Bohm, david quantum theory (prentice hall, 1951)

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 appl

PRENTICE-HALL PHYSICS SERIES

D ONALD H M ENZEL, Editor

First printing February, 1951 Second printing January, 1952 Third printing September, 1952 Fourth printing June 1955 Fifth printing January, 1956 Sixth printing May, 1958 Seventh printing January, 1959 Eighth printing April, 1960 Ninth printing January, 1961 Tenth printing February, 1963

PRINTED IN THE UNITED STATES OF AMERICA

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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 experi­mental 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 caus­ality 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 con­cepts 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 d<il- emp ha sis on mathematics is 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 quan­tum theory Numerous discussions with students and faculty at Prince­ton 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 reason­ing, 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

2 FURTHER DEVELOPMENTS OF THE EARLY QUANTUM

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

P A R T 11

Mathematical Formulation ol the Quantum Theory

9 WAVE FUNCTIONS, OPERATORS, AND SCHRODINGER'S

10 FLUCTUATIONS, CORRELATIONS, AND EIGENFUNCTIONS 199

vii

viii CONTENTS

P A R T Ill

Applications to Simple Systems Further Extensions

ol Quantum Theory Formulation

11 SOLUTIONS OF WAVE EQUATIONS FOR SQUARE TIALS • • • • • • • • · · · · ·

POTEN-12 THE CLASSICAL LIMIT OF QUANTUM THEORY THE WKB APPROXIMATION

'13 THE HARMONIC OSCILLATOR

14 ANGULAR MOMENTUM AND THE THREE-DIMENSIONAL WAVE EQUATION

15 SOLUTION OF RADIAL EQUATION, THE HYDROGEN

2 2 9

264

296

310

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, DEPENDENT AND

CONTENTS ix

P A R T V I Quantum Theory ol the Process ol Measurement

22 QUANTUM THEORY OF THE PROCESS OF MEASUREMENT 583

23 RELATIONSHIP BETWEEN QUANTUM AND CLASSICAL CONCEPTS

INDEX • • • • •

624

629

PART I

PH Y S I CAL 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 phys­ical 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 replace­ment 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 ele­ments 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 prescrip­tion, justified only by the fact that the results of its abstruse mathematical calculations happen to agree with experiment

1

2 PHYSICAL FORMULATION 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 philo­sophical 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 sig­nificance 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 er­spersed 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 and­ing 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 throughout various part s of t his book: Bohr, N., Atomic Theory and the Description of Nature London: Cambridge University Press, 1934

Born, M., Atomic Physics Glasgow: Blackie & Son, Ltd., 1945

Born, M., Mechanics of the Atom London: George Bell & Sons, Ltd., 1927 Dirac, P A M., The Principles of Quantum Mechanics · Oxford: Clarendon Press,

Pauling, L., and E Wilson, Introduction to Quantum Mechanics New York: McGraw-Hill Book Company, Inc., 1935

Richtmeyer, F K., and E H Kennard, Introduction to Modern Physics Ne w

York: McGraw-Hill Book Company, Inc , 1933

PHYSICAL FORMULATION OF THE QUANTUM THEORY 3

Rojansky, V., Introductory Quantum Mechanics New York: Prentice - Hall , 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 distribu­tion 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 fre­quency 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 v2, 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) = v3f(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

I

I /. -

Frn 1

EMPIRICAL CURVE

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 , _, KTv2 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 Experi­mentally, 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 QUANTUM THEORY 7

deal with this classical probl em will also constitute some pr eparation for its later use in the problems of quantum theory

2 Electromagnetic Energy According to classical electrodynamics, empty space containing electromagnetic radiation possesses energy In fact, this radiant energy is responsible for the ability of a hollow cavity

to absorb heat In terms of the electric field, B(x, y, z, t), and the mag­netic field, 3C(x, y, z, t), the energy can be shown to be*

E = l_ J (82 + JC2) dr

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 dis­tributed 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 oscil­lator, 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

