QUANTUM THEORY

By

DAVID BOHM
Palmer Physical Laboratory
Princeton University

PRENTICE-HALL, INC.
Englewood Cliffs, N. J.


PRENTICE-HALL PHYSICS SERIES
DONALD H. M ENZEL, Editor

C OPYRIGHT, 1951, BY
PRENTICE-HALL, INC.
Englewood Cliffs, N . J.

ALL

RIGHTS

RESERVED.

NO

PART

OF

THIS

BOOK

MAY BE REPRODUCED IN ANY FORM, BY MIMEO­
GRAPH OR ANY OTHER MEANS, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHERS.

First printing...........February,
Second printing..........January,
Third printing......... September,
Fourth printing ..............June.
Fifth printing............January,
Sixth printing...............May,
Seventh printing.........January,
Eighth printing............. April,
Ninth printing........... January,
Tenth printing.......... February,

1951
1952
1952
1955
1956
1958
1959
1960
1961
1963


PRINTED IN THE UNITED STATES OF AMERICA

747 8 7-C


PREFACE
THE QUANTUM THEORY is the result of long and successful eff orts of
physicists to account cor rectly for an extremely wide range of experi­
mental results, which the previously existing classical theory could not
even begin to explain. It is not generally realized, however, that the
quantum theory represents a radical change, not only in the content of
scientific knowledge, but also in the fundamental conceptual framework
in terms of which such knowledge can be expressed. The true extent of
this change of conceptual framework has perhaps been obscured by the
contrast between the relatively pictorial and easily imagined terms in
which classical theory has always been expressed, with the very abstract
and mathematical form in which quantum theory obtained its original
development. So strong is this contrast that an appreciable number of
physicists were led to the conclusion that the quantum properties of
matter imply a renunciation of the possibility of their being understood
in the customary imaginative sense, and that instead, there remains only
a self-consistent mathematical formalism which can, in some mysterious
way, predict correctly the numerical results of actual experiments.
Nevertheless, with the further development of the physical interpretation
of the theory (primarily as a result of the work of Niels Bohr) , it finally
became possible to express the results of the quantum theory in terms of
comparatively qualitative and imaginative concepts, which are, however,
of a totally different nature from those appearing in the classical theory.
To provide such a formulation of the quantum theory at a relatively
elementary level is the central aim of this book.

The precise nature of the new quantum-theoretical concepts will be
developed throughout the book, principally in Chapters 6, 7, 8, 22, and 23,
but the most important conceptual changes can be briefly summarized
here. F irst, the classical concept of a continuous and precisely defined
trajectory is fundamentally altered by the introduction of a description
o f motion in terms of a series of indivisible transitions. Second, the
rigid d eterminism of classical theory is replaced by the concept of caus­
ality as an approximate and statistical trend. Third, the classical
assumption that elementary particles have an " intrinsic " nature which
can ne ver change is replaced by the assumption that they can act either
like waves or like partic les, depending on how they are treated by t he
surrounding environment. The application of these three new con­
cepts re sults in the breakdown of an assumption which lies behind much
iii


iv

PREFA CE

of our customary language and way of thinking ; namely, that the world
can correctly be analyzed into distinct parts, each having a separate
existence, but working together according to exact causal laws to form
the whole. Instead, quantum concepts imply that the world acts more
like a single indivisible unit, in whic h even the " intrinsic " nature of each
part (wave or particle) depends to some degree on its relationship to its
surroundings. It is only at the microscopic (or quantum) level, however,
that the indivisible unity of the various parts of the world produces
significant effects, so that at the macroscopic (or classical) level, the
parts act, to a very high degree of approximation, as if they did have a

completely separate existence.
It has been the author's purpose throughout this book to present
the main ideas of the quantum theory in non-mathematical terms.
Experience shows, however, that some mathematics is needed in order to
e xpress these ideas in a more precisely defined form, and to indicate how
typical problems in the quantum theory can be solved. The general
p lan adopted in this book has therefore been to supplement a basically
qualitative and physical presentation of fundamental principles with a
b road range of specific applications that are worked out in considerable
mathematical detail.
In accordance with the general plan outlined above, unusual emphasis
i s placed (especially in Part I) on showing how the quantum theory can
be developed in a natural way, starting from the previously existing
c lassical theory and going step by step through the experimental facts
and theoretical lines of reasoning which led to replacement of the classical
theory by th e quantum theory. In this way, one avoids the need for
i ntroducing the basic principles of quantum theory in terms of a com­
p lete set of abstract mathematical postulates, justified only by the fact
t hat complex calculations based on these postulates happen to agree with
experiment. Although the trea tment adopted in this book is perhaps not
as neat mathematically as the postulational approach, it has a threefold
advantage. First, it shows more clearly why such a radically new kind of
theory is needed. Second, it makes the physical meaning of the theory
clearer. Third, it is less rigid in its conceptual structure, so that one
can see more easily how small modifications in the theory can be made
if complete agreement with experiment is not immediately obtained.
Although the qualitative and physical development of the quantum
theory takes place mainly in Parts I and VI, a systematic effort is made
throughout the whole book to explain the results of mathematical
calculations in qualitative and physical terms. It is hoped, moreover,

