《高等量子力学》课程参考教材：An Introduction to Advanced Quantum Physics（Hans P. Paar）.pdf_大学文库

This edition first published 2010  2010 John Wiley and 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.wiley.com. 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. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom. Library of Congress Cataloging-in-Publication Data Paar, Hans P. An introduction to advanced quantum physics / Hans P. Paar. p. cm. Includes bibliographical references and index. ISBN 978-0-470-68676-8 (cloth) – ISBN 978-0-470-68675-1 (pbk.) 1. Quantum theory – Textbooks. I. Title. QC174.12P33 2010 530.12 – dc22 2009054392 A catalogue record for this book is available from the British Library. ISBN: H/bk 978-0-470-68676-8 P/bk 978-0-470-68675-1 Typeset in 10/12 Sabon by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall

Preface Over the years, material that used to be taught in graduate school made its way into the undergraduate curriculum to create room in graduate courses for new material that must be included because of new developments in various fields of physics. Advanced Quantum Physics and its relativistic extension to Quantum Field Theory are a case in point. This book is intended to facilitate this process. It is written to support a second course in Quantum Physics and attempts to present the material in such a way that it is accessible to advanced undergraduates and starting graduate students in Physics or Electrical Engineering. This book consists of two parts. Part 1, comprising Chapters 1 through 5, contains the material for a second course in Quantum Physics. This is where concepts from classical mechanics, electricity and magnetism, statistical physics, and quantum physics are pulled together in a discussion of the interaction of radiation and matter, selection rules, symmetries and conservation laws, scattering, relativistic quantum physics, questions related to the validity of quantum physics, and more. This is material that is suitable to be taught as part of an undergraduate quantum physics course for physics and electrical engineering majors. Surprisingly, there is no undergraduate textbook that treats this material at the undergraduate level, although it is (or ought to be) taught at many institutions. In Part 2, comprising Chapters 6 through 8, we present elementary Quantum Field Theory. That this material should be studied by undergraduates is controversial but I expect it will become accepted practice in the future. This material is intended for undergraduates that are interested in the topics discussed and need it, for example, in a course on elementary particle physics or condensed matter. Traditionally such a course is taught in the beginning of graduate school. When teaching particle physics to advanced undergraduate students I felt that the time was ripe for an elementary introduction to quantum field theory, concentrating on only those topics that have an

PREFACE 5 UNITS AND METRIC It is customary in advanced quantum physics to use natural units.These are cgs units with Planck's constant divided by 2 and the velocity of light set to unity.Thus we set =c=1. In natural units we have for example that the Bohr radius of the hydrogen atom is 1/am with a=/(4x)th fine-structure constant and m the mass of the electron.Likewise we have that the energy of the hydrogen atom is am/(2n2)and the classical radius of the electron is a/m. When results of calculations have to be compared with experiment, we must introduce powers of h and c.This can be done easily with dimensional analysis u 9GeV-mb g,for example,that the product he=0.197 GeVfm and(hc2=0.3 In summations I use the Einstein convention that requires one to sum over repeated indices,1 to 3 for Latin and 1 to 4 for Greek letter indices. Three-vectors are written bold faced such as x for the coordinate vector. We use the 'East-coast metric',introduced by Minkovski and made popular by Pauli in special relativity.Its name is obviously US-cen tric.In it ,four vectors have an com nent The alte mative is to use ativity. s is called th West-co me made popular by the textbook on Field Theory by Bjorken and Drell,but is overkill for our purposes.Thus=(x,it),pu=(p,iE),and k=(k,iw) with k the wave vector and the angular frequ ency.a traveling wave can be written as e mple 2 expi(kxx+ y+k: )Squaring four-vecto we get for 2 2 unfortunate negative)and -0 for electromagn etic radiation When integratin 1g four-dimensional space we use dx which one might think is equal to dxidt but this is not so.By dx we mean d3xdt(the West-coast metric does not have this inconsistency,sorry).With this convention the gamma matrices in the Dirac equation are all Hermitian. We also find that e can write the mmutation relations of p and x on the one hand and E and t on th e other in relativistic ly covariant form as [pa,=v/i where is the Kronecker delta.Thus it is seen that the (at first sight odd)difference in sign of the two original commutation relations is required by relativistic invariance. The partial derivatives a/ax will often be abbreviated to a.For example the Lor ntz condition VA+/at=0can be written as aA=0,shov ving that the covarian Furtherm re we have that a ==(V-0-/0t-)o,an expression that is useful in writing down a Lorentz invariant wave equation for the function

PREFACE xi UNITS AND METRIC It is customary in advanced quantum physics to use natural units. These are cgs units with Planck’s constant divided by 2π and the velocity of light set to unity. Thus we set = c = 1. In natural units we have for example that the Bohr radius of the hydrogen atom is 1/αm with α = e2/(4π) the fine-structure constant and m the mass of the electron. Likewise we have that the energy of the hydrogen atom is αm/(2n2) and the classical radius of the electron is α/m. When results of calculations have to be compared with experiment, we must introduce powers of and c. This can be done easily with dimensional analysis using, for example, that the product c = 0.197 GeVfm and (c) 2 = 0.389 GeV2mb. In summations I use the Einstein convention that requires one to sum over repeated indices, 1 to 3 for Latin and 1 to 4 for Greek letter indices. Three-vectors are written bold faced such as x for the coordinate vector. We use the ‘East-coast metric’, introduced by Minkovski and made popular by Pauli in special relativity. Its name is obviously US-centric. In it, fourvectors have an imaginary fourth component. The alternative is to use a metric tensor as in General Relativity. This is called the ‘West-coast metric’, made popular by the textbook on Field Theory by Bjorken and Drell, but is overkill for our purposes. Thus xµ = (x, it), pµ = (p, iE), and kµ = (k, iω) with k the wave vector and ω the angular frequency. A traveling wave can be written as exp ikx = exp i(kxx + kyy + kzz − ωt). Squaring four-vectors, we get for example p2 = pµpµ = p2 − E2 which is −m2 (unfortunately negative) and k2 = 0 for electromagnetic radiation. When integrating in four-dimensional space we use d4x which one might think is equal to d3xidt but this is not so. By d4x we mean d3xdt (the West-coast metric does not have this inconsistency, sorry). With this convention the gamma matrices in the Dirac equation are all Hermitian. We also find that we can write the commutation relations of p and x on the one hand and E and t on the other in relativistically covariant form as [pµ, xν ] = δµν /i where δµν is the Kronecker delta. Thus it is seen that the (at first sight odd) difference in sign of the two original commutation relations is required by relativistic invariance. The partial derivatives ∂/∂xµ will often be abbreviated to ∂µ. For example, the Lorentz condition ∇A + ∂φ/∂t = 0 can be written as ∂µAµ = 0, showing that the Lorentz condition is relativistically covariant. Furthermore we have that ∂2φ = ∂µ∂µφ = (∇2 − ∂2/∂t 2)φ, an expression that is useful in writing down a Lorentz invariant wave equation for the function φ