1 a:re

c at V·JC=O

(4) (5)

v x 1 as

4 :re= -car,+ 11")

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 :

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 QUANTUM 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 + ! ay;

c at The above is called a " gauge transformation." } (10)

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, sup­pose 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

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 representa­tion 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

div E = - ! div c at aa - V2cfi = o But since div a = 0, we obtain

1.4) THE ORIGIN OF THE QUANTUM THEORY 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 :

subject to the condition that

X = V X a

8= 1 aa

c at div a= 0

( 1 1) (12)

(13) Finally, we obtain the partial differential equation defining a in empty space by inserting (11), (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 condi­tions, 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 demon­strated 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 condi­tions 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 solu­tion 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:

of the simpler mathematical expressions to which they lead

"There will be, of course, the usual regularity conditions that prevent a f ro m being infin ite or discontinuous

t The function must be piecewise continuous,

1.5] THE ORiGI N OF THE QUANTUM THEORY 1 1 Equations (16) are obtained with the aid of the following orthogonal­ity relations :*

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

l, m,n

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

*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 sponds to no magnetic field, and to a spatially uniform electric field (s = -! cat Cla)

corre-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 PHYSICAL FORMULATION OF THE QUANTUM THEORY [1.6

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

L; hence the wavelength is 'JI = 27r/k, or

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 · r + bk(t) sin k · r]

k where the summation extends over all permissible values of k

Pro blern 1 : Prove the above theorem, using the orthogonality relations (17)

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,,

is in the x direction ; but for µ = 2, it is in the y direction All possible

1.7] THE ORIGIN OF THE QUANTUM THEORY 1 3

Gk vectors can then be represented as a sum of some Gk,i vector, and some other Gk,2 vector Hence, the most general vector potential, subject to the condition that div G = 0, is given by

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 G These expressions are :

Problem 2: Derive the above expressions for E and :JC

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

g 11" 82 dr = S1rC2 � � k,, k',,.' 0 0 0 dx dy dz

(ak,µ • Qk',p' + ilk,,.· bk.,,.• COS k • T COS k' • T + bk,µ• bk',µ' sin k• T sin k' • T )

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,, • ak,,.' = 0 unless µ = µ' When µ :P µ', the two vectors are, by definition, perpendicular to each other Thus, the above expression reduces to

14 PHYSICAL FORMULATION OF THE QUA NTUM THEORY [1.8

Thus, the electromagnetic energy in the cavity is (with L3 = V)

(2) The energy associated with each ak,µ (or bk,,.) has the same mathe­matical form as that of a material harmonic oscillator A harmonic oscillator of mass m, angular frequency w, has energy

The analogy with a material oscillator can be carried further For example, with material oscillators, we introduce a momentum p = m:t

Here the momentum is

We can then introduce a Hamiltonian function

For the ak,µ we get

Similar terms may be introduced for the bk,µ·

The correct equations of motion are obtained from the Hamiltonian equations

ak,µ = iJpk,,,

* See also eq (22)

and Pk,,.= aH

1.9] THE ORIGIN OF THE QUANTUM THEORY 1 5 which yield eq (22), obtained originally by direct substitution into MaxJ well's equations

(25)

and similarly for the b's

The ak,, and bk,, are, as we have seen, analogous to the co-ordinates of separate noninteracting harmonic oscillators In a sense, the ak,, and bk,, may also be regarded as the co-ordinates of the radiation field This

is because, once they are given, the field is specified everywhere through

eq (20) There are an infinite number of these co-ordinates, because there are an infinite number of possible values of k But the infinity is discrete, or countable, as distinguished from the continuous infinity of points on a line The main advantage of the Fourier series is that it enables us to describe the fields over a continuous region of space by means of a discrete infinity of co-ordinates

