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ESSENTIAL QUANTUM MECHANICS,This page intentionally left blank. Essential Quantum,GARY E BOWMAN,Department of Physics and Astronomy. Northern Arizona University,Great Clarendon Street Oxford OX2 6DP. 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 in,Oxford New York. Auckland Cape Town Dar es Salaam Hong Kong Karachi. Kuala Lumpur Madrid Melbourne Mexico City Nairobi,New Delhi Shanghai Taipei Toronto.

With o ces in, Argentina Austria Brazil Chile Czech Republic France Greece. Guatemala Hungary Italy Japan Poland Portugal Singapore. South Korea Switzerland Thailand Turkey Ukraine Vietnam. Oxford is a registered trade mark of Oxford University Press. in the UK and in certain other countries,Published in the United States. by Oxford University Press Inc New York,c Gary E Bowman 2008. The moral rights of the author have been asserted,Database right Oxford University Press maker. First published 2008, 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 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 book in any other binding or cover. and you must impose the same condition on any acquirer. British Library Cataloguing in Publication Data,Data available. Library of Congress Cataloging in Publication Data. Data available, Typeset by Newgen Imaging Systems P Ltd Chennai India. Printed in Great Britain,on acid free paper by,Biddles Ltd King s Lynn. ISBN 978 0 19 922892 8 Hbk,ISBN 978 0 19 922893 5 Pbk.

1 3 5 7 9 10 8 6 4 2,Preface ix,1 Introduction Three Worlds 1. 1 1 Worlds 1 and 2 1,1 2 World 3 3,1 3 Problems 4,2 The Quantum Postulates 5. 2 1 Postulate 1 The Quantum State 6, 2 2 Postulate 2 Observables Operators and Eigenstates 8. 2 3 Postulate 3 Quantum Superpositions 10,2 3 1 Discrete Eigenvalues 11. 2 3 2 Continuous Eigenvalues 12,2 4 Closing Comments 15.

2 5 Problems 16,3 What Is a Quantum State 19,3 1 Probabilities Averages and Uncertainties 19. 3 1 1 Probabilities 19,3 1 2 Averages 22,3 1 3 Uncertainties 24. 3 2 The Statistical Interpretation 26,3 3 Bohr Einstein and Hidden Variables 28. 3 3 1 Background 28,3 3 2 Fundamental Issues 30,3 3 3 Einstein Revisited 32. 3 4 Problems 33,4 The Structure of Quantum States 36.

4 1 Mathematical Preliminaries 36,4 1 1 Vector Spaces 36. 4 1 2 Function Spaces 39,4 2 Dirac s Bra ket Notation 41. 4 2 1 Bras and Kets 41,4 2 2 Labeling States 42,4 3 The Scalar Product 43. 4 3 1 Quantum Scalar Products 43,4 3 2 Discussion 45. vi Contents,4 4 Representations 47,4 4 1 Basics 47.

4 4 2 Superpositions and Representations 48,4 4 3 Representational Freedom 50. 4 5 Problems 52,5 Operators 53,5 1 Introductory Comments 54. 5 2 Hermitian Operators 56,5 2 1 Adjoint Operators 56. 5 2 2 Hermitian Operators De nition and Properties 57. 5 2 3 Wavefunctions and Hermitian Operators 59,5 3 Projection and Identity Operators 61. 5 3 1 Projection Operators 61,5 3 2 The Identity Operator 62.

5 4 Unitary Operators 62,5 5 Problems 64,6 Matrix Mechanics 68. 6 1 Elementary Matrix Operations 68,6 1 1 Vectors and Scalar Products 68. 6 1 2 Matrices and Matrix Multiplication 69,6 1 3 Vector Transformations 70. 6 2 States as Vectors 71,6 3 Operators as Matrices 72. 6 3 1 An Operator in Its Eigenbasis 72,6 3 2 Matrix Elements and Alternative Bases 73.

6 3 3 Change of Basis 75,6 3 4 Adjoint Hermitian and Unitary Operators 75. 6 4 Eigenvalue Equations 77,6 5 Problems 78,7 Commutators and Uncertainty Relations 82. 7 1 The Commutator 83,7 1 1 De nition and Characteristics 83. 7 1 2 Commutators in Matrix Mechanics 85,7 2 The Uncertainty Relations 86. 7 2 1 Uncertainty Products 86, 7 2 2 General Form of the Uncertainty Relations 87.