that the mathematics has been simplified sufficiently to allow the reader
to follow the general line of reasoning without spending too much time on
mathematical details . F inally, it should be st ated that the relative
di not intended for the purpose of r educing


PREFACE

t he amount of thinking needed for a thorough grasp of the theory.
Instead, it is hoped that the reader will thereby be stimulated to do more
thinking, and thus to provide himself with a general point of view which
serves to orient him for further reading and study in this fascinating field.
An appreciable part of the material in this book was suggested by
remarks made by Professor J. R. Oppenheimer in a series of lectures on
quantum theory delivered at the University of California at Berkeley,
and by notes on part of these lectures taken by Professor B. Peters. A
series of lectures by Niels Bohr, entitled "Atomic Theory and the
Description of Nature " were of crucial importance in supplying the
general philosophical basis needed for a rational understanding of quan­
tum theory. Numerous discussions with students and faculty at Prince­
ton University were very helpful in clarifying the presentation. Dr. A.
Wightman, in particular, contributed significantly to the clarificat ion of
Chapter 22, which deals with the quantum theory of measurements.
Members of the author's quantum theory class in 1947 and 1948 per­
formed invaluable work, checking both the mathematics and the reason­
ing, while the manuscript was being written. Finally, the author wishes
to express his gratitude to M. Weinste in, who read and criticized the
manuscript, and who supplied many very useful suggestions, and to
L. Schmid who edited the manuscript and read the proofs.

DAVID BOHM



CO N T E N T S

P A RT I

Physical Formulation ol the Quantum Theory

1. THE ORIGIN OF THE QUANTUM THEORY

•

2. FURTHER DEVELOPMENTS OF THE EARLY QUANTUM
THEORY

5

23

3. WAVE PACKETS AND DE BROGLIE WAVES.

59

4. THE DEFINITION OF PROBABILITIES.

81

5. THE UNCERTAINTY PRINCIPLE.

99

•

•

6. WAVE VS. PARTICLE PROPERTIES OF MATTER

116

7. SUMMARY OF QUANTUM CONCEPTS INTRODUCED

141

8. AN ATTEMPT TO BUILD A PHYSICAL PICTURE OF THE
QUANTUM NATURE OF MATTER

•

•

144

P A R T 11

Mathematical Formulation ol the Quantum Theory

9. WAVE FUNCTIONS, OPERATORS, AND SCHRO DINGER'S
EQUATION

.

•

.

.

.

•

.

.

173

10. FLUCTUATIONS, CORRELATIONS, AND EIGENFUNCTIONS 199
vii


CONTENTS

viii

P A R T Ill

Applications to Simple Systems.

Further Extensions

ol Quantum Theory Formulation

11. SOLUTIONS OF WAVE EQUATIONS FOR SQUARE POTENTIALS

•

•

•

•

•

•

.

•

•

.

·

·

·

·

·

1 2. THE CLASSICAL LIMIT OF QUANTUM THEORY.

22 9

THE WKB

APPROXIMATION.

264

'13. THE HARMONIC OSCILLATOR

296

14. ANGULAR MOMENTUM AND THE THREE-DIMENSIONAL

310

WAVE EQUATION

15. SOLUTION

OF

RADIAL

EQUATION,

THE HYDROGEN

ATOM, THE EFFECT OF A MAGNETIC FIELD.

.

334

16. MATRIX FORMULATION OF QUANTUM THEORY

3 61

17. SPIN AND ANGULAR MOMENTUM.

387

•

•

•

•

•

P A R T IV

Methods of Approximate Solution of SchrOc/inger's Equation

18. PERTURBATION

THEORY,

TIME-DEPENDENT AND TIME-

INDEPENDENT

407

19. DEGENERATE PERTURBATIONS

462

20. SUDDEN AND ADIABATIC APPROXIMATIONS

496

PART V

Theory of Scattering

21. THEORY OF SCATTERING.

.

.

.

.

.

.

