com Quantum Mechanics Second Edition www.com Quantum Mechanics Concepts and Applications Second Edition Nouredine Zettili Jacksonville State University, Jacksonville, USA A John Wiley and Sons, Ltd.com Copyright 2009 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. 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, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
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Library of Congress Cataloging-in-Publication Data Zettili, Nouredine. Quantum Mechanics: concepts and applications / Nouredine Zettili. Includes bibliographical references and index. ISBN 978-0-470-02678-6 (cloth: alk.
paper) – ISBN 978-0-470-02679-3 (pbk.12 – dc22 2008045022 A catalogue record for this book is available from the British Library Produced from LaTeX files supplied by the author Printed and bound in Great Britain by CPI Antony Rowe Ltd, Chippenham, Wiltshire ISBN: 978-0-470-02678-6 (H/B) 978-0-470-02679-3 (P/B) www.com Contents Preface to the Second Edition xiii Preface to the First Edition xv Note to the Student xvi 1 Origins of Quantum Physics 1 1.2 Particle Aspect of Radiation .3 Wave Aspect of Particles .1 de Broglie’s Hypothesis: Matter Waves .2 Experimental Confirmation of de Broglie’s Hypothesis .3 Matter Waves for Macroscopic Objects .4 Particles versus Waves .1 Classical View of Particles and Waves .2 Quantum View of Particles and Waves .3 Wave–Particle Duality: Complementarity .4 Principle of Linear Superposition .5 Indeterministic Nature of the Microphysical World .1 Heisenberg’s Uncertainty Principle .6 Atomic Transitions and Spectroscopy .1 Rutherford Planetary Model of the Atom .2 Bohr Model of the Hydrogen Atom .1 Localized Wave Packets .2 Wave Packets and the Uncertainty Relations .3 Motion of Wave Packets .com vi CONTENTS 2 Mathematical Tools of Quantum Mechanics 79 2.2 The Hilbert Space and Wave Functions .1 The Linear Vector Space .2 The Hilbert Space .3 Dimension and Basis of a Vector Space .4 Square-Integrable Functions: Wave Functions .5 Uncertainty Relation between Two Operators .6 Functions of Operators .7 Inverse and Unitary Operators .8 Eigenvalues and Eigenvectors of an Operator .9 Infinitesimal and Finite Unitary Transformations .5 Representation in Discrete Bases .1 Matrix Representation of Kets, Bras, and Operators .2 Change of Bases and Unitary Transformations .3 Matrix Representation of the Eigenvalue Problem .6 Representation in Continuous Bases .4 Connecting the Position and Momentum Representations .7 Matrix and Wave Mechanics. 155 3 Postulates of Quantum Mechanics 165 3.2 The Basic Postulates of Quantum Mechanics .3 The State of a System .2 The Superposition Principle .4 Observables and Operators .5 Measurement in Quantum Mechanics .1 How Measurements Disturb Systems .3 Complete Sets of Commuting Operators (CSCO) .4 Measurement and the Uncertainty Relations .com CONTENTS vii 3.6 Time Evolution of the System’s State .1 Time Evolution Operator .2 Stationary States: Time-Independent Potentials .3 Schrödinger Equation and Wave Packets .4 The Conservation of Probability .5 Time Evolution of Expectation Values .7 Symmetries and Conservation Laws .1 Infinitesimal Unitary Transformations .2 Finite Unitary Transformations .3 Symmetries and Conservation Laws .8 Connecting Quantum to Classical Mechanics .1 Poisson Brackets and Commutators .2 The Ehrenfest Theorem .3 Quantum Mechanics and Classical Mechanics. 209 4 One-Dimensional Problems 215 4.2 Properties of One-Dimensional Motion .4 Symmetric Potentials and Parity .3 The Free Particle: Continuous States .4 The Potential Step .5 The Potential Barrier and Well .2 The Case E V0 : Tunneling .3 The Tunneling Effect .6 The Infinite Square Well Potential .1 The Asymmetric Square Well .2 The Symmetric Potential Well .7 The Finite Square Well Potential .1 The Scattering Solutions (E V0 ) .2 The Bound State Solutions (0 E V0 ) .8 The Harmonic Oscillator .3 Energy Eigenstates in Position Space .4 The Matrix Representation of Various Operators .5 Expectation Values of Various Operators .9 Numerical Solution of the Schrödinger Equation .com viii CONTENTS 5 Angular Momentum 283 5.2 Orbital Angular Momentum .3 General Formalism of Angular Momentum .4 Matrix Representation of Angular Momentum .5 Geometrical Representation of Angular Momentum .6 Spin Angular Momentum .1 Experimental Evidence of the Spin .2 General Theory of Spin .3 Spin 1 2 and the Pauli Matrices .7 Eigenfunctions of Orbital Angular Momentum .1 Eigenfunctions and Eigenvalues of L z .3 Properties of the Spherical Harmonics. 325 6 Three-Dimensional Problems 333 6.2 3D Problems in Cartesian Coordinates .1 General Treatment: Separation of Variables .2 The Free Particle .3 The Box Potential .4 The Harmonic Oscillator .3 3D Problems in Spherical Coordinates .1 Central Potential: General Treatment .2 The Free Particle in Spherical Coordinates .3 The Spherical Square Well Potential .4 The Isotropic Harmonic Oscillator .5 The Hydrogen Atom .6 Effect of Magnetic Fields on Central Potentials.
