LANDAU
LIFSHITZ

Quantum Mechanics
Non-relativistic
Second

Theory

edition, revised

and enlarged

Course of Theoretical Physics
Volume 3

o
£

Landau and

M. Lifshitz
Institute of Physical Problems USSR
Academy of Sciences
D.

E.

3
£

O
3

O
V)

Pergamon

Pergamon Press


QUANTUM MECHANICS
Non-relativistic

L

D.

LANDAU

and

E.

Theory
M.

UFSHITZ

This second edition, published to meet

the consistent demand for this important and
informative book, has been considerably revised
and enlarged, the basic plan and style of the
first edition, however, being retained. The
volume gives a comprehensive treatment of
non-relativistic quantum mechanics, and an
introduction to its application to atomic and
molecular phenomena. Thetopics dealtwith
include the basic concepts, Schrodinger's
equation, angular momentum, motion in
centrally symmetric fields perturbation theory,
the quasi-classical case, spin, identity of
particles, atoms, diatomic and polyatomic
molecules, the theory of symmetry, elastic and
inelastic collisions,
field. In this

second

and motion

in

a magnetic

edition, extensive

changes

the sections dealing with the

theory of the addition of angular momenta and
with collision theory and a new chapter on
nuclear structure has been added. The discussion
is intended to display the physical significance
of the theory, and to be complete and
self-contained.
As with other volumes in this series, a list of
which —together with statements aboutthem by
will
knowledgeable and impartial reviewers
be found on the back cover of this jacket, this
book has been written to a level that will prove
invaluable to those carrying out undergraduate

have been made

in

—

and post-graduate study.

2217

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COURSE OF THEORETICAL PHYSICS
Volume


3

QUANTUM MECHANICS
Non-relativistic

Second

edition, revised

Theory

and enlarged

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OTHER TITLES IN THE SERIES

Vol.3.

MECHANICS
THE CLASSICAL THEORY OF FIELDS
QUANTUM MECHANICS—Non-Relativistic Theory

Vol. 4.

RELATIVISTIC PHYSICS

Vol. 5.


STATISTICAL PHYSICS

Vol. 6.

FLUID MECHANICS
THEORY OF ELASTICITY
ELECTRODYNAMICS OF CONTINUOUS MEDIA
PHYSICAL KINETICS

Vol.1.
Vol. 2.

Vol. 7.
Vol. 8.
Vol. 9.

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QUANTUM MECHANICS
NON-RELATIVISTIC THEORY
by

LANDAU

L. D.

and

INSTITUTE OF PHYSICAL PROBLEMS,


Volume

E.

M. LIFSHITZ

U.S.S.R.

ACADEMY OF SCIENCES

3 of Course of Theoretical Physics

Translated from the Russian by
J.

B.

Second

SYKES and

J.

edition, revised

S.

BELL


and enlarged

PERGAMON PRESS
OXFORD

•

LONDON
PARIS

•

•

EDINBURGH
FRANKFURT

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•

NEW YORK


Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.l
Pergamon Press (Scotland)
Pergamon Press

Inc.,


Ltd., 2

&

3

Teviot Place, Edinburgh

44-01 21st Street, Long Island City,

Pergamon Press S.A.R.L., 24 rue des
Pergamon Press

GmbH,

New

Ecoles, Paris 5e

Kaiserstrasse 75, Frankfurt-am-Main

Sole distributors in the U.S.A.

Addison- Wesley Publishing Company, Inc.
Reading, Massachusetts

Copyright

© 1958 and 1965


Pergamon Press Ltd.

First published in English

1958

2nd impression 1959
3rd impression 1962

Second

{revised) edition

Library of Congress Card

Printed in Great Britain by J.

1965

Number 57-14444

W. Arrowsmith

2217/65

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1


York 11101

Ltd., Bristol.


CONTENTS
Page

From

the Preface to the

first

English edition

Preface to the second English edition

Notation
I.

xi
xii
xiii

THE BASIC CONCEPTS OF QUANTUM MECHANICS

§2.

The

The

§3.

Operators

§4.

Addition and multiplication of operators

13

§5.

The continuous spectrum
The passage to the limiting case of classical mechanics
The wave function and measurements

20

§1.

§6.
§7.

uncertainty principle

1

principle of superposition


6
8

II.

§8.

§9.

15

21

ENERGY AND MOMENTUM

The Hamiltonian operator
The differentiation of operators with

25
respect to time

26

§10.

Stationary states

27


§11.

Matrices

30

§12.

Transformation of matrices

§13.

The Heisenberg representation
The density matrix

§14.

35
of operators

37
38

§15.

Momentum

41

§16.


Uncertainty relations

46

§17.

Schrodinger's equation

§18.

III.

SCHRODINGER'S EQUATION
50

§20.

The fundamental properties
The current density
The variational principle

§21.

General properties of motion in one dimension

60

§22.


63

§23.

The
The

§24.

Motion

§25.

The

§19.

of Schrodinger's equation

potential well
linear oscillator

in a

58

67

homogeneous


field

transmission coefficient
IV.

53

55

73

75

ANGULAR MOMENTUM

momentum

§26.

Angular

§27.

Eigenvalues of the angular

§28.

Eigenfunctions of the

§29.