511


CONTENTS

ix

P A RT V I

Quantum Theory ol the Process ol Measurement

22. QUANTUM THEORY OF THE PROCESS OF MEASUREMENT
23. RELATIONSHIP
CONCEPTS.
INDEX

•

•

•

•

583

BETWEEN QUANTUM AND CLASSICAL
624

•

629



PART I

PH Y S I C AL F O R M U LAT I ON O F T H E

QUANTUM THEORY

MonERN QUANTUM THEORY

is unusual in two respects. First, it embodies
a set of physical ideas that differ completely with much of our everyday
experience, and also with most experiments in physics on a macroscopic
scale. Second, the mathematical apparatus needed to apply this theory
to even the simplest examples is much less familiar than that required in
corresponding problems of classical physics. As a result, there has been

a tendency to present the quantum theory as being inseparable from the
mathematical problems that arise in its applications. This approach
might be likened to introducing Newton's laws of motion to a student of
elementary physics, as problems in the theory of differential equations.
In this book, special emphasis is placed on developing the guiding phys­
ical principles that are useful not only when it is necessary to apply our
ideas to a new problem, but also when we wish to forsee the general
properties of the mathematical solutions without carrying out extensive
calculations. The development of the special mathematica l techniques
that are necessary for obtaining quantitative results in complex problems
should take place, for the most part, either in a mathematics course or in
a special course concerned with the mathematics of quantum theory.
It seems impossible, however, to develop quantum concepts extensively
without Fourier analysis. It is, therefore, presupposed that the reader
is moderately familiar with Fourier analysis.
In the first part of this book, an unusual amount of attention is given
to the steps by which the quantum theory may be developed, starting
with classical theory and with specific experiments that led to the replace­
ment of classical theory by the quantum theory. The experiments are
presented not in historical order, but rather in what may be called a
lo gical order. An historical order would contain many confusing ele­
ments that would hide the inherent unity that the quantum theory
possesses. In this book, the experimental and theoretical developments
are presented in such a way as to emphasize this unity and to show that
each new step is either based directly on experiment or else follows
logically from the previous steps. In this manner, the quantum theory
can be made to seem less like a strange and somewhat arbitrary prescrip­
tion, justified only by the fact that the results of its abstruse mathematical
calculations happen to agree with experiment.
1

2

PHYSICAL FORMU LATION OF THE QUANTUM THEORY

As an int egral part of our plan for developing t he t heory on a basis
t hat is not t oo abst ract for a beginner, a complet e account of t he relat ion
bet ween quant um t heory and t he previously exist ing classical t heory is
given. Wherever possible t he meaning of t he quant um t heory is illus­
t rat ed in simple physical t erms. Moreover, t he final chapt er of Part I
point s out broad regions of everyday experience in which we c ont inually use
ways of t hinking t hat are closer to quant um-t heoret ical t han t o classical
concept s. In t his chapt er, we also discuss in det ail some of t he philo­
sophical implicat ions of t he quant um t heory, and show t hat t hese lead
t o a st riking modificat ion in our general view of t he world, as compared
wit h t hat suggest ed by classical t heory.
The reader will not ice t hat most of t he problems are int erspersed
t hroughout t he t ext . These problems should be read as part of t he t ext ,
because t he result s obt ained from t hem are oft en used direct ly in t he
d evelopment of ideas. It is usually possible t o underst and t he sig­
nificance of t he result s wit hout solving t he problems, but t he reader is
st rongly urged t o t ry t o solve t hem. The main advant age of t he int er­
spersed problems is t hat t hey make t he reader t hink more specifically
about t he subject previously discussed, t hus facilit at ing his underst and­
ing of the subject .
Supplementary References

The following list of supplement ary text s will prove very helpful t o
the reader and will be referred t o t hroughout various part s of t his book:

Bohr, N., Atomic Theory and the Description of Nature.
University Press, 1934.
Born, M., Atomic Physics.

London: Cambridge

Glasgow: Blackie & Son, Ltd ., 1 945.

Born, M., Mechanics of the Atom.

London: George Bell & Sons, Ltd., 1927.

Dirac, P. A. M., The Principles of Quantum Mechanics. · Oxford: Clarendon Press,
1947.
Heisenberg, W., The Physical Principles of the Quantum Theory.
versity of Chicago Press, 1930.
Kramers, H. A., Die Grundlagen der Quantentheorie.
Verlagsgesellschaft, 1938.
Mott, N. F., An Outline of Wave Mechanics.
Press, 1 934.

