Quantum Mechanics. A Shorter Course of Theoretical Physics

373 Pages·1974·4.935 MB·English

by  L D Landau and E. M. Lifshitz (Auth.)

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
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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
f = (m\f\ri) matrix elements of the quantity/
mn
Η Hamiltonian
Ε non-relativistic energy
ω = (E—E)/h transition frequency
ηηι n m
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, v..which take the values 0,1,2,3.
9
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. On observing the passage of the beam through
both slits, we should expect, on the basis of ordinary classical ideas,
a pattern which is a simple superposition of the other two: each elec-
tron, moving in its path, passes through one of the slits and has no
effect on the electrons passing through the other slit. The phenomenon
of electron diffraction shows, however, that in reality we obtain a
diffraction pattern which, owing to interference, does not at all corre-
spond to the sum of the patterns given by each slit separately. It is
clear that this result can in no way be reconciled with the idea that
electrons move in paths.
Thus the mechanics which governs atomic phenomena —quantum
mechanics or wave mechanics—must be based on ideas of motion
which are fundamentally different from those of classical mechanics.
In quantum mechanics there is no such concept as the path of a par-
ticle. This forms the content of what is called the uncertainty principle,
one of the fundamental principles of quantum mechanics, discovered
by W. Heisenberg in 1927.f
t It is of interest to note that the complete mathematical formalism of quantum
mechanics was constructed by W. Heisenberg and E. Schrodinger in 1925-6, be-
fore the discovery of the uncertainty principle, which revealed the physical con-
tent of this formalism.