Matrix elements of vectors

91

§30.

Parity of a state

95

§31.

Addition of angular momenta

97

81

momentum
angular momentum

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85
88


Contents


vi

MOTION IN A CENTRALLY SYMMETRIC FIELD

V.

symmetric

Page
101

§32.

Motion

§33.

Free motion (spherical polar co-ordinates)

104

§34.

Resolution of a plane wave

111

§35.

"Fall" of a particle to the centre


113

§36.

Motion

in a

§37.

Motion

in a

in a centrally

Coulomb
Coulomb
VI.

§38.

field

field (spherical polar co-ordinates)

116

field (parabolic co-ordinates)


125

PERTURBATION THEORY
129

Perturbations independent of time

133

§39.

The

§40.

Perturbations depending on time

§41.

Transitions under a perturbation acting for a

§42.

Transitions under the action of a periodic perturbation

146

§43.


Transitions in the continuous spectrum

147

secular equation

136

§44.

The

§45.

Potential energy as a perturbation

finite

time

140

150

uncertainty relation for energy

153

THE QUASI-CLASSICAL CASE


VII.

158

§46.

The wave

§47.

Boundary conditions

§48.

Bohr and Sommerfeld's quantisation

§49.

Quasi-classical motion in a centrally symmetric

§50.

Penetration through a potential barrier

171

177

function in the quasi-classical case


161

in the quasi- classical case

162

rule
field

167

§51.

Calculation of the quasi-classical matrix elements

§52.

The

§53.

Transitions under the action of adiabatic perturbations

§54.

Spin

188

§55.


Spinors

191

§56.

Spinors of higher rank

196

§57.

§58.

The
The relation between

198

§59.

Partial polarisation of particles

§60.

Time

transition probability in the quasi-classical case


VIII.

SPIN

wave functions of particles with arbitrary spin

reversal

spinors and tensors

and Kramers' theorem
IX.

181

185

200

204
206

IDENTITY OF PARTICLES
209

§61.

The

§62.


Exchange interaction

212

§63.

Symmetry with

216

§64.
§65.

principle of indistinguishability of similar particles

respect to interchange

Second quantisation. The case of Bose statistics
Second quantisation. The case of Fermi statistics

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221

227


Contents


X.
§66.

Atomic energy

vii

THE ATOM

Page
231

levels

§67.

Electron states in the atom

232

§68.

Hydrogen-like energy levels

236

§69.

The self-consistent field
The Thomas-Fermi equation

Wave functions of the outer electrons

241

§70.
§71.

237

near the nucleus

246

§72.

Fine structure of atomic levels

247

§73.

The

252

§74.

X-ray terms

§75.


Multipole moments

261

§76.

The
The

Stark effect

265

Stark effect in hydrogen

269

§77.

periodic system of D.

I.

Mendeleev

259

XI.


THE DIATOMIC MOLECULE

§78.

Electron terms in the diatomic molecule

277

§79.

279

§80.

The
The

§81.

Valency

§82.

Vibrational and rotational structures of singlet terms in the diatomic

intersection of electron terms
relation

between molecular and atomic terms


282
286

molecule

293

§83.

Multiplet terms.

Case a

§84.

Multiplet terms.

Case b

§85.

Multiplet terms. Cases c and

§86.

Symmetry of molecular terms

299
303


d

307

309

§87.

Matrix elements for the diatomic molecule

312

§88.

A-doubling

316

§89.

The

§90.

Pre-dissociation

interaction of atoms at large distances

XII.


319

322

THE THEORY OF SYMMETRY

§91.

Symmetry transformations

§92.

Transformation groups

335

§93.

Point groups

338

§94.

Representations of groups

347

§95.


Irreducible representations of point groups

354

§96.

Irreducible representations and the classification of terms

358

332

§97.

Selection rules for matrix elements

361

§98.

Continuous groups

364

§99.

Two-valued representations of

finite


point groups

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367


Contents

viii

XIII.

POLYATOMIC MOLECULES
Page

§100.

The

§101.

Vibrational energy levels

378

§102.

Stability of symmetrical configurations of the molecule


380

§103.

Quantisation of the rotation of a rigid body

383

§104.

The interaction between the vibrations and the
The classification of molecular terms

§105.

classification of

molecular vibrations

371

rotation of the molecule 389

394

ADDITION OF ANGULAR MOMENTA

XIV.
§106.


3/-symbols

401

§107.

Matrix elements of tensors

408

§108.

6/-symbols

412

§109.

Matrix elements for addition of angular momenta

418

XV.

MOTION IN A MAGNETIC FIELD

§110.

Schrodinger's equation in a magnetic field


421

§111.

Motion

424

in a uniform magnetic field

§112.

The Zeeman

§113.

Spin in a variable magnetic

§114.

The

427

effect

434

field


current density in a magnetic field

XVI.

435

NUCLEAR STRUCTURE

Isotopic invariance

438

§116.

Nuclear forces

442

§117.

The

§118.

Non-spherical nuclei

456

§119.


Isotopic shift

461

§120.

Hyperfine structure of atomic levels

463

§121.

Hyperfine structure of molecular

466

§115.

shell

447

model

XVII.

levels

THE THEORY OF ELASTIC COLLISIONS
469


§122.