Chicago: Uni­

Leipzig: AJrn.demische

London: Cambridge University

Mott, N. F., and I. N. Sneddon, Wave Mechanics and Its Applications.
Clarendon Press, 1948.

Oxford:

Pauli, W., Die Allgemeinen Prinzipen der Wellenmechanik. Ann Arbor, Mich.:
Edwards Bros., Inc . , 1946. Reprinted from Handbuch der Physik, 2. Aufl.,
Band 24. 1 . Teil.
Pauling, L., and E. Wilson, Introduction to Quantum Mechanics.
McGraw-Hill Book Company, Inc., 1935.

New York:

Richtmeyer, F. K., and E. H. Kennard, Introduction to Modern Physics.
York: McGraw-Hill Book Company, Inc. , 1933.

New


PHYSICAL FORMULATION OF TH E QUA NTUM TH EORY

Rojansky, V., Introductory Quantum Mechanics.
1938.

New York: Prentice-Hall , Inc . ,

Ruark, A. E., and H. C. Urey, Atoms, Molecules, and Quanta.
McGraw-Hill Book Company, Inc., 1930.
Schiff, L., Quantum Mechanics.
1949.

3

New York:

New York: McGraw-Hill Book Company, Inc.



CHAPTER 1

The Origin oF the Quantum Theory
The Rayleigh-Jeans Law

1. Blackbody Radiation in Equilibrium. Historically, the quantum
theory began with the attempt to account for the equilibrium distribu­
tion of electromagne tic radiation in a hollow cavity. We shall, therefore,
begin with a brief d escription of the characteristics of this distribution of
radiation. The radiant energy originates in the walls of the cavity, which
continually emit waves of every possible frequency and direction, at a
rate which increases very rapidly with the temperature. The amount
of radiant energy in the cavity does not, however, continue to increase
indefinitely with time, because the process of emission is opposed by the
process of absorption that takes place at a rate proportional to the
intensity of radiation already present in the cavity. In the state of
thermodynamic equilibrium, the amount of energy U(v) dv, in the fre­
quency range between v and v + dv, will be determined by the condition
that the rate at which the walls emit this frequency shall be balanced by
the rate at which they absorb this frequency. It has been demonstrated
both experimentally and theoretically, * that after equilibrium has been
reached, U (v) depends only on the temperature of the walls, and not on
the material of which the walls are made nor on their structure.
To observe this radiation, we make a hole in the wall. If the hole is

very small compared with the size of the cavity, it produces a negligible
change in the distribut ion of radiant energy inside the cavity. The
intensity of radiation per unit solid angle coming through the hole is then

readily shown to be J(v)

=

4: U(v) , where

c

is the velocity of light. t

Measurements disclose that, at a particular temperature, the function
U(v) follows a curve resembling the solid curve of Fig. 1. At low fre­
quencies the energy is proportional to v 2 , while at high frequencies it
drops off exponentially. As the temperature is raised, the maximum is
shifted in the direction of higher frequencies ; this accounts for the change
i n the color of the radiation emitted by a body as it gets hotter.
By thermodynamic argumentst Wien showed that the distribution
Richtmeyer and Kennard. (See list of references on p. 2.)
t See Richtmeyer and Kennard for a derivation of this formula and also for a
more complete account of blackbody radiation. The term "blackbody" arose
5
*


6

PHYSICAL FORMULATION OF THE QUANTUM TH EORY

[1.1

must be of the form U(v) = v 3f(v/T) . The function f, however, cannot
be determined from thermodynamics alone. Wien obtained a fairly
good, but not perfect, fit to the empirical curve with the formula
(Wien's law)

( 1)

Here K is Boltzmann's constant, and h is an experimentally determined
constant (which later turned out to be the famous quantum of action) . *

/.:__RAYLEIGH-JEANS
I

I

I
I
/.---

U(V)

EMPIRICAL
--CURVE

Frn. 1

Classical electrodynamics, on the other hand, leads to a perfectly
d efinite and quite incorrect form for U( v) . This theoretical distribution,
which will be derived in subsequent sections, is given by
U(v) dv

,..._,

KTv 2 dv

(Rayleigh-Jeans law)

(2)