7 2 3 Interpretations 88,7 2 4 Re ections 91,7 3 Problems 93. 8 Angular Momentum 95,8 1 Angular Momentum in Classical Mechanics 95. 8 2 Basics of Quantum Angular Momentum 97,8 2 1 Operators and Commutation Relations 97. Contents vii,8 2 2 Eigenstates and Eigenvalues 99,8 2 3 Raising and Lowering Operators 100. 8 3 Physical Interpretation 101,8 3 1 Measurements 101.

8 3 2 Relating L2 and Lz 104,8 4 Orbital and Spin Angular Momentum 106. 8 4 1 Orbital Angular Momentum 106,8 4 2 Spin Angular Momentum 107. 8 5 Review 107,8 6 Problems 108,9 The Time Independent Schro dinger Equation 111. 9 1 An Eigenvalue Equation for Energy 112,9 2 Using the Schro dinger Equation 114. 9 2 1 Conditions on Wavefunctions 114,9 2 2 An Example the In nite Potential Well 115.

9 3 Interpretation 117,9 3 1 Energy Eigenstates in Position Space 117. 9 3 2 Overall and Relative Phases 118,9 4 Potential Barriers and Tunneling 120. 9 4 1 The Step Potential 120,9 4 2 The Step Potential and Scattering 122. 9 4 3 Tunneling 124,9 5 What s Wrong with This Picture 125. 9 6 Problems 126,10 Why Is the State Complex 128,10 1 Complex Numbers 129.

10 1 1 Basics 129,10 1 2 Polar Form 130, 10 1 3 Argand Diagrams and the Role of the Phase 131. 10 2 The Phase in Quantum Mechanics 133,10 2 1 Phases and the Description of States 133. 10 2 2 Phase Changes and Probabilities 135,10 2 3 Unitary Operators Revisited 136. 10 2 4 Unitary Operators Phases and Probabilities 137. 10 2 5 Example A Spin 12 System 139,10 3 Wavefunctions 141. 10 4 Re ections 142,10 5 Problems 143,11 Time Evolution 145.

11 1 The Time Dependent Schro dinger Equation 145,11 2 How Time Evolution Works 146. 11 2 1 Time Evolving a Quantum State 146,11 2 2 Unitarity and Phases Revisited 148. viii Contents,11 3 Expectation Values 149,11 3 1 Time Derivatives 149. 11 3 2 Constants of the Motion 150,11 4 Energy Time Uncertainty Relations 151. 11 4 1 Conceptual Basis 151,11 4 2 Spin 12 An Example 153.

11 5 Problems 154,12 Wavefunctions 157,12 1 What is a Wavefunction 158. 12 1 1 Eigenstates and Coe cients 158,12 1 2 Representations and Operators 159. 12 2 Changing Representations 161,12 2 1 Change of Basis Revisited 161. 12 2 2 From x to p and Back Again 161,12 2 3 Gaussians and Beyond 163. 12 3 Phases and Time Evolution 165,12 3 1 Free Particle Evolution 165.

12 3 2 Wavepackets 167,12 4 Bra ket Notation 168,12 4 1 Quantum States 168. 12 4 2 Eigenstates and Transformations 170,12 5 Epilogue 171. 12 6 Problems 172,A Mathematical Concepts 175,A 1 Complex Numbers and Functions 175. A 2 Di erentiation 176,A 3 Integration 178,A 4 Di erential Equations 180. B Quantum Measurement 183,C The Harmonic Oscillator 186.

C 1 Energy Eigenstates and Eigenvalues 186,C 2 The Number Operator and its Cousins 188. C 3 Photons as Oscillators 189,D Unitary Transformations 192. D 1 Unitary Operators 192,D 2 Finite Transformations and Generators 195. D 3 Continuous Symmetries 197,D 3 1 Symmetry Transformations 197. D 3 2 Symmetries of Physical Law 197,D 3 3 System Symmetries 199.

Bibliography 201, While still a relatively new graduate student I once remarked to my advi. sor Jim Cushing that I still didn t understand quantum mechanics To this. he promptly replied You ll spend the rest of your life trying to understand. quantum mechanics Despite countless books that the subject has spawned. since it rst assumed a coherent form in the 1920s quantum mechanics. remains notoriously even legendarily di cult Some may believe students. should be told that physics really isn t that hard presumably so as not to. intimidate them I disagree what can be more demoralizing than struggling. mightily with a subject only to be told that it s really not that di cult. Let me say it outright then quantum mechanics is hard In writing. this book I have not found any magic bullet by which I can render. the subject easily digestible I have however tried to write a book that is. neither a popularization nor a standard text a book that takes a modern. approach rather than one grounded in pedagogical precedent a book that. focuses on elucidating the structure and meaning of quantum mechanics. leaving comprehensive treatments to the standard texts. Above all I have tried to write with the student in mind The pri. mary target audience is undergraduates about to take or taking their rst. quantum course But my hope is that the book will also serve biologists. philosophers engineers and other thoughtful people people who are fasci. nated by quantum physics but nd the popularizations too simplistic and. the textbooks too advanced and comprehensive by providing a foothold. on real quantum mechanics as used by working scientists. Popularizations of quantum mechanics are intended not to expound. the subject as used by working scientists but rather to discuss quantum. weirdness such as Bell s theorem and the measurement problem in terms. palatable to interested non scientists As such the mathematical level of. such books ranges from very low to essentially nonexistent. In contrast the comprehensive texts used in advanced courses often. make daunting conceptual and mathematical demands on the reader. Preparation for such courses typically consists of a modern physics course. but these tend to be rather conceptual Modern physics texts generally. take a semi historical approach discussing topics such as the Bohr atom. and the Compton e ect Formalism is minimized and description empha. sized the highly abstract mathematical and physical concepts of quantum. mechanics remain largely untouched There is thus a rather large gap to be. bridged and students in advanced courses may nd that they must solve. problems and learn new applications even while the framework of quantum. mechanics remains unclear, Neither popularization nor standard text this book is intended to serve. in a variety of settings as a primary text in a short course a supple. mentary text in a standard course a vehicle for independent study or. a reference work Knowledge of elementary calculus and basic complex. analysis should provide su cient mathematical background a condensed. discussion of these topics appears in Appendix A, The book s modernity is re ected in its overall style and tenor but. also in some broad themes such as the early and extensive use of Dirac. notation and the fact that neither wavefunctions nor the time independent. Schro dinger equation are granted privileged status Another such theme is. the adoption of the statistical interpretation a very useful and lucid way. to understand how quantum mechanics works in actual practice Because. the statistical interpretation is really a broad framework rather than an. interpretation per se it is easily imported into other approaches as the. student may nd necessary, Notable by their absence from the book are many standard topics such. as perturbation theory scattering and the Hydrogen atom This is in keep. ing with a central motivating idea that to properly understand the many. and varied applications of quantum mechanics one must rst properly. understand its overall structure This implies a focus on fundamentals such. as superposition and time evolution with the result that they may then be. developed in a more detailed and explanatory style than in advanced texts. Some authors seem to believe that if they provide a clear elegant terse. explanation one time any remaining confusion is the student s responsi. bility I disagree Having taught and learned physics for many years at. many levels I nd that there are myriad ways to misunderstand the sub. ject so I have tried to make this book especially explanatory and useful. for the student Common variations in terminology and notation are clar. i ed e g the terms quantum state state vector and wavefunction And. I discuss not only what is right but what is wrong For example although. position space and momentum space are standard topics students often. fail to realize that there is but one quantum state which may be cast. into various representations Such potential stumbling blocks are explicitly. pointed out and explained, The great majority of problems are to my knowledge new Most are.

intended to help develop conceptual understanding A vast array of addi. tional problems may be found in other quantum texts The time honored. physics dictum that one doesn t understand the physics unless one can. solve problems bears repeating here But so does its lesser known cousin. just solving problems without the capacity to lucidly discuss those prob. lems and the attendant concepts and ideas may also indicate insu cient. understanding,Preface xi, In part because this book is intended to transcend the traditional. physics audience a few words about studying the subject are in order. Much of our intellectual heritage from art and music to social political. and historical thought concerns our human experience of the world By its. Essential Quantum Mechanics GARY E BOWMAN Department of Physics and Astronomy Northern Arizona University 1 3 Great Clarendon Street Oxford OX2 6DP 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 in Oxford NewYork Auckland CapeTown DaresSalaam HongKong

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