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 other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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