Reference to Fig. 1 shows that the Rayleigh-Jeans law is in agreement
with experiment at low frequencies, but gives too much radiation for
high frequencies. In fact, if we attempt to integrate over all frequencies
to find the total energy, the result diverges, and we are led to the absurd
conclusion that the cavity contains an infinite amount of energy. Experi­
mentally, the correct curve begins to deviate appreciably from the
Rayleigh-Jeans law where hv becomes of the order of KT. Hence, we must
try to develop a theory that leads to the classical results for hv < KT, but
which deviates from classical theory at higher frequencies.
Before we proceed to discuss the way in which the classical theory
must be modified, however, we shall find it instru ctive to examine in
some detail the derivation of the Rayleigh-Jeans law. In the course of
this deviation we shall not only gain insight int o the ways in which
classical physics fails, but we shall also be led to introduce certain classical
physical concepts that are very helpful in the understanding of the
quantum theory. In addition, the introduction of Fourier analysis to
because the radiation from a hole in such a cavity is identical with that coming from

a perfectly black object .
* Wien did not actually introduce Planck's constant, h, but instead the const;i,nt

h/K,


1.3]

THE ORIGIN OF THE QUA NTUM TH EORY

7

deal with this classical probl em will also constitute some pr eparation
for its later use in the problems of quantum theory.
According to classical electrodynamics,
2. Electromagnetic Energy.
empty space containing electromagnetic radiation possesses energy. In
fact, this radiant energy is responsible for the ability of a hollow cavity
and the mag­
to absorb heat. In terms of the electric field, B(x, y, z,
netic field, 3C(x, y, z,
the energy can be shown to be*

t),

t),

E = l_

811"

J (82

+

JC 2 ) dr

(3)

where dr signifies integration over all the space avail able to the fields.
Our problem, then, is to determine the way in which this energy is dis­
tributed among the various frequencies present in the cavity when the
walls are at a given temperature. The first step will be to use Fourier
analysis for the fields and to express the energy as a sum of contributions
from each frequency. In so doing, we shall see that the radiation field
behaves, in every respect, like a collection of simple harmonic oscillators,
the so-called " radiation oscillators." We shall then apply statistical
mechanics to these oscillators and determine the mean energy of each
oscillator when it is in equilibrium with the walls at the temperature T.
F inally, we shall determine the number of oscillators in a given frequency
r ange and, by multiplying this number by the mean energy of an oscil­
lator, we shall obtain the equilibrium energy corresponding to this
frequency, i.e., the Rayleigh-Jeans law.
3. Electromagnetic Potentials. We begin with a brief review of
electrodynamics. The partial differential equations of the electromagnetic
field, according to Maxwell, are given by
VXE=

1 :re

c

a
at

V·JC=O

1
.
v x :re= -car,+ 411")

as

(4)

--­

(5)

v.

8

= 411"p

(6)
(7)

wherej is the current density and p is the charge density. We can show
from (4) and (5) that the most general electric and magnetic field can be

expressed in terms of the vector and scalar potentials, a and c/>, in the
following way :
3C=VXa
(8)
and

8=

1 aa

cat

Vcp

(9)

When 8 and 3C are expressed in this form, (4) and (5) are satisfied identi­
cally, and the equations for a and cl> are then obtained by the substitution
of relations (8) and (9) into (6) and (7) .
Now, eqs. (8) and (9) do not define the potentials uniq uely in te rms
* Richtmeyer and Kennard,

-

Chap. 2,

-


8

PHYSICAL FORMULATION OF THE QUA NTUM THEORY

[1.3

of the fields. If, for example, we add an arbitrary vector, -Vi/;, to the
vector potential, the magnetic field is not changed because V X Vi/; = 0.
If we simultaneously add the quantity

�:

t

1/; to the scalar potential, the

electric field is also unchanged. Thus, we find that the electric and
magnetic fields remain invariant under the following transformation of
the potentials :*
a'=a - Vi/;
ct/ = cp + !
c

ay;
at

}

(10)

The above is called a " gauge transformation."

We can utilize the invariance of the fields to a gauge transformation
for the purpose of simplifying the expressions for 8 and :JC. A common
choice is to make div a = 0. To show that this is always possible, sup­
pose that we start with an arbitrary set of potentials, a(x, y, z, t) and
cfi(x, y, z, t). We then make the gauge transformation of eq. (10) to a
new set of potentials, A' and cfi'. In order to obtain div a' = O, we must
choose 1/; such that
div a - V 21/; = 0
But the above is just Poisson's equation defining 1/; in terms of the speci­
fied function, div a. Its solution can always be obtained and is, in fact,
e qual to
div a(x', y', z', t) dx' dy' dz'
__!__
1/; =
_

411"

JJJ

[r - r'l

Thus, we prove that a gauge transformation that yields div a' = 0 can
always b e carried out.
We now show that in empty space the choice div a = 0 also leads to
cfi = 0 and, therefore, to a considerable simplification in the representa­
tion of the electric field. To do this, we substitute eq. (9) into (7),
setting p = 0 since, by hypothesis, there are no charges in empty space.
The result is
aa

- V 2cfi = o
div E= - ! div
c

But since div

a

=

0, we obtain

at

v2c1i = o

This is, however, simply Laplace's equation. It is well known that the
only solution of this equation that is regular over all of space is cfi = 0.
(All other solutions imply the existence of charge at some points in space
and, therefore, a failure of Laplace's equation at these point. s.) W�
* 8 and :JC are the only physically signific ant quantities connected with the electro­
magnetic field.


