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A Shorter Course of Theoretical Physics Vol. 1. Mechanics and Electrodynamics Vol. 2. Quantum Mechanics Vol. 3. Macroscopic Phys

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A Shorter Course of Theoretical Physics Vol. 1. Mechanics and Electrodynamics Vol. 2. Quantum Mechanics Vol. 3. Macroscopic Phys A Shorter Course of Theoretical Physics IN THREE VOLUMES Vol. 1. Mechanics and Electrodynamics Vol. 2. Quantum Mechanics Vol. 3. Macroscopic Physics A SHORTER COURSE OF THEORETICAL PHYSICS VOLUME 2 QUANTUM MECHANICS BY L. D. LANDAU AND Ε. M. LIFSHITZ Institute of Physical Problems, U.S.S.R. Academy of Sciences TRANSLATED FROM THE RUSSIAN BY J. B. SYKES AND J. S. BELL PERGAMON PRESS OXFORD · NEW YORK · TORONTO · SYDNEY Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Copyright © 1974 Pergamon Press Ltd. 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, without the prior permission of Pergamon Press Ltd. First edition 1974 Library of Congress Cataloging in Publication Data Landau, Lev Davidovich, 1908-1968. A shorter course of theoretical physics. Translation of Kratkii kurs teoreticheskoi riziki. CONTENTS: v. 1. Mechanics and electrodynamics. —v. 2. Quantum mechanics. 1. Physics. 2. Mechanics. 3. Quantum theory. I. Lifshits, Evgenii Mikhaflovich, joint author. II. Title. QC21.2.L3513 530 74-167927 ISBN 0-08-016739-X (v. 1) ISBN 0-08-017801-4 (v. 2) Translated from Kratkii kurs teoreticheskoi fiziki, Kniga 2: Kvantovaya Mekhanika IzdateFstvo "Nauka", Moscow, 1972 Printed in Hungary PREFACE THIS book continues with the plan originated by Lev Davidovich Landau and described in the Preface to Volume 1: to present the minimum of material in theoretical physics that should be familiar to every present-day physicist, working in no matter what branch of physics. Part I, dealing with non-relativistic quantum theory, follows our Quantum Mechanics (Volume 3 of the Course of Theoretical Physics). This has been abridged by dropping completely some sections that are of interest only to specialists, as well as numerous details of technique that are intended for those whose profession lies in theoretical physics. This considerable abridgement has naturally meant rewriting a fairly large part of the book. I have nevertheless tried to keep unchanged the manner and style of the exposition, and in no place to allow a simplification by popularising; the only simplification is by the omis- sion of detail. In Part I, the words "it can be shown" hardly occur: the results given are accompanied by their derivations. This is, however, less true of Part II. The treatment here is based on the Relativistic Quantum Theory by Berestetskii, Pitaevskii and myself (Volume 4 of the Course), but only the fundamentals of quantum electrodynamics are presented. Here again I have sought to proceed in such a way as to show as clearly as possible the physical hypotheses and logical structure of the theory; but many applications of the theory are mentioned only by way of their results, on account of the frequent complexity of the calculations needed to solve specific problems in this field. In the choice of materials for Part II I have also been guided to some extent by the content of Landau's lectures ix χ Preface on quantum electrodynamics at Moscow University in 1959-60; my thanks are due to A. S. Kompaneets, Ν. I. Bud'ko and P. S. Kondratenko for making available their notes of these lectures. The final chapter on Feynman diagrams differs somewhat in style, both as regards its greater complexity and in being concerned with methods rather than physical results. I felt it necessary, however, to provide the reader with at least an idea of the origin and significance of the concepts of the diagram technique, which are an indispensable part of the equipment of theoretical physics at the present time. (I do not seek to describe the use of this technique for the solution of practical problems.) This chapter can be omitted, if the reader so wishes, without affecting the study of the remainder. This book was published in the original Russian almost exactly ten years after the fateful day of 7 January 1962, when a road accident cut short Lev Davidovich Landau's work as a scientist and a teacher. Not one of the readers of the Shorter Course has had the joy of attend- ing Landau's lectures. I should like to think that in these books it will be possible to convey to them something of his spirit as a teacher, his striving for clarity, his effort to make simple what was complex and so to reveal the laws of nature in their true simplicity and beauty. Ε. M. Lifshitz PUBLISHER'S NOTE As is the general rule in the volumes in the Course of Theoretical Physics, references to original papers give simply the author's name and the date. χ Preface on quantum electrodynamics at Moscow University in 1959-60; my thanks are due to A. S. Kompaneets, Ν. I. Bud'ko