The

§123.

An investigation of the

§124.

The

§125.

Born's formula

479

§126.

The

486

general theory of scattering

general formula

unitary condition for scattering


quasi-classical case

472
475

§127.

Scattering at high energies

489

§128.

Analytical properties of the scattering amplitude

492

§129.

The
The

dispersion relation

497

scattering of slow particles

500


§130.

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Contents

ix

Page
505

§131.

Resonance scattering

§132.

Resonance

§133.

Rutherford's formula

516

§134.

The system


519

§135.

Collisions of like particles

§136.

Resonance scattering of charged

§137.

Elastic collisions

§138.

Scattering with spin-orbit interaction

XVIII.

at

low energies

at a quasi-discrete level

511

of wave functions of the continuous spectrum


between

523

526

particles

fast electrons

and atoms

531

535

THE THEORY OF INELASTIC COLLISIONS

§139.

Elastic scattering in the presence of inelastic processes

§140.

Inelastic scattering of

§141.

The


§142.

Breit

§143.

Interaction in the final state in reactions

§144.

Behaviour of cross-sections near the reaction threshold

§145.

Inelastic collisions

§146.

The

§147.

Inelastic collisions

§148.

Scattering

slow particles


scattering matrix in the presence of reactions

and Wigner's formula

between

542
548

550
554

fast electrons

562

and atoms

effective retardation

565
571

580

between heavy

particles


and atoms

by molecules

584
587

MATHEMATICAL APPENDICES
§a.

Hermite polynomials

§b.

The Airy

§c.

Legendre polynomials

598

§d.

The confluent hypergeometric function
The hypergeometric function
The calculation of integrals containing

600


§e.
§f.

593

function

596

605
confluent hypergeometric

functions

607

INDEX

611

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:

FROM THE PREFACE TO THE FIRST ENGLISH EDITION
The

present book


one of the

is

on Theoretical

series

Physics, in

which we

endeavour to give an up-to-date account of various departments of that science.
The complete series will contain the following nine volumes
1.

Mechanics.

The

2.

(non-relativistic theory).
6.

Fluid mechanics.

media.

7.


Physical

9.

classical theory

4. Relativistic

of

fields.

3.

quantum theory.

Theory of elasticity.

8.

Quantum mechanics
5. Statistical physics.

Electrodynamics of continuous

kinetics.

Of these, volumes 4 and 9 remain to be written.
The scope of modern theoretical physics is very wide, and we have, of

course, made no attempt to discuss in these books all that is now included in
the subject. One of the principles which guided our choice of material was
not to deal with those topics which could not properly be expounded without
same time giving a detailed account of the existing experimental results.
For this reason the greater part of nuclear physics, for example, lies outside the

at the

scope of these books. Another principle of selection was not to discuss very
complicated applications of the theory. Both these criteria are, of course,
to

some extent

subjective.

We have tried to deal as fully as possible with those topics that are included.
For this reason we do not, as a
but simply name their authors.
work which contains matters not
plexity

lie

"on the borderline"

rule, give references to the original papers,

We


give bibliographical references only to

which by

their

com-

as regards selection or rejection.

We

have

fully

expounded by

us,

which might be of use for reference.
however, the bibliography given makes no pre-

tried also to indicate sources of material

Even with

these limitations,
tence of being exhaustive.
attempt to discuss general topics in such a


We

way that the

physical signifi-

exhibited as clearly as possible, and then to build up the

cance of the theory
mathematical formalism.
is

rigour" of exposition,

In doing

which

so,

we do not aim

in theoretical physics often

at "mathematical

amounts

to self-


deception.

The present volume is devoted to non-relativistic quantum mechanics. By
theory" we here mean, in the widest sense, the theory of all
quantum phenomena which significantly depend on the velocity of light. The
"relativistic

volume on

this subject

(volume 4) will therefore contain not only Dirac's

theory and what is now known as quantum electrodynamics, but
also the whole of the quantum theory of radiation.
relativistic

Institute of Physical Problems

L. D.

USSR Academy

E.

of Sciences

August 1956


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M.

Landau
Lifshitz


PREFACE TO THE SECOND ENGLISH EDITION
For

second edition the book has been considerably revised and enbut the general plan and style remain as before. Every chapter has
been revised. In particular, extensive changes have been made in the sections
dealing with the theory of the addition of angular momenta and with collision
theory. A new chapter on nuclear structure has been added; in accordance
with the general plan of the course, the subjects in question are discussed only
to the extent that is proper without an accompanying detailed analysis of the
this

larged,

experimental results.

We should like to express our thanks to all our many colleagues whose
comments have been utilised in the revision of the book. Numerous comments were received from V. L. Ginzburg and Ya. A. Smorodinskii. We are
especially grateful to L. P. Pitaevskii for the great help which he has given in
checking the formulae and the problems.
Our sincere thanks are due to Dr. Sykes and Dr. Bell, who not only
translated excellently both the first and the second edition of the book, but
also made a number of useful comments and assisted in the detection of

various misprints in the
Finally,

we

first edition.

are grateful to the

Pergamon

Press,

which always acceded

to

our requests during the production of the book.
L. D.
E.

October 1964

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M.