1.4)

THE ORIGI N OF THE QUA NTUM TH EORY

9

should note, however, that the condition </> = 0 follows only in empty
space because, in the presence of charge, eq. (7) leads to 'V2</> = - 411"p,
which is Poisson's equation. This equation has nonzero regular solutions,
provided that p is not everywhere zero.
We conclude, then, that in empty space we obtain the following
expressions for the fields :
( 1 1)
X = V X a
1 aa
(12)
8=
c at
subject to the condition that

div a= 0

(13)

Finally, we obtain the partial differential equation defining a in
empty space by inserting ( 1 1 ) , (12) , and (13) into (6) , provided that we
also assume that j = 0, as is necessary in the absence of matter. We
obtain
(14)
Equations (11), (12) , (13), and (14) , together with the boundary condi­
tions, completely determine the electromagnetic fields in a cavity that
contains no charges or currents.
4. Boundary Conditions. As pointed out in Sec. 1, it has been demon­
strated both experimentally and theoretically* that the equilibrium
distribution of energy density in a hollow cavity does not depend on the
shape of the container or on the material in the walls. Hence, we are

at liberty to choose the simplest possible boundary conditions consistent
with equilibrium. We shall choose a set of boundary conditions that
are somewhat artificial from an experimental point of view, but that
greatly simplify the mathematical treatment.
Let us imagine a cube of side L with very thin walls of some material
that is not an electrical conductor. We then imagine that this structure
is repeated periodically through space in all directions, so that space is
filled up with cubes of side L. Let us suppose, further, that the fields
are the same at corresponding points of every cube.
We now assert that these boundary conditions will yield the same
equilibrium radiation density as will any other boundary conditions at
the walls. t To prove this, we need only ask why the equilibrium condi­
tions are independent of the type of boundary. The answer is that, from
* The theoretical proof depends on the use of statistical mechanics. See, for
example, R. C. Tolman, The Principles of Statutical Mechanics. Oxford, Clarendon
Press, 1938.
t With these conditions, no walls are actually necessary, but the thermodynamic
results are the same as for an arbitrary wall, including, for example, a perfect reflector
or a perfect absorber.


10

PHYSICAL FORMULATION OF THE QUANTUM THEORY

[1.5

the thermodynamic viewpoint, the wall merely serves to prevent the
system from gaining or losing energy. Making the fields periodic must
have the same effect because each cube can neither gain energy from the

other cubes nor lose it to them; if this were not so, the system would
cease to be periodic. Thus, we have a boundary condition that serves
the essential function of keeping the energy in any individual cube
constant. Although artificial, it must give the right answer, and it will
make the calcub.tio::.::, easier by simplifying the Fourier analysis of the
fields.
6. Fourier Analysis. Now, a (x, y, z, t) may be any conceivable solu­
tion of Maxwell's equations, with the sole restriction, imposed by our
boundary conditions, that it must be periodic in space with period L/n,
where n is an integer. * It is a well-known mathematical theorem that
an arbitrary periodic function, t f(x, y, z, t), can be represented by means
of a Fourier series in the following manner:

f (x, y, z, t)
271'
271'
(lx+my+nz) ] (15)
sm
(lx+my+nz)+bz,m,n
(t)
= � az,m,n (t) cos
L
L
�[
l,m,n
where l, m, n are integers running from
to
including zero. Any
choice of a's and b's leading to a convergent series defines a function,
f(x, y, z, t), which is periodic in the sense that it takes on the. same value

each time x, y, or z changes by L. For a given function, f(x, y, z, t) it can
be shown that the a1,m,,.(t) and the bi,m,n (t) are given by the following
•

-

formulas :

az,m,n (t) + a_z,-m,-n (t)
2 [L f L f L
= 3 } 0 0 0 dx dy dz
L
J J

cos

}f L }f }f dx dy dz

sin

bz,m,n (t) - b_z,-m,-n (t)
=

2
L

a

0

L

0

L

0

oo

oo,

{ (lx + my + nz)f(x, y, z, t)

2

{ (lx + my + nz)f(x, y, z, t)

(16)