385 7 Rotations and Addition of Angular Momenta 391 7.1 Rotations in Classical Physics .2 Rotations in Quantum Mechanics .3 Properties of the Rotation Operator .5 Representation of the Rotation Operator .6 Rotation Matrices and the Spherical Harmonics .3 Addition of Angular Momenta .1 Addition of Two Angular Momenta: General Formalism .2 Calculation of the Clebsch–Gordan Coefficients .com CONTENTS ix 7.3 Coupling of Orbital and Spin Angular Momenta .4 Addition of More Than Two Angular Momenta .5 Rotation Matrices for Coupling Two Angular Momenta .4 Scalar, Vector, and Tensor Operators .3 Tensor Operators: Reducible and Irreducible Tensors .4 Wigner–Eckart Theorem for Spherical Tensor Operators .1 Many-Particle Systems .1 Schrödinger Equation .3 Systems of Distinguishable Noninteracting Particles .2 Systems of Identical Particles .1 Identical Particles in Classical and Quantum Mechanics .4 Constructing Symmetric and Antisymmetric Functions .5 Systems of Identical Noninteracting Particles .3 The Pauli Exclusion Principle .4 The Exclusion Principle and the Periodic Table. 484 9 Approximation Methods for Stationary States 489 9.2 Time-Independent Perturbation Theory .1 Nondegenerate Perturbation Theory .2 Degenerate Perturbation Theory .3 Fine Structure and the Anomalous Zeeman Effect .3 The Variational Method .4 The Wentzel–Kramers–Brillouin Method .2 Bound States for Potential Wells with No Rigid Walls .3 Bound States for Potential Wells with One Rigid Wall .4 Bound States for Potential Wells with Two Rigid Walls .5 Tunneling through a Potential Barrier .com x CONTENTS 10 Time-Dependent Perturbation Theory 571 10.2 The Pictures of Quantum Mechanics .1 The Schrödinger Picture .2 The Heisenberg Picture .3 The Interaction Picture .3 Time-Dependent Perturbation Theory .2 Transition Probability for a Constant Perturbation .3 Transition Probability for a Harmonic Perturbation .4 Adiabatic and Sudden Approximations .5 Interaction of Atoms with Radiation .1 Classical Treatment of the Incident Radiation .2 Quantization of the Electromagnetic Field .3 Transition Rates for Absorption and Emission of Radiation .4 Transition Rates within the Dipole Approximation .5 The Electric Dipole Selection Rules .1 Scattering and Cross Section .1 Connecting the Angles in the Lab and CM frames .2 Connecting the Lab and CM Cross Sections .2 Scattering Amplitude of Spinless Particles .1 Scattering Amplitude and Differential Cross Section .3 The Born Approximation .1 The First Born Approximation .2 Validity of the First Born Approximation .4 Partial Wave Analysis .1 Partial Wave Analysis for Elastic Scattering .2 Partial Wave Analysis for Inelastic Scattering .5 Scattering of Identical Particles. 650 A The Delta Function 653 A.1 One-Dimensional Delta Function .1 Various Definitions of the Delta Function .2 Properties of the Delta Function .3 Derivative of the Delta Function .2 Three-Dimensional Delta Function .com CONTENTS xi B Angular Momentum in Spherical Coordinates 657 B.1 Derivation of Some General Relations .2 Gradient and Laplacian in Spherical Coordinates .3 Angular Momentum in Spherical Coordinates. 659 C C++ Code for Solving the Schrödinger Equation 661 Index 665 www.com xii CONTENTS www.com Preface Preface to the Second Edition It has been eight years now since the appearance of the first edition of this book in 2001.
During this time, many courteous users—professors who have been adopting the book, researchers, and students—have taken the time and care to provide me with valuable feedback about the book. In preparing the second edition, I have taken into consideration the generous feedback I have received from these users. To them, and from the very outset, I want to express my deep sense of gratitude and appreciation. The underlying focus of the book has remained the same: to provide a well-structured and self-contained, yet concise, text that is backed by a rich collection of fully solved examples and problems illustrating various aspects of nonrelativistic quantum mechanics.
The book is intended to achieve a double aim: on the one hand, to provide instructors with a pedagogically suitable teaching tool and, on the other, to help students not only master the underpinnings of the theory but also become effective practitioners of quantum mechanics. Although the overall structure and contents of the book have remained the same upon the insistence of numerous users, I have carried out a number of streamlining, surgical type changes in the second edition. These changes were aimed at fixing the weaknesses (such as typos) detected in the first edition while reinforcing and improving on its strengths. I have introduced a number of sections, new examples and problems, and new material; these are spread throughout the text.
Additionally, I have operated substantive revisions of the exercises at the end of the chapters; I have added a number of new exercises, jettisoned some, and streamlined the rest. I may underscore the fact that the collection of end-of-chapter exercises has been thoroughly classroom tested for a number of years now. The book has now a collection of almost six hundred examples, problems, and exercises. Every chapter contains: (a) a number of solved examples each of which is designed to illustrate a specific concept pertaining to a particular section within the chapter, (b) plenty of fully solved problems (which come at the end of every chapter) that are generally comprehensive and, hence, cover several concepts at once, and (c) an abundance of unsolved exercises intended for home- work assignments.
Through this rich collection of examples, problems, and exercises, I want to empower the student to become an independent learner and an adept practitioner of quantum mechanics. Being able to solve problems is an unfailing evidence of a real understanding of the subject. The second edition is backed by useful resources designed for instructors adopting the book (please contact the author or Wiley to receive these free resources). The material in this book is suitable for three semesters—a two-semester undergraduate course and a one-semester graduate course.
A pertinent question arises: How to actually use xiii www.com xiv PREFACE the book in an undergraduate or graduate course(s)? There is no simple answer to this ques- tion as this depends on the background of the students and on the nature of the course(s) at hand.