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Quantum Mechanics for Beginners OUP CORRECTED PROOF – FINAL, 10/3/2020, Spi OUP CORRECTED PROOF – FINAL, 10/3/2020, Spi OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi Quantum Mechanics for Beginners OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi Quantum Mechanics for Beginners with applications to quantum communication and quantum computing M. Suhail Zubairy Texas A&M University 1 OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © M. Suhail Zubairy 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020934341 ISBN 978–0–19–885422–7 (hbk.) ISBN 978–0–19–885423–4 (pbk.) DOI: 10.1093/oso/9780198854227.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi Dedicated to my beloved Zoya, Aliya, Qasim, Sameer, and Khalid OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi Preface The laws of quantum mechanics were formulated about a hundred years ago, replacing the classical laws of Newton and Maxwell. Since then, quantum mechanics has been applied remarkably successfully to understand a very wide range of observations and systems. The success of the laws of quantum mechanics in predicting and explaining essentially all the known physical phenomena is astounding. However, in spite of the great success, it remains a mysterious theory and the concepts of wave–particle duality, complementarity, the prob- abilistic nature of measurement, quantum interference, and quantum entanglement are still hotly discussed. However, it is not just the remarkable success in explaining all the known phenomena that makes quantum mechanics a fascinating subject. It is truly amazing that, even today, a mere knowledge of the basic postulates can lead to startling new ideas and devices. For example, just the knowledge of the principle of complementarity can lead to perfectly secure communication systems, or the understanding of a beam splitter for a single photon can lead to a highly counterintuitive communication protocol with no particle present in the transmission channel, or the resource of quantum entanglement can lead to novel quantum computing algorithms. Therefore it becomes possible to convey not only the foundations of quantum mechanics but also some mind-boggling applications, such as in quantum communication and quantum computing, with just elementary knowledge of basic physics and mathematics. With this background it is interesting to ask whether it is possible to convey the basic concepts of quantum mechanics and its amazing applications to someone with a limited knowledge of physics and mathematics. In the fall of 2018, I offered a course on Quantum Mechanics to incoming freshman students at Texas A&M University. These students, just out of high school, took this course before they took the usual Mechanics and Electricity/Magnetism courses. This book grew out of the lecture notes of this course. The main objective of this book is to present an introduction to quantum mechanics in an almost self-contained way for someone with a high school physics and mathematics background. The book challenges the common perception that quantum mechanics is a highly math- ematical and abstract subject that is inaccessible to anyone without an advanced knowledge of mathematics. This book, except the last chapter on the Schrödinger equation, is entirely algebra-based. An effort is made to derive some amazing results from very simple ideas and ele- mentary mathematical tools. Ideally every chapter offers results that are highly counterintuitive and interesting. This book can be used as a text for a course on quantum mechanics or quantum informatics at the undergraduate level. However it can also be a useful and accessible book for those who are not familiar with but want to learn some of the fascinating recent and ongoing developments in areas related to the foundation of quantum mechanics and its applications to areas such as quantum communication and quantum computing. The book is divided into four parts. After an introductory chapter, some basic mathematical tools such as complex numbers, vector analysis, and introduction to probability as well as a classical description of particles and waves are introduced in the next three chapters. In the OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi viii PREFACE next eight chapters, basic concepts of quantum mechanics are discussed, such as wave–particle duality, complementarity, the Heisenberg uncertainty relation, quantum interference and entanglement, no-cloning theorem, as well as issues at the foundations of quantum mechanics such as the delayed-choice quantum eraser, the Schrödinger’s cat and EPR paradoxes, and Bell theorem. In the following chapters, these fundamentals of quantum mechanics are applied to applications in areas such as secure quantum communication, quantum teleportation, counterfactual communication, and quantum computation. In the last part, the Schrödinger equation is introduced with its relation to Newtonian dynamics and its applications for a particle inside a box and the hydrogen atom. Each chapter is followed by a short bibliography, guiding an interested reader to some relevant books and papers. In some instances, the original papers are included in the list. No attempt has, however, been made to give an exhaustive list of references. A number of problems are also given at the end of each chapter for the students in case the book is followed as a text for a course. I owe my gratitude to several individuals for their support and encouragement in the preparation of this book. First and foremost, I thank Marlan Scully for his long friendship and many fruitful collaborations that helped shape my thinking about aspects of the foundations of quantum mechanics. A person most responsible for this book is David Lee who first proposed the idea and remained an enthusiastic supporter and inspiration throughout the writing of this book. I am also grateful to Peter McIntyre whose enthusiastic support, as the Head of the Department, for teaching an unprecedented course on quantum mechanics to freshmen, was vital to this project. Robert Brick and Wenchao Ge graciously read parts of the book and gave me their unvarnished, but extremely helpful, comments. I also thank Jiru Liu and Chaofan Zhou for their help with proofreading the manuscript. Special thanks are due to Sonke Adlung and Harriet Konishi of the Oxford University Press and Cheryl Brant of SPi Global for all their help during the publication process. Finally I wish to acknowledge the loving support of my family members, Sarah, Neo, Sahar, Shani, Raheel, and Reema. My deepest gratitude is however reserved for my wife, Parveen. She has been relentless in her support not just during the writing of this book, but for all the projects, big and small, during my life. M. Suhail Zubairy College Station, Texas October 9, 2019 OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi Contents 1 What is this Book About? 1 1.1 From Classical to Quantum Mechanics 2 1.2 Outline of the Book 5 PART I: INTRODUCTORY TOPICS 11 2 Mathematical Background 13 2.1 Complex Numbers 13 2.2 Trigonometry 16 2.3 Vector and Scalar Quantities 20 2.4 Elements of Probability Theory 25 3 Particle Dynamics 32 3.1 Classical Trajectory 32 3.2 Linear Momentum 35 3.3 Kinetic and Potential Energy 37 3.4 Inelastic and Elastic Collisions 39 3.5 Angular Motion 39 3.6 Angular Momentum 44 3.7 Motion of an Electron in Electric and Magnetic Fields 45 4 Wave Theory 50 4.1 Wave Motion 50 4.2 Young’s Double-slit Experiment 57 4.3 Diffraction 61 4.4 Rayleigh Criterion 66 OUP CORRECTED PROOF – FINAL, 10/3/2020, SPi x CONTENTS PART II: FUNDAMENTALS OF QUANTUM MECHANICS 71 5 Fundamentals of Quantum Mechanics 73 5.1 Quantization of Energy 73 5.2 Wave–Particle Duality 74 5.3 End of Certainty—Probabilistic Description 75 5.4 Heisenberg Uncertainty Relations and Bohr’s Principle of Complementarity 76 5.5 Coherent Superposition and Quantum Entanglement 78 6 Birth of Quantum Mechanics—Planck, Einstein, Bohr 81 6.1 Brief History of Light 81 6.2 Radiation Emitted by Heated Objects 84 6.3 Einstein and the Photoelectric Effect 88 6.4 History of the Atom till the Dawn of the Twentieth Century 90 6.5 The Rutherford Atom 92 6.6 The Hydrogen Spectrum 93 6.7 Quantum Theory of the Atom: Bohr’s Model 94 7 De Broglie Waves: Are Electrons Waves or Particles? 100 7.1 De Broglie waves 100 7.2 Wave–Particle Duality—A Wavefunction
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