2

These formulas illustrate the fact that only the sum of the a's and the
difference of the b's are determined by the function f.
From the above, we conclude that f may be specified completely in
terms of the quantities az,m,n + a-l,-m, -n and bz,m,n - b_z,-m,-n, but we
prefer to retain the specification in terms of the az,m.n and bz,m,n because
of the simpler mathematical expressions to which they lead.
"There will be, of course, the usual regularity conditions that prevent a fro m

being infin ite or discontinuous.

t The function must be piecewise continuous,


1.5]

11

THE ORiGI N OF TH E QUANTUM TH EORY

[ L [L [L dx dy dz cos 2{, (lx + my + nz)
} } }
sin � (l'x + m'y + n'z)
f1L loL loL dx dy dz cos � (lx + my + nz)

Equations (16) are obtained with the aid of the following orthogonal­
ity relations :*
0

0

0

unless
in which case
is L 3 .

( )

(

)

2ir
L

=

(17a)

(l'x + m'y + n'z) 0
l - l'
l = -l'
m = -m'
m = m'
or
n = -n'
n = n'
it is L3/2, except when l = m n = 0, in which case it
cos

=

=

loL loL loL sin � (lx + my + nz) sin � (l'x + m'y + n'z)

unless

0

(ml -= l'm')
n = n'

or

(ml == -l'-m')

=

0 (17b)

n = -n'

in which case it is L3 /2. [It is suggested that the reader prove (17a and
b) as an exercise, and use the results to obtain (16) .]
Fourier analysis, in the preceding form, enables us to represent an
arbitrary function as a sum of standing plane waves of all possible wave­
lengths and amplitudes. The entire treatment is essentially the same
as that used with waves in strings and organ pipes, except that it is
three-dimensional.
Let us now expand the vector potential in a Fourier series. Because a
is a vector, involving three components, each ai,m,n and bz,m,n also has three
componerits and, hence, must be represented as a vector :

a = � [ ai,m,n(t) cos
l, m,n

� (lx+my+nz) + b1,m,,.(t) sin � (lx+my+nz) ]

We assume that a0,0,0 is zero in the above series. t

(s - !catCla) .

* For the origin of the term " orthogonality " see Chap. 16, Sec. 10; also Chap. 10,
Sec . 24.
1 This follows from the fact that the part of a which is constant in space corre-

sponds to no magnetic field, and to a spatially uniform electric field

=

Such a field requires a charge distribution somewhere to produce it, i .e . , at the bound­
aries, and since we assume that no such distribution is present, we set ao,o,o = 0.


12

[1 .6

PHYSICAL FORMULATION OF THE QUANTUM TH EORY

=

We now introduce the propagation vector k, defined as follows :
"'

2irm

L

k2

k
•

k11 _
- L

k =2irl
=

(2{)2 (l2 + m2 + n 2)

n

27r

L

(18)

By orienting our co-ordinate axes in such a way that the z axis is directed
along the k vector, we obtain l = m = 0, and k = 2ir/L. From the
definition of k, it follows that k/271" is the number of waves in the distance
L; hence the wavelength is 'JI. = 27r/k, or

k = 27r/X

(19)

In this co-ordinate system a typical wave takes the form cos 27rnz/L.
Thus, the vector k is in the direction in which the phase of the wave
changes. Going back to arbitrary co-ordinate axes, we conclude that k
is a vector in the direction of propagation of the wave. Its magnitude
is 271"/X, and it is allowed to take on only the values permitted by integral
l, m, and n in eq. (18) .
With this simplification of notation, we obtain
a

=

! [ak(t) cos k + b k (t) sin k
k

·

r

·

r]

(20)

where the summation extends over all permissible values of k.
6. Polarization of Waves. Let us now apply the condition div a
to (20). We have
div a =

Ik ( - k

·

ak sin k r + k
·

·

bk cos k

·

r)

=

=

O

0

It is a well-known theorem that if a Fourier series is identically zero, then
all of the coefficients, ak and bk, must vanish.
Pro blern 1 :

Prove the above theorem, using the orthogonality relations ( 1 7) .