Landau
Lifshitz



NOTATION
Operators are denoted by a circumflex

dq element in configuration space
fnm

= /" =

(n\f\m) matrix elements of the quantity/ (see definition in §11)

%m = (E

n — Em)!^ transition frequency

{/,

g}

= fg — g/ commutator

of two operators

i? Hamiltonian

S,
Si

&?


electric

and magnetic

fields

phase shifts of wave functions

eua antisymmetric unit tensor

"± =

"a;

i ^"y

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—

CHAPTER

I

THE BASIC CONCEPTS OF QUANTUM MECHANICS
The uncertainty principle
When we attempt to apply classical mechanics and electrodynamics to explain
§1.


atomic phenomena, they lead to results which are in obvious conflict with
experiment. This is very clearly seen from the contradiction 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 electro-

magnetic waves continually.

By this

emission, the electrons

energy, and this would eventually cause

them

would

lose their

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

mena occurring


in particles of very small

phenomena

mass

—that

is,

pheno-

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 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.

the behaviour of material particles

—

Thus, under certain conditions,

in this case, the electrons

—displays


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
features belonging to

beam of electrons^ through one of the slits, the other being
we obtain, on a continuous screen placed behind the slit, some pat-

passage of a
covered,

tern of intensity distribution;
slit

and covering the

passage of the

first,

we

same way, by uncovering the second

in the

obtain another pattern.


beam through both

slits,

ordinary classical ideas, a pattern which

two

:

each electron, moving in

its

is

we should

On

expect,

observing the

on the

basis of

a simple superposition of the other


path, passes through one of the

slits

and

f 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 construct it in such a way that the connection between the basic
principles of quantum mechanics and the experimentally observed phenomena is most clearly shown
J The beam is supposed so rarefied that the interaction of the particles in it plays no part.
1

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—
2

The Basic Concepts of Quantum Mechanics

§1

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 correspond to the sum

by each slit separately. It is clear that this result can
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 particle. 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
of the patterns given
in

—

In that

it

rejects the ordinary ideas of classical mechanics, the uncertainty

might be said to be negative in content. Of course, this principle
in itself does not suffice as a basis on which to construct a new mechanics of
particles.
Such a theory must naturally be founded on some positive assertions, which we shall discuss below (§2).
However, in order to formulate
these assertions, we must first ascertain the statement of the problems which
confront quantum mechanics. To do so, we first examine the special nature
of the interrelation between quantum mechanics and classical mechanics. A
more general theory can usually be formulated in a logically complete manner,
independently of a less general theory which forms a limiting case of it. Thus,
relativistic mechanics can be constructed on the basis of its own fundamental
principles, without any reference to Newtonian mechanics. It is in principle

principle

impossible, however, to formulate the basic concepts of

quantum mechanics
without using classical mechanics. The fact that an electron^ has no definite
path means that it has also, in itself, no other dynamical characteristics.
Hence it is clear that, for a system composed only of quantum objects,
it would be entirely impossible to construct any logically independent
mechanics. The possibility of a quantitative description of the motion of an
electron requires the presence also of physical objects which obey classical
||

mechanics to a sufficient degree of accuracy. If an electron interacts with
such a "classical object", the state of the latter is, generally speaking, altered.
The nature and magnitude of this change depend on the state of the electron,

and therefore may serve

to characterise

it

quantitatively.

In this connection the "classical object" is usually called apparatus, and
its interaction with the electron is spoken of as measurement.
However, it
must be emphasised that we are here not discussing a process of measurement
in which the physicist-observer takes part. By measurement, in quantum

mechanics, we understand any process of interaction between classical and
f 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, before the discovery of the uncertainty
principle, which revealed the physical content of this formalism.
X In this and the following sections we shall, for brevity, speak of "an electron", meaning in general
any object of a quantum nature, i.e. a particle or system of particles obeying quantum mechanics and
not classical mechanics.
We refer to quantities which characterise the motion of the electron, and not to those, such as the
I
charge and the mass, which relate to it as a particle these are parameters.
;

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The uncertainty principle

§1

quantum

objects, occurring apart

The importance

3

from and independently of any observer.
quantum mechanics was


of the concept of measurement in

by N. Bohr.
denned "apparatus" as a physical object which is governed, with
sufficient accuracy, by classical mechanics.
Such, for instance, is a body
of large enough mass. However, it must not be supposed that apparatus is
necessarily macroscopic. Under certain conditions, the part of apparatus may
also be taken by an object which is microscopic, since the idea of "with
sufficient accuracy" depends on the actual problem proposed.
Thus, the
motion of an electron in a Wilson chamber is observed by means of the
cloudy track which it leaves, and the thickness of this is large compared with
atomic dimensions; when the path is determined with such low accuracy,
elucidated

We have

the electron

is

an entirely

classical object.

Thus quantum mechanics

occupies a very unusual place among physical
contains classical mechanics as a limiting case, yet at the same

requires this limiting case for its own formulation.

theories:

time

it

it

We may now

formulate the problem of quantum mechanics. A typical
problem consists in predicting the result of a subsequent measurement from
the known results of previous measurements. Moreover, we shall see later
that, in comparison with classical mechanics, quantum mechanics, generally
speaking, restricts the range of values which can be taken by various physical
quantities (for example, energy)
that is, the values which can be obtained
as a result of measuring the quantity concerned. The methods of quantum
mechanics must enable us to determine these admissible values.
The measuring process has in quantum mechanics a very important property: it always affects the electron subjected to it, and it is in principle
:

impossible to

it,

make its effect arbitrarily small,


for a given accuracy of measureexact the measurement, the stronger the effect exerted by
and only in measurements of very low accuracy can the effect on the mea-

ment.