How many independent co-ordinates are there for each permissible value of k? First, there are two polarization directions ; then we have, for each k and y, an ak,, and a bk,, Thus, it would seem, at first sight, that we need four independent co-ordinates for each value of k But, from eq (16), we see that it is necessary to specify only the combinations ak,, + a-k,,, and bk,, - b-k,,., so that the number of variables necessary is reduced by a factor of two This means that for each k there are two independent co-ordinates

9 Number of Oscillators We must now find the number of oscil­lators with frequencies between v and (v + dv) Since v = kc/2?r, the problem is equivalent to that of finding the number between k and k + dk Now, for any reasonable value of k, the number of waves fitting into

a box is usually very large For example, at moderate temperatures, most of the radiation is in the infrared, with wavelengths ,.,.,10-4 cm Hence, when k changes in such a way that one more wavelength fits into the box, only a very small fractional shift of k results It is possible, therefore, to choose the interval dk so small that no important physical quantity (such as the mean energy) changes appreciably within it, yet

so large that very many radiation oscillators are included This means that the number of oscillators can be treated as virtually continuous, so that we can represent it in terms of a density function

We must now find the number of oscillators in the volume dk,, dk11 dkz

If we imagine a space in which the co-ordinates are l, m, and n, there will

be one oscillator every time l, m, and n take on separate integral values Hence, there is one oscill&tor per unit cube of l, m, n space, so that the density in this space is unity To go to k space, we use eq (18), obtaining

v

oNi = dl dm dn = (27r)3 dk,, dk11 dk, (26)

16 PHYSICAL FORMULATION OF THE QUANTUM THEORY [1 1 0

It is now convenient to adopt polar co-ordinates in k space We define k2 = k� + k; + k; Then the element of volume becomes

k2 dk dn; where dn is the element of solid angle Since we are not interested in the direction of k, we integrate over dn, obtaining 411"k2 dk

for the element of volume, and

This gives the number of permissible values of k in the range between

J1 and J1 + dJ1 As shown in the section discussing the significance of the a's and b's, there are two independent coordinates for each k, correspond­ing to the two directions of polarization Thus, for the total number of oscillators between JI and JI+ dv, we find

811"V 5N = 25N1 = - v2 dJ1 (29)

c3

10 Equipartition of Energy To calculate the mean energy possessed

by each oscillator when it is in thermodynamic equilibrium with the walls, we shall apply classical statistical mechanics to these oscillators Although this theory was derived for material oscillators alone, the derivation involved only the formal properties of the equations of motion Any other systems acting formally like material oscillators must, there­fore, have the same equilibrium distribution of energy It is shown in classical statistical mechanics * that in any assembly of independent, noninteracting systems (such as our assembly of radiation oscillators) , the probability that a co-ordinate lies between q and q + dq, and that the corresponding momentum lies between p and p + dp, is equal to

A e-E/•T dp dq

E denotes total energy, kinetic and potential ; and A denotes a normaliz­ing factor, defined by the requirement that the total probability integrate out to unity or

/_"', /_"',.A e-E/•T dp dq = 1 For a perfect gas, E = p2 /2m, and we obtain the familiar Maxwell­Boltzmann distribution of velocities

1 1 0] THE ORIGIN OF THE QUANTUM THEORY 1 7

I t i s convenient to transform these equations t o new variables defined by

The mean value of the energy E is obtained by integration of EW(E)

over all energies This means that we weight each energy according to its probability We get

lo 00 E e-E/•T dE lo " e-•E de

lo 00 e-E!•T dE lo" e-• de (30)

where E = E/KT Thus, we prove that the average energy of each oscillator is KT This is an example of the theorem of equipartition of energy *

Collecting the information obtained from (29) and (30), we get the Rayleigh-Jeans law :

U(v) dv = E oN = -3 KTv2 dv (31)

c

Because this law disagrees with experiments, we conclude that the con­

* See Richtmeyer and Kennard, p 161

1 8 PHYSICAL FORMULATION OF THE QUANTUM THEORY [1.11 cepts of classical physics are in some way inadequate to describe the interaction of matter and radiation