and P. S. Kondratenko for making available their notes of these lectures. The final chapter on Feynman diagrams differs somewhat in style, both as regards its greater complexity and in being concerned with methods rather than physical results. I felt it necessary, however, to provide the reader with at least an idea of the origin and significance of the concepts of the diagram technique, which are an indispensable part of the equipment of theoretical physics at the present time. (I do not seek to describe the use of this technique for the solution of practical problems.) This chapter can be omitted, if the reader so wishes, without affecting the study of the remainder. This book was published in the original Russian almost exactly ten years after the fateful day of 7 January 1962, when a road accident cut short Lev Davidovich Landau's work as a scientist and a teacher. Not one of the readers of the Shorter Course has had the joy of attend- ing Landau's lectures. I should like to think that in these books it will be possible to convey to them something of his spirit as a teacher, his striving for clarity, his effort to make simple what was complex and so to reveal the laws of nature in their true simplicity and beauty. Ε. M. Lifshitz PUBLISHER'S NOTE As is the general rule in the volumes in the Course of Theoretical Physics, references to original papers give simply the author's name and the date. NOTATION Ψ time-dependent wave function ψ wave function without time factor Operators are denoted by a circumflex Transposed operators are denoted by a tilde ~ Hermitian conjugate operators are denoted by a superscript fmn = (m\f\ri) matrix elements of the quantity/ Η Hamiltonian Ε non-relativistic energy ωηηι = (En—Em)/h transition frequency ε relativistic particle energy, including rest energy dq element in configuration space dV = dx dy dz element in ordinary space Ω normalisation volume xi xii Notation Four-dimensional vector indices are denoted (in Part II) by Greek letters λ, μ, v9..which take the values 0,1,2,3. In Part II, relativistic units are used; they are defined in the first footnote to §76. References to Mechanics and Electrodynamics are to Volume 1 of the Shorter Course. CHAPTER 1 THE BASIC CONCEPTS OF QUANTUM MECHANICS §1. The uncertainty principle When we attempt to apply classical mechanics and electrodynamics to explain atomic phenomena, they lead to results which are in obvious conflict with experiment. This is very clearly seen from the contradic- tion obtained on applying ordinary electrodynamics to a model of an atom in which the electrons move round the nucleus in classical orbits. During such motion, as in any accelerated motion of charges, the electrons would have to emit electromagnetic waves continually. By this emission, the electrons would lose their energy, and this would eventually cause them to fall into the nucleus. Thus, according to classical electrodynamics, the atom would be unstable, which does not at all agree with reality. This marked contradiction between theory and experiment indicates that the construction of a theory applicable to atomic phenomena —that is, phenomena occurring in particles of very small mass at very small distances—demands a fundamental modification of the basic physical concepts and laws. As a starting-point for an investigation of these modifications, it is convenient to take the experimentally observed phenomenon known as electron diffraction* It is found that, when a homogeneous beam t The phenomenon of electron diffraction was in fact discovered after quantum mechanics was invented. In our discussion, however, we shall not adhere to the historical sequence of development of the theory, but shall endeavour to con- struct it in such a way that the connection between the basic principles of quantum mechanics and the experimentally observed phenomena is most clearly shown. 3 4 The Bask Concepts of Quantum Mechanics §1 of electrons passes through a crystal, the emergent beam exhibits a pattern of alternate maxima and minima of intensity, wholly similar to the diffraction pattern observed in the diffraction of electromagnetic waves. Thus, under certain conditions, the behaviour of material particles—in this case, the electrons—displays features belonging to wave processes. How markedly this phenomenon contradicts the usual ideas of motion is best seen from the following imaginary experiment, an idealisation of the experiment of electron diffraction by a crystal. Let us imagine a screen impermeable to electrons, in which two slits are cut. On observing the passage of a beam of electrons through one of the slits, the other being covered, we obtain, on a continuous screen placed behind the slit, some pattern of intensity distribution; in the same way, by uncovering the second slit and covering the first, we obtain another pattern.
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