The more

sured object be small. This property of measurements is logically related
to the fact that the dynamical characteristics of the electron appear only as a
result of the measurement itself. It is clear that, if the effect of the measuring
process on the object of it could be made arbitrarily small, this would mean
that the measured quantity has in itself a definite value independent of the

measurement.

Among the various kinds of measurement, the measurement of the coordinates of the electron plays a fundamental part. Within the limits of
applicability of quantum mechanics, a measurement of the co-ordinates of an
electron can always be performed-]- with any desired accuracy.
Let us suppose that, at definite time intervals At, successive measurements of
the co-ordinates of an electron are made.

on a smooth curve.

On the

contrary, the

The

results will not in general lie


more

accurately the measurements

f Once again we emphasise that, in speaking of "performing a measurement", we refer to the
interaction of an electron with a classical " apparatus", which in no way presupposes the presence of
an external observer.

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;

The Basic Concepts of Quantum Mechanics

4
are

made, the more discontinuous and disorderly

will

their results, in accordance with the non-existence of a

A fairly smooth

path

is


obtained only

if

§1

be the variation of
path of the electron.

the co-ordinates of the electron are

measured with a low degree of accuracy, as for instance from the condensation of vapour droplets in a Wilson chamber.
If now, leaving the accuracy of the measurements unchanged, we diminish
the intervals At between measurements, then adjacent measurements, of
course, give neighbouring values of the co-ordinates. However, the results
of a series of successive measurements, though they lie in a small region of
space, will be distributed in this region in a wholly irregular manner, lying on
no smooth curve. In particular, as A* tends to zero, the results of adjacent
measurements by no means tend to lie on one straight line.
This circumstance shows that, in quantum mechanics, there is no such
concept as the velocity of a particle in the classical sense of the word, i.e. the
which the difference of the co-ordinates at two instants, divided by
the interval At between these instants, tends as At tends to zero. However,
limit to

we

shall see laier that in


quantum mechanics,

nevertheless, a reasonable

definition of the velocity of a particle at a given instant can

and

we

be constructed,

mechBut whereas in classical mechanics a particle has definite co-ordinates
and velocity at any given instant, in quantum mechanics the situation is
entirely different. If, as a result of measurement, the electron is found to have
definite co-ordinates, then it has no definite velocity whatever. Conversely,
if the electron has a definite velocity, it cannot have a definite position in
space. For the simultaneous existence of the co-ordinates and velocity would
mean the existence of a definite path, which the electron has not. Thus, in
quantum mechanics, the co-ordinates and velocity of an electron are quantities
which cannot be simultaneously measured exactly, i.e. they cannot simultaneously have definite values. We may say that the co-ordinates and velocity
of the electron are quantities which do not exist simultaneously. In what
follows we shall derive the quantitative relation which determines the possibility of an inexact measurement of the co-ordinates and velocity at the same
this velocity passes into the classical velocity as

pass to classical

anics.

instant.


A complete
anics

is

description of the state of a physical system in classical

effected

by stating

all its

mech-

co-ordinates and velocities at a given instant

with these initial data, the equations of motion completely determine the
behaviour of the system at all subsequent instants. In quantum mechanics
such a description is in principle impossible, since the co-ordinates and the
corresponding velocities cannot exist simultaneously. Thus a description
of the state of a

quantum system

is

effected


quantities than in classical mechanics,

by means

i.e. it is

of a smaller

less detailed

number

than a

of

classical

description.

A very important consequence follows from this regarding the nature of the
predictions

made

in

quantum mechanics.

suffices to predict the future


Whereas a classical description
motion of a mechanical system with complete

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The uncertainty principle

§1

5

quantum mechanics evidently
cannot be enough to do this. This means that, even if an electron is in a state
described in the most complete manner possible in quantum mechanics, its
behaviour at subsequent instants is still in principle uncertain. Hence quantum mechanics cannot make completely definite predictions concerning the
future behaviour of the electron. For a given initial state of the electron, a
subsequent measurement can give various results. The problem in
quantum mechanics consists in determining the probability of obtaining various results on performing this measurement. It is understood, of course,
accuracy, the less detailed description given in

that in

some

cases the probability of a given result of

equal to unity,


i.e.

measurement may be
measurement is unique.
quantum mechanics may be divided into two

certainty, so that the result of that

All measuring processes in

In one, which contains the majority of measurements, we find those
which do not, in any state of the system, lead with certainty to a unique
result.
The other class contains measurements such that for every possible
result of measurement there is a state in which the measurement leads with
certainty to that result. These latter measurements, which may be called
predictable, play an important part in quantum mechanics. The quantitative
characteristics of a state which are determined by such measurements are
what are called physical quantities in quantum mechanics. If in some state
a measurement gives with certainty a unique result, we shall say that in this
classes.

state the corresponding physical quantity has a definite value. In future we
shall always understand the expression "physical quantity" in the sense given

here.