Planck's Hypothesis

11 The Quantization of the Radiation Oscillators In searching for

a modification of the above treatment that would reduce the contribution

of the high frequencies to the energy, Planck was led to make an assump­tion equivalent to the following : The energy of an oscillator of natural frequency v is restricted to integral multiples of a basic unit hv This basic unit is not the same for all oscillators, since it is proportional to the frequency The energy of an oscillator is, then, E = nhv, where n is any integer from 0 to co With this assumption, Planck obtained an exact fit, within experimental error, to the observed distribution of radiation

According to classical mechanics, there should be no restrictions what­ever on the energy an oscillator may possess Our experience with oscil­lators, such as radio waves, clock springs, and pendulums, seem to verify this prediction How, then, can Planck's hypothesis be consistent with all these well-known results ? The answer is that h is a very small quan­tity, equal to about 6.6 X 10-27 erg-sec Hence, even for microwaves having a frequency as high as 1010 cps, the basic unit of energy is only 6.6 X 10-17 erg which is not detectable except by use of the most sensitive apparatus now available With clock springs and pendulums of period of the order of 1 sec, the basic unit of energy is obviously so small that in such relatively gross observations as can now be made, the allowed values of energy seem to be continuous With light waves, however, v , _, 1015, and

hv , _, 10-12 erg, a value that can be detected with sensitive instruments Hence, as we go to higher frequencies, where the basic unit becomes larger, quantization of the energy levels is easier to observe

To obtain Planck's distribution of energy, we need to know what the probability is that the oscillator has an energy corresponding to its nth allowed value Now, when n is very large, so that the discrete character

of the energy becomes unimportant (as with radio waves, for example)

we must obtain a result that is consistent with classical mechanics, which

is known to be correct in this region The simplest way of obtaining agreement is to choose a probability that is the same function of the energy as in the classical theory, * namely, e -Ef •T For a given energy,

En = nhv, the probability is then

W(n) , _, e-En/•T = e-nhv/•T

* This choice involves an assumption that is justified in part by its success in accounting for the energy distribution in a blackbody A systematic development

of the theory of quantum statistics (see Tolman, The Principles of Statistical Mechanics)

shows, however, that no other probability distribution would lead to thermodynamic equilibrium

1.13] THE ORIGIN OF THE QUA NTUM THEORY

To normalize this, we write *

To evaluate this sum, we can write

We can apply this result, obtaining

dal-e-a

Multiplying by oN, we find the Planck distribution

811" v e-hv/•T U(v) = - hv3 -o -

This leads to the Wien law In between, there is excellent agreement with experiment at all temperatures Hence, despite the strangeness of Planck's hypothesis, there is evidently something to it

The decrease in mean energy of the high-frequency oscillators arises because of the great amount of energy required to bring them to the first excited state, which is a state of rare occurrence As v is lowered or T

raised, it becomes more likely that the oscillator will gain a quantum of energy After the oscillator is excited to a high quantum number n,

its behavior will be essentially classical, because the basic unit of energy

is then much less than the mean available energy KT

13 Material vs Radiation Oscillators Planck's original idea was not to quantize the radiation oscillators as we have done previously Instead, he assumed that the radiation was in equilibrium with material 0scillators in the walls of the container, and that these material oscillators

*We use the expansion l _ = � xn

1 -x n=O �

20 PHYSICAL FORMULATION OF THE QUA NTUM THEORY [1 1 4

could give up or absorb radiant energy only in quanta with E = nhv

With this assumption, he obtained exactly the same distribution of radiant energy The quantization of the radiation oscillators was a later idea that has, as we shall see, many far-reaching consequences Moreover, the step of quantizing the radiation oscillators is almost imperative to explain the fact that the blackbody spectrum is independent

of the materials of which the walls are composed

The only alternative possibility is that all matter, and not only har­monic oscillators, can accept or emit radiation only in quanta of size,