From the above i t follows that k ak( t) = k bk(t) = 0. Thus, ak ( t)
and bk(t) are perpendicular to k, as are also the electric and magnetic
fields belonging to the kth wave. Since the vibrations are normal to the
direction of propagation, the waves are transverse. The direction of the
electric field is also called the direction of polarization.
To describe the orientation of ak let us return to the set of co-ordinat1:1
axes in which the z axis is in the direction of k. The vector ak can have
only x and y components, and if we specify the values of these, we shall
have specified both the magnitude and the direction of ak.
We designate the direction of the vector ak by the subscript µ, writing
ak,µ, where µ is allowed to take on the values 1 and 2. Forµ
1, ak,,.
=
in
the
x
direction
;
but
for
µ
2,
it
is
in
the
y
direction.
All
possible

is
·

·

=


13

THE ORIGIN OF THE QUANTUM THEORY

1.7]

Gk,i

Gk Gk,
2

vectors can then be represented as a sum of some
vector, and some
other
vector. Hence, the most general vector potential, subject to
the condition that div = 0, is given by
G

=

G
k

k,µ [Gk,p(t) cos k· r + bk,p(t) sin k

(21)

r]

•

Here the summation extends over all permissible k vectors and over the
two possible values of µ.
It can be verified from (14) and (21) that the
satisfies the following
differential equation :
(22)

Gk,,.

Gk,,.

terms oscillate with simple harmonic motion
which shows that the
and with angular frequency, w =
7. Evaluation of the Electromagnetic Energy. The first step in
evaluating the electromagnetic energy is to express 8 and :JC in terms of
the Fourier series for
These expressions are :

Problem 2:

=

G.

.
.
- c1 �
Gk,p
cos k · r + bk,,. sm k r)
(
�
k,µ
�
k,,. ( - k X Gk,,, sin k · r + k X bk,,, cos k

8 =

:JC

kc.

.

•

•

r)

Derive the above expressions for E and :JC.

Let us now evaluate the following over the cube of side L:

1
�
111" f 82 dr
k,,. � fL fL fã L+dx dy dz
(ak,à ã Qk',p' k ã k' bk,àã bk',à' sin kã sin k' ã )
+ ilk,,.Ã bk.,,.ã cos k · r sin k' · r + bk,,.· ilk',,.' sin k · r cos k' · r
With the aid of eqs. (17) we see that all integrals vanish except when
k
k', and that all terms involving ilk,,.· bk,µ' are zero. Furthermore,
ak,,.ã ak,,.' 0 unless à à'. When à à', the two vectors are, by
definition, perpendicular to each other. Thus, the above expression

g

=

S1rC 2

k',,.'

COS

=

T

0

COS

0

T

T

f

---g;-82dr

=

T

:P

=

=

reduces to

0

£3
[ 21 (ak,,.) 2 + 2lebk,· ,.) 2 ]
811"c2 �
�

k,,.
.

With a similar method, which involves somewhat more algebra, we obtain

2
1
f JC---g;- 811" �k,,. k2 [2 (ak,,,)2
dr _ L3

Problem 3 :

�

+

Derive the above expression for f:JC2 dr.

1
2

(bk,,.) 2 ]


14

[1.8

PHYSICAL FORMULATION OF THE QUA NTUM THEORY

Thus, the electromagnetic energy in the cavity is (with L3

=

V)

(23)
8. Meaning of Preceding Result for Electromagnetic Energy. The
following are the most important properties of eq. (23):
(1) The energy is a sum of separate terms, one for each
and one
for each
This means that different wavelengths and polarizations
do not interact with each other, because the interaction of any two
systems always requires that the energy of one should depend on the
state of the other. Here we see that the energy in each wave of propaga­
tion vector k and polarization direction µ is proportional only to the
square of
and
and not to any of the other a's or b's. A similar
result holds for each of the b's.
(2) The energy associated with each
(or
has the same mathe­
matical form as that of a material harmonic oscillator. A harmonic
oscillator of mass
angular frequency has energy

ak,,.,

bk,p·

cik,,. ak,,.,

ak,µ bk,,.)
m,
w,
= m (:t2 + w 2x2)
By analogy, we can write
v
m =--•
811"c2 w =kc
The frequency is then f= w/211" =
= We know, of course,
that an electromagnetic wave of wavelength has just the above fre­
E

2

kc/211"

c/X.
X

quency. * This shows that our harmonic oscillator analogy gives the
right description of the way in which the a's oscillate.
The analogy with a material oscillator can be carried further. For
example, with material oscillators, we introduce a momentum
m:t.
Here the momentum is

p=

We can then introduce a Hamiltonian function

p2 + mw 2x2
H=For the ak,µwe get
H= 811"£3c2 (Pk,µ) 2 £3811" 2 (ak,µ) 2
Similar terms may be introduced for the bk,µ·
The correct equations of motion are obtained from the Hamiltonian
equations
a.k,µ iJaH
Pk,,.= aH
pk,,,
2m

2

=

* See also eq. (22) .

and

--

2

+

k

2

(24)