We shall often find in what follows that by no

means every set of physical

quantum mechanics can be measured simultaneously, i.e. can
have definite values at the same time. We have already mentioned one

quantities in
all

example, namely the velocity and co-ordinates of an electron. An important
is played in quantum mechanics by sets of physical quantities
having
the following property:
these quantities can be measured simultaneously,
but if they simultaneously have definite values, no other physical quantity
(not being a function of these) can have a definite value in that state.
We
shall speak of such sets of physical quantities as complete sets; in
particular
cases a complete set may consist of only one quantity.
part

Any

description of the state of an electron arises as a result of some meashall now formulate the meaning of a complete description of
a state in quantum mechanics. Completely described states occur as a
result

surement.

We

of the simultaneous measurement of a complete set of physical

quantiFrom the results of such a measurement we can, in particular, determine the probability of various results of any subsequent measurement,
regardless of the history of the electron prior to the first measurement.
ties.

In quantum mechanics we need concern ourselves in practice only with
completely described states, and from now on (except in §14) we shall understand by the states of a quantum system just these completely described
states.

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The Basic Concepts of Quantum Mechanics

6
§2.

The

§2

principle of superposition

Passing

now

to an exposition of the fundamental mathematical formalism

quantum mechanics, we shall denote by q the set of co-ordinates of a quantum system, and by dq the product of the differentials of these co-ordinates.
This dq is often called an element of volume in the configuration space of the

of

system;

for one particle, dq coincides with

an element of volume

dV

in

ordinary space.

quantum mechanics lies in the
system can be described, at a given moment, by a
definite (in general complex) function Y(#) of the co-ordinates. The square
of the modulus of this function determines the probability distribution of the
2
values of the co-ordinates: |Y| d# is the probability that a measurement
performed on the system will find the values of the co-ordinates to be in the

The

basis of the mathematical formalism of

fact that

any


state of a

Y

is called the wave function
element dq of configuration space. The function
of the system (sometimes also the probability amplitude), jA knowledge of the wave function allows us, in principle, to calculate the
probability of the various results of any other measurement (not of the coordinates) also. All these probabilities are determined by expressions biand Y*. The most general form of such an expression is
linear in

Y

jjV{q)Wffl
(2.1)

t

where the function (f>(q, q) depends on the nature and the result of the measurement, and the integration is extended over all configuration space. The
probability

YY*

of various values of the co-ordinates

is itself

an expression

of this type. J

it the wave function, in general varies
can be regarded as a function of
function
with time. In this sense the wave
some initial instant, then, from
is
known
at
time also. If the wave function
description of a state, it is in
complete
of
concept
the very meaning of the

The

state of the system,

and with

The actual dependence
determined by equations which will be de-

principle determined at every succeeding instant.

of the wave function on time

is

rived later.

The sum

of the probabilities of

all

possible values of the co-ordinates of

the system must, by definition, be equal to unity. It is therefore necessary
2
that the result of integrating |Y| over all configuration space should be equal
to unity:

J>|»df
This equation

is

what

is

=

l.

called the normalisation condition for

(2.2)

wave functions.

If the integral of |Y| converges, then by choosing an appropriate constant
can always be, as we say, normalised. Sometimes,
coefficient the function
2

Y

first introduced into quantum mechanics by Schrodinger in 1926.
8(?-? ) 8(q'-q ), where 8 denotes the delta function,
obtained from (2.1) when<£(g, q')
required.
defined in §5 below; q denotes the value of the co-ordinates whose probability is

f It
%

was

It is

=

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The principle of superposition

§2

7

however, wave functions are used which are not normalised; moreover, we
may diverge, and then
cannot be
normalised by the condition (2.2). In such cases |T| 2 does not, of course,
determine the absolute values of the probability of the co-ordinates, but the

Y

shall see later that the integral of |TJ 2

ratio of the values of |T| 2 at

mines the

two

different points of configuration space deter-

relative probability of the corresponding values of the co-ordinates.

by means of the wave function, and having a
form (2.1), in which
appears multiplied
by Y*, it is clear that the normalised wave function is determined only to
within a constant phase factor of the form e ia (where a is any real number),

whose modulus is unity. This indeterminacy is in principle irremovable;
it is, however, unimportant, since it has no effect upon any physical results.
The positive content of quantum mechanics is founded on a series of
propositions concerning the properties of the wave function. These are as
Since

all

quantities calculated

T

direct physical meaning, are of the

follows.

Suppose that, in a state with wave function VF^), some measurement leads
with certainty to a definite result (result 1), while in a state with
2 (#) it
leads to result 2. Then it is assumed that every linear combination of
x
and 2 i.e. every function of the form cJ 1 +c2xF2 (where cx and c% me con-

Y

Y

Y

T

,

stants), gives a state in

which that measurement leads to

either result 1 or

Moreover, we can assert that, if we know the time dependence of
the states, which for the one case is given by the function T^, t), and for the
other by 2 (<7, t), then any linear combination also gives a possible dependence
of a state on time. These propositions can be immediately generalised to
any number of different states.
The above set of assertions regarding wave functions constitutes what is
result 2.