E = hv But this means that all radiation ever emitted has energy restricted to E = hv; even if some were present with other energies it could not, by hypothesis, interact with matter and, hence, would be undetectable This hypothesis is, then, equivalent to the statement that all radiation oscillators have their energies restricted to E = nhv

14 Quantization of Material Oscillators We can consider the specific heats of solids, to determine whether Planck's quantum hypothesis applies to material oscillators In a crystal, for example, each atom is in equilibrium when it lies in its proper lattice position and, if disturbed, it can oscillate about the equilibrium position with a motion that is approxi-

mately simple harmonic for small oscillations

Einstein proposed that this curve could be explained by assuming that the molecular oscillators are quantized with E = nhv In contrast

to the radiation oscillators, which can have all possible frequencies, the material oscillators have only one frequency, which is the characteristic frequency of the substance Applying Planck's result for a given fre­quency, eq 32, we obtain

E = _h_v_e_-_h._1._r_ 1

- e-h>/•T and iJE aT = (hv)2 KT2 (1 -e-hv/•T e-h•/•T)2

This formula clearly predicts a specific heat per molecule of " at high

1 1 4] THE ORIGIN OF THE QUANTUM THEORY

temperatures, and approaching zero at very low temperat ur es as

The reason for the discrepancy at very low temperatures was explained

by Debye * The oscillations of each atom are actually not independent

of the oscillations of the others, but are coupled to them because of the forces between molecules The description in terms of independent oscillations is, therefore, not completely accurate

To describe the coupled oscillations of the molecules, we may con­sider, for example, a one-dimensional string of particles Suppose that each particle interacts only with its two nearest neighbors It can then

be shown t that waves are propagated through this system resembling those propagated through a chain, except that here the waves are both longitudinal and transverse, whereas the waves in a chain are transverse only When the wavelength is large compared with the distance between particles, the propagation differs very little from that in a continuous string ; but as the length of the waves approaches the mean distance between particles, the law of propagation changes For wavelengths shorter than the mean distance between particles, propagation becomes impossible

Problem 4 : In the one-dimensional string of particles specified above, let the equilibrium distance between particles be a Suppose the force on the nth particle

to be

Fn = -mwo2[ ( xn - Xn-1 ) + ( x, - Xn+1 ) ] = min

Here x, is the deviation of the nth particle from its equilibrium position

Find solutions of the form x, = A ne'"'', and show that we can choose A n = e"'",

where a is a suitable constant whose relationship to w is obtained by solving the equations

Show that for low frequencies the oscillations resemble sound waves, and that

w � 21rv /'>-., where v = woa is the speed of sound in the system Show also that there

is a maximum possible frequency

In three dimensions, a similar treatment can be given and, in this way,

we can describe the propagation of sound waves through a crystal As was done with the electromagnetic field, we can adopt the amplitudes of the possible sound waves as co-ordinates to describe the state of the system Since these co-ordinates oscillate harmonically with the time, the energies of the associated oscillators must be quantized t In com­puting the energy, however, we must take into account the fact that only

a finite number of wavelengths is permissible, and also that the relation

* See Richtmeyer and Kennard, p 450

t F Seitz, The Modern Thewy of Solids New York : McGraw-Hill Book Com­

]>any, Inc., 1946, pp 121 and 125

t See Secs 10 and 13

22 PHYSICAL FORMULATION OF THE QUA NTUM THEORY [1 1 s

between frequency and wavelength becomes more complex as we approach wavelengths comparable with interatomic spacing When all these factors have been taken into account, the quantum hypothesis leads to excellent general agreement with experimental specific heats at all temperatures Thus, in addition to quanta of electromagnetic energy,

we now have evidence for the existence of quanta of sound energy

16 Summary We may conclude that all systems which oscillate harmonically are quantized with E = nhv, whether these systems be material oscillators, sound waves, or electromagnetic waves Since we assume that all systems can interact with each other, the quantization of any one type of harmonic oscillator requires a similar quantization of all other types If experiments had not verified the existence of this unity, the quantum theory would have had to be abandoned, or at least funda­mentally modified

