Quantum Mechanics Downloaded from www.worldscientific.com QUANTUM MECHANICS For Electrical Engineers
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Quantum Mechanics Downloaded from www.worldscientific.com QUANTUM MECHANICS For Electrical Engineers Isaak Mayergoyz University of Maryland, USA World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI TOKYO
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Quantum Mechanics Downloaded from www.worldscientific.com Library of Congress Cataloging-in-Publication Data Names: Mayergoyz, I. D., author. Title: Quantum mechanics for electrical engineers / Isaak Mayergoyz (University of Maryland, USA). Description: Hackensack, New Jersey : World Scientific, 2016. Includes bibliographical references and index. Identifiers: LCCN 2016038481 ISBN 9789813146907 (hardcover) ISBN 9789813148017 (pbk) Subjects: LCSH: Quantum theory. Classification: LCC QC174.12.M376 2016 DDC 530.12--dc23 LC record available at https://lccn.loc.gov/2016038481 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be 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. Printed in Singapore
Quantum Mechanics Downloaded from www.worldscientific.com To my grandson Jacob
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Preface Quantum Mechanics Downloaded from www.worldscientific.com You have in your hands a book on quantum mechanics. This book is designed as a graduate course for electrical engineers, and it reflects the experience of the author in teaching quantum mechanics to graduate students in the Electrical and Computer Engineering Department of the University of Maryland, College Park. This book can also be used for teaching of quantum mechanics to graduate students in materials science and engineering departments as well as to applied physicists. The development of quantum mechanics in the late 1920s has been universally acclaimed as one of the greatest achievements of modern physics. This development eventually resulted in the emergence of new technologies such as lasers, nuclear power, semiconductor electronics, superconducting devices, spintronics, quantum computing, etc. This, in turn, led to the penetration of quantum mechanics into electrical and computer engineering curriculum in order to provide the theoretical foundation for graduate courses dealing with the above-mentioned technologies. This state of affairs is similar to the penetration of electromagnetic field theory into electrical engineering curriculum that occurred about eighty years ago. There are many excellent books on quantum mechanics. However, most of these books are written for physicists, and this is reflected in the selection of the topics covered in those books. The selection of many topics covered in this book is based on their relevance to engineering applications. In other words, this book is designed to serve as the theoretical foundation for graduate courses in quantum optics and lasers, semiconductor electronics, applied superconductivity and quantum computing, which are currently being taught in many electrical and computer engineering departments. To achieve the above goal, the decision was made to cover (along with traditional subjects) the following topics: tunneling, including resonant vii
viii Quantum Mechanics for Electrical Engineers Quantum Mechanics Downloaded from www.worldscientific.com tunneling and Josephson tunneling in superconductors; Landau levels and their relation to the integer quantum Hall effect; Bloch waves, band structures, effective mass Schrödinger equation and semiclassical transport; energy gap formation in superconductors; quantum transitions in two-level systems and Rabi flopping frequency; Berry phase and Berry curvature; density matrix and optical Bloch equation for two-level systems; Wigner function and quantum transport; and exchange interaction and spintronics. The book also contains a short chapter covering the very basic facts related to Lagrangian and Hamiltonian mechanics. The reason is that electrical engineers are usually unfamiliar with this form of classical mechanics, whereas the Lagrangian and Hamiltonian mechanics have many intimate connections with the mathematical formalism of quantum mechanics. The bibliography contains a short list of books on quantum mechanics and related topics. This list is not exhaustive but rather suggestive. A special effort has been made to produce a relatively short book that can be used for one semester course in quantum mechanics. Naturally, this could only be achieved by omitting many topics. In particular, many important facts related to atomic, molecular and nuclear physics are not discussed in the book. In undertaking this project, the intention was to produce a studentfriendly textbook on quantum mechanics. It has been realized that this is a very difficult task because many facts and concepts of quantum mechanics are very remote from (and contradictory to) our everyday macroscopic experience. Furthermore, the mathematical formalism of quantum mechanics is quite sophisticated. For these reasons, every effort was made to introduce quantum mechanical concepts and related mathematical formalism in a straightforward way and to strive to achieve clarity and precision in their exposition. It is for students to judge to what extent these efforts have been successful. I wish to express my sincere gratitude to Dr. Nathan A. Moody who carefully assembled and edited the first version of my lecture notes on quantum mechanics. I am greatly indebted to Dr. Patrick C. McAvoy for reading the manuscript and providing the valuable suggestions for its improvement. I am also very grateful to Priscilla Tang for all her help in the preparation of the book manuscript. My thanks to Chelsea Chin from World Scientific Publishing Company for her assistance and patience. I gratefully acknowledge the financial support derived from the Alford L. Ward Professorship that made this project possible.
Contents Quantum Mechanics Downloaded from www.worldscientific.com Preface 1. Basics of Lagrangian and Hamiltonian Mechanics 1 1.1 What is Quantum Mechanics................ 1 1.2 The Principle of Least Action and Lagrange Equations.. 3 1.3 Conservation Laws...................... 9 1.4 Hamiltonian Equations and Poisson Brackets....... 14 Problems 20 2. Fundamentals of Quantum Mechanics 23 2.1 Hilbert Space and Hermitian Operators.......... 23 2.2 Postulates of Quantum Mechanics............. 32 2.3 Operators of Coordinates and Momentum......... 42 2.4 Angular Momentum Operators............... 51 2.5 Hamiltonian Operator.................... 63 Problems 73 vii 3. Tunneling 75 3.1 Tunneling Through a Rectangular Barrier......... 75 3.2 Resonant Tunneling..................... 84 3.3 Josephson Tunneling in Superconductors.......... 91 Problems 102 4. Problems with Discrete Energy Spectrum 103 4.1 The Potential Well...................... 103 4.2 The Harmonic Oscillator................... 114 ix
x Quantum Mechanics for Electrical Engineers 4.3 The Hydrogen Atom..................... 127 4.4 Landau Levels. Quantum Hall Effect............ 137 4.5 Model Problem for Energy Gap Formation in Superconductors....................... 147 Problems 157 Quantum Mechanics Downloaded from www.worldscientific.com 5. Periodic Potentials 159 5.1 Bloch Function and Energy Bands............. 159 5.2 Kronig-Penney Model.................... 167 5.3 Effective Mass Schrödinger Equation and Semiclassical Transport........................... 173 Problems 182 6. Matrix Form of Quantum Mechanics. Perturbation Theory. Density Matrix 183 6.1 Matrix Form of Quantum Mechanics............ 183 6.2 Time-Independent Perturbation Theory.......... 196 6.3 Time-Dependent Perturbations. Adiabatic Perturbations and Berry s Phase...................... 209 6.4 Density Matrix. Optical Bloch Equation. Wigner Function and Quantum Transport.............. 225 Problems 237 7. Spin. Identical Particles. Second Quantization. Quantization of Electromagnetic Field 239 7.1 Spin.............................. 239 7.2 Identical Particles. Exchange Interaction.......... 254 7.3 Second Quantization. (Representation of Occupation Numbers)............................ 270 7.4 Quantization of Electromagnetic Field........... 280 Problems 292 Bibliography 295 Index 297