Y

called the principle of superposition of states, the chief positive principle of

quantum mechanics.

In particular,

it follows at once from this principle
by wave functions must be linear in Y.
Let us consider a system composed of two parts, and suppose that the state
of this system is given in such a way that each of its parts is completely

that all equations satisfied

described.f Then we can say that the probabilities of the co-ordinates
qx of
the first part are independent of the probabilities of the co-ordinates
of
the
q%
second part, and therefore the probability distribution for the whole system
should be equal to the product of the probabilities of its parts. This means
that the wave function
12 (ft, qz ) of the system can be represented in the form
of a product of the wave functions Y^ft) and
) of its parts:
2

Y

Y (#

2

^i 2 (?i,?2)=Yi(?i)Y 2 (?2 ).
If the

two parts do not

interact,

of the system and those of

its

(2.3)

then this relation between the wave function

parts will be maintained at future instants also,

f This, of course, means that the state of the whole system is completely described also. However,
that the converse statement is by no means true: a complete description of the state

we emphasise

of the whole system does not in general completely determine the states of
also §14).

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its

individual parts (see


/

The Basic Concepts of Quantum Mechanics

8

i.e.

we can

write

Tu(fo
§3.

§3

?i, t)

= YjCfc, T
/)

2 ( ?2 ,

0-

(2.4)

Operators

Let us consider some physical quantity / which characterises the state
of a quantum system. Strictly, we should speak in the following discussion
not of one quantity, but of a complete set of them at the same time.
However, the discussion is not essentially changed by this, and for brevity
and simplicity we shall work below in terms of only one physical quantity.
The values which a given physical quantity can take are called in quantum

mechanics

its

eigenvalues,

and the

set of these is referred to as the spectrum

In classical mechanics, generally speakrun through a continuous series of values. In quantum mech-

of eigenvalues of the given quantity.
ing, quantities

anics also there are physical quantities (for instance, the co-ordinates)

eigenvalues occupy a continuous range ; in such cases

whose

we speak of a continuous

spectrum of eigenvalues. As well as such quantities, however, there exist in
quantum mechanics others whose eigenvalues form some discrete set; in

such cases we speak of a discrete spectrum.
We shall suppose for simplicity that the quantity / considered here has a
discrete spectrum; the case of a continuous spectrum will be discussed in §5.

The eigenvalues of the quantity / are denoted by/n where the suffix n takes
We also denote the wave function of the system, in
the values 0, 1, 2, 3
,

Y

The wave functions
the state where the quantity / has the value n by
n
Each of these
quantity/.
physical
given
of
the
eigenfunctions
the
called
n are
,

.

Y

functions

is

supposed normalised, so that

J>n
If the

system

is

in

some

=

|*d«z

(3.1)

l.

arbitrary state with

wave function T, a measure-

one of the eigenprinciple of superposition, we can assert
the
with
accordance
In

fn
must be a linear combination of those eigenfuncthat the wave function
to the values fn that can be obtained, with probcorrespond
which
tions
TC
ability different from zero, when a measurement is made on the system and
Hence, in the general case of an arbitrary state,
it is in the state considered.
can be represented in the form of a series
the function

ment

of the quantity / carried out

values

on

it

will give as a result

.

Y

T

T

Y=Sa Yw
w

where the summation extends over

all n,

(3.2)

,

and the a n are some constant

coeffi-

cients.

reach the conclusion that any wave function can be, as we say,
set of
expanded in terms of the eigenfunctions of any physical quantity.
complete
called
a
is
made
can
be
expansion

an
such
which
of
functions in terms

Thus we

A

(or closed) set.

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9

Operators

§3

The
ing

expansion (3.2) makes

(i.e.

it

possible to determine the probability of find-

the probability of getting the corresponding result on measurement),

wave function Y, any given value/n of the quantity
For, according to what was said in the previous section, these probabili-

in a system in a state with
/.

Y

must be determined by some expressions bilinear in
and Y*, and
must be bilinear in an and a n*. Furthermore, these expressions
must, of course, be positive. Finally, the probability of the value fn must
become unity if the system is in a state with wave function
n and
must become zero if there is no term containing n in the expansion (3.2)
of the wave function Y. This means that the required probability must be
ties

therefore

Y=Y

,

Y

if all the coefficients a n except one (with the given n) are zero, that one
being unity; the probability must be zero, if the an concerned is zero. The
only essentially positive quantity satisfying these conditions is the square of the
modulus of the coefficient an Thus we reach the result that the squared
modulus \a n % of each coefficient in the expansion (3.2) determines the probability of the corresponding value fn of the quantity / in the state with wave
function Y. The sum of the probabilities of all possible values fn must be
equal to unity; in other words, the relation

unity

.

\

S

n

la w 2
|

=

1

(3.3)

must hold.

bilinear in

Y were not normalised,

then the relation (3.3) would not
would then be given by some expression
and Y*, and becoming unity when
was normalised. Only

If the function

The sum £

hold either.

Y

the integral J

YY* dq

is

\a n 2
\

Y

Thus

such an expression.

the equation

= JYY*d?

SflA*

(3.4)

must hold.