CHAPTER 2

Further Developments of the Early Quantum Theory

The New Concepts of the Quantum Theory 'IHTJS FAR, quantum restrictions on allowed energies have arisen only in connection with harmonic oscillators We shall see, however, that the results of many experiments, together with the systematic and logical development of the quantum hypothesis, lead to the conclusion that all matter is subj ect to quantum restrictions This conclusion thus enables

ll! to explain correctly a wide variety of experimental data for which the results of the classical theory are either wrong or ambiguous As exam­ples, we shall deal with the photoelectric effect, the Compton effect, the energy levels of material systems, and the laws governing the emission and absorption of radiation In all these examples, we shall also study

in detail how the quantum laws approach the classical limit

1 Photoelectric Effect We begin with a discussion of the photo­electric effect The study of blackbody radiation would enable one to deduce indirectly that electromagnetic waves can change their energy only in units of hv; it would certainly now seem desirable to verify directly whether this statement is true by a study of the emission and absorption

of radiation The earliest experimental investigations of this problem were concerned with the photoelectric effect These experiments showed that electrons are emitted from a metal surface * that is irradiated with light or ultraviolet rays ; also, that their kinetic energy is independent of the intensity of the radiation, but depends only on the frequency in the following manner :

Here v is the frequency of the incident radiation, and W is the work func­tion of the metal or, in other words, the energy needed to remove the electron from the interior of the metal

Einstein was the first person to relate this result to Planck's hypothe­sis (1905) Perusal of the data showed that h was a universal constant, and eaual to the h appearing in Planck's theory This agreement is a strong confirmation of the hypothesis that the radiation field can change energy only in units of hv If the constant had not been obtained, the theory would have been in serious difficulties

* Some electrons are liberated from the layers below the surface ; these lose energy

as a result of having to penetrate the meta.I

2l

24 PHYSICAL FORMULATION OF THE QUANTUM THEORY [!.1 The next important task is to try to determine why the electron absorbs energy only in quanta, independently of the intensity of the radiation In this connection, it is worth noting that with radiation of very low intensity we simply obtain a correspondingly low rate of emis­sion of photoelectrons

The simplest interpretation of this phenomenon is that light consists

of particles * which, because they are localized objects, can transfer all their energy to the photoelectrons during a collision This idea is strengthened by experiments in which very low-intensity beams are directed at a photographic plate ;t we obtain dark spots at random posi­tions, with an average density proportional to the intensity of the light

In the limit of a very intense beam, the distribution of spots gets so dense that it is practically continuous

When the beam is so intense that it seems to be continuous, it must in some way become the equivalent of what is described as a light wave in classical physics Such a " classical" wave has a certain rate at which energy is incident on any surface per unit area per unit time Let us call the rate S Then, when many quanta are present (intense beam or low frequencies), this rate must also be equal to the mean number, N,

of incident quanta per unit area per second times their energy hv Thus, we have

of the two slits may cancel each other

It would be impossible, however, to explain interference if one assumed that light was made up of localized particles Such particles would have

to go through either one slit or the other, and the opening of a second slit

* A particle is an object that can always be localized within a certain minimum legion, which we call its size

i Ri1ark and Urey (See list of references on p 2.)

2.1 ] FURTHER DEVELOPMENTS OF EARLY QUANTUM THEORY 25

could hardly prevent a particle from reaching a certain point to which the particle would be free to go if this second slit were closed On the other hand, the assumption of wavelike properties for light not only explains this particular experiment, but also a whole host of other experi­ments involving radiation-from radio waves to x rays It is, therefore, certainly desirable to try to understand the appearance of quanta in terms of the wave theory of light, if possible

To do this, we now consider the classical account of what happens in the photoelectric effect When radiation strikes an electron vibrating within an atom, it transfers energy to the electron If the electric field oscillates at a frequency that is resonant with the frequency of the electron

in the atom, the electron will absorb energy from the light wave until

it is liberated One could try to explain the photoelectric effect by assum­ing that the properties of the atom are such that the electron would keep

on gaining energy until it had picked up an amount equal to hv, after which it would be ejected If atoms had these properties then, with very weak light, the photoelectric effect should not be observed for a long time, since it would take a long time to store the necessary quantum

of energy Experiments were conducted, however, with metallic-dust particles and very weak light These dust particles were so small that

it would have taken many hours to store hv of energy; yet, some photo­electrons were found to appear instantaneously

To explain the above result, we could suppose that the metal con­tained electrons with all sorts of energies and, when the metal was struck

by a light wave, a few electrons of appropriate energy could be liberated immediately If we consider a case in which hv » W, however, it seems unlikely that electrons with so large a surplus of energy would remain indefinitely inside the metal, until their release was triggered by a light wave of exactly the right frequency Moreover, it has been found that

no matter how we try to release an electron from a metal (for example, by bombardment of the metal by protons or by other electrons) , we must always supply the same minimum energy, equal to the work function W

Similarly, it has been found that electrons cannot be liberated from atoms

of a gas, unless a certain minimum energy equal to the ionization poten­tial I is supplied (see the discussion of the Franck-Hertz experiments in Sec 15) Yet, some electrons are liberated instantaneously by very weak light from gas atoms with a kinetic energy equal to hv - I In view of all this evidence we must, therefore, rule out the possibility of explaining the photoelectric effect by assuming that some electrons initially possess nearly aH the energy with which they escape

If electrons in metals had such a range of energies, then it would be difficult to make the quantum hypothesis self-consistent, because only part of a quantum would have to be absorbed to liberate a typical elec­tron According to Planck's hypothesis, however, the radiation oscil·

26 PHYSICAL FORMULATION OF THE QUA NTUM THEORY [2 3 lators can supply a minimum of a full quantum in each absorption process What would then happen to the rest of the quantum if only part of it were absorbed by the electron ?

These particular efforts fail to explain the photoelectric effect in terms

of a process of gradual accumulation of energy, and every similar attempt that has ever been carried out has also failed This means that the wave theory is unable to account for the sudden appearance of finite amounts

of energy on a single electron We are, therefore, in a quandary One set of experiments suggests that light is a particle that can be localized, and the other suggests, with equal emphasis, that it is a wave Which approach leads to the correct picture? The answer is, neither

Before we can obtain a correct theory of the wave-particle duality

of the properties of light, we shall see that it is necessary to make radical changes in some of our most fundamental concepts dealing with the properties of matter and energy These new concepts will be developed through the remainder of this book, but primarily in Chaps 6, 8, and 22

For the present, however, we merely state that light must be regarded

as existing in the form of fundamental units, or quanta, which can, in some circumstances, act like particles and, in other circumstances, like waves We find a strong analogy here to the fable of the seven blind men who ran into an elephant One man felt the trunk and said that

" an elephant is a rope" ; another felt the leg and said that " an elephant

is obviously a tree," and so on The question that we have to answer

is : Can we find a single concept that will unify our different experiences with light, just as our concept of the elephant unifies the experiences of the seven blind men?

2 Differences between Classical and Quantum Laws of Physics

Our first step in the program of developing the new concepts needed in quantum theory will be to bring out two crucial differences between the kind of physical law obtained in classical theory and the kind suggested

by experience with quantum phenomena The first difference is that whereas classical theory always deals with continuously varying quantities,

quantum theory must also deal with discontinuous or indivisible processes

The second difference is that whereas classical theory completely deter­mines the relationship between variables at an earlier time and those

at a later time (i.e., it is completely causal), quantum laws determine only

probabilities of f uture events in terms of given conditