . . 94
4.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Berry’s Phase, Aharonov–Bohm and Sagnac Effects 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Derivation of Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Origin and Properties of Berry’s Phase . . . . . . . . . . . . . . . . . . . 108
5.4 Classical Analogue of Berry’s Phase . . . . . . . . . . . . . . . . . . . . . 109
5.5 Berry’s Phase in Solid State Physics . . . . . . . . . . . . . . . . . . . . . 111
5.6 Examples and Effects of Berry’s Phase . . . . . . . . . . . . . . . . . . . . 113
5.7 Applications of Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.8 Experimental Verification of Berry’s Phase . . . . . . . . . . . . . . . . . 116
5.9 Pancharatnam’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.10 Cumulants Associated with Geometric Phases . . . . . . . . . . . . . . . 118
5.11 The Aharonov–Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.12 Sagnac Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Phase Space Picture and Canonical Transformations 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Squeeze and Rotation in Phase Space . . . . . . . . . . . . . . . . . . . . 132
6.3 Linear Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Time Evolution of the Wigner Function . . . . . . . . . . . . . . . . . . . 139
6.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.7 Advantages of the Wigner Function . . . . . . . . . . . . . . . . . . . . . 146
6.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Quantum Entanglement 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 States in Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Quantum Entangled States . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Contents ix
7.6 Separability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.7 Multipartite Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.8 Quantifying Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.9 Applications of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8 Quantum Decoherence 179
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2 Decoherence and Interference Damping . . . . . . . . . . . . . . . . . . . 180
8.3 Interaction of a Detector on the Double-Slit Experiment . . . . . . . . . . 181
8.4 Decoherence Due to Phase Randomization . . . . . . . . . . . . . . . . . 182
8.5 Position Decoherence Due to Environmental Scattering . . . . . . . . . . 185
8.6 Master Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.7 Decoherence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.8 Decoherence Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.9 The Role of Decoherence in the Interpretation of Quantum Mechanics . . 195
8.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9 Quantum Computers 201
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 What is a Quantum Computer? . . . . . . . . . . . . . . . . . . . . . . . 201
9.3 Why is a Quantum Computer? . . . . . . . . . . . . . . . . . . . . . . . . 204
9.4 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.5 Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.6 Testing Quantum Computers Using Grover’s Algorithm . . . . . . . . . . 220
9.7 Features of Quantum Computation . . . . . . . . . . . . . . . . . . . . . . 221
9.8 Quantum Computation Through NMR . . . . . . . . . . . . . . . . . . . 221
9.9 Why is Making a Quantum Computer Extremely Difficult? . . . . . . . . 222
9.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10 Quantum Cryptography 229
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.2 Standard Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.3 Quantum Cryptography–Basic Principle . . . . . . . . . . . . . . . . . . . 231
10.4 Types of Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . . . 233
10.5 Multiparty Quantum Secret Sharing . . . . . . . . . . . . . . . . . . . . . 239
10.6 Applications of Quantum Cryptography . . . . . . . . . . . . . . . . . . . 241
10.7 Implementation and Limitations . . . . . . . . . . . . . . . . . . . . . . . 242
10.8 Fiber-Optical Quantum Key Distribution . . . . . . . . . . . . . . . . . . 243
10.9 Quantum Cheque Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
10.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248