On

the other hand, multiplying by

the function

Y*

(the

this

with

expansion

(3.4),

Y*

=

tt

*

= San*jYn*Yd?

in the expansion of the function

an
substitute here

*
n *Yn of
obtain

we

,

from which we derive the following formula determining the

we

tf

we have

SaB«

If

S

integrating,

= S an*j Yn*Y dq.

j YY* dq

Comparing

Y the

complex conjugate of Y), and

from

coefficients

Y in terms of the eigenfunctions Yn
= jYYn*d 2

(3.2),

«n

we

(3.5)

.

obtain

= Sa w fY Yn*d9
TO

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an

:

,


The Basic Concepts of Quantum Mechanics

10

from which

it is

evident that the eigenfunctions must satisfy the conditions

JT m Vn*dq = 8 nm

=

where $ nm

T

= m and Snm = for n
TwTn* with m ^ n vanish

for n

1

of the products

§3

#

(3.6)

,

The fact that the integrals

m.

called the orthogonality of the

is

T

Thus the set of eigenfunctions n forms a complete set of
n
normalised and orthogonal (or, for brevity, orthonormal) functions.
We shall now introduce the concept of the mean value f of the quantity/
in the given state. In accordance with the usual definition of mean values,

functions

.

we

define / as the sum of all the eigenvalues fn of the given quantity, each
multiplied by the corresponding probability |a n 2
Thus
.

|

/= IfnWWe

(3.7)

/ in the form of an expression which does not contain the
a n in the expansion of the function T, but this function itself.
Since the products a n a n * appear in (3.7), it is clear that the required expression must be bilinear in

and Y*. We introduce a mathematical operashall write

coefficients

T

which we denotef by /and define as follows. Let (/Y) denote the result
of the operator / acting on the function Y. We define / in such a way that
the integral of the product of (/Y) and the complex conjugate function Y*
tor,

is

mean

equal to the

value /:

/=jV(/T)d?
It

is

(3.8)

.

easily seen that, in the general case, the operator

For, using the expression (3.5) for a n
definition (3.7) of the mean value in the form
integral operator.

/= \fnana* = J Y*(S anfnWn
Comparing

this

the function

with

(3.8),

we

we

/ is a linear J
we can rewrite the

6q.

see that the result of the operator

Y has the form
(/Y)

If

)

,

= San /MYw

(3.9)

.

substitute here the expression (3.5) for an

/ acting on

,

we

find that /is an integral

operator of the form

= JK(q,
tfr)

where the function K(q,

q') (called

K(q,

f

By

+

An operator is

convention,

we

F 1 and *F 2

X

=

it

by

letters

with circumflexes.

has the properties

=M+/Y, and/(«T) = afY,

are arbitrary functions

is

S/.Y.-foT^fe).

shall always denote operators

said to be linear if

/(^i+T2 )
where

q')

the kernel of the operator)

(3.io)

and a

is

an arbitrary constant.

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(3.11)


Operators

§3

11

Thus, for every physical quantity in quantum mechanics, there
corresponding linear operator.
It is

seen from (3.9) that,

if

the function

T

is

is

a definite

one of the eigenfunctions \Fn

(so that all the a n except one are zero), then, when the operator / acts on

this function is simply multiplied by the corresponding eigenvalue fn

it,

:

fa =/«^n-

(3.12)

(In what follows we shall always omit the parentheses in the expression
(/T), where this cannot cause any misunderstanding; the operator is taken
to act

on the expression which follows

Thus we can say

it.)

that the eigen-

functions of the given physical quantity /are the solutions of the equation

where / is a constant, and the eigenvalues are the values of this constant for
which the above equation has solutions satisfying the required conditions.
Of course, while the operator /is still defined only by the expressions (3.10)
and (3.11), which themselves contain the eigenfunctions ^I^, no further conclusions can be drawn from the result we have obtained. However, as we
shall see below, the form of the operators for various physical quantities can
be determined from direct physical considerations, and then the above property of the operators enables us to find the eigenfunctions and eigenvalues

by solving the equations p¥
f¥.
The values which can be taken by real physical quantities are obviously
real.
Hence the mean value of a physical quantity must also be real, in any

=

state.

Conversely,

state, its

the

if

the

mean

eigenvalues also are

mean

functions

value of a physical quantity

show

is

real in every

note that
values coincide with the eigenvalues in the states described by the

Tn

From the

all real;

to

this, it is sufficient to

.

mean values are real, we can draw some conclusions
concerning the properties of operators. Equating the expression (3.8) to its
complex conjugate, we obtain the relation
fact that the

J

T*(/Y) dg

= j Y(f*Y*) dg,

(3.13)

where /* denotes the operator which is the complex conjugate of /. j- This
an arbitrary linear operator, so that it is
a restriction on the form of the operator /. For an arbitrary operator / we
can find what is called the transposed operator/, defined in such a way that
relation does not hold in general for

JY(fl>)dq=f<l>(jY)dq,
where *F and
the function
(3.13)

By

t
that for


two

different functions.

If

we

(3.14)

take, as the function

O,

T* which is the complex conjugate of Y, then a comparison with

shows that we must have

definition, if for^the operator

which we have /*^t*

=

/=/••
/we h&vefifi =

<f>,

(3.15)
then the complex conjugate operator

<f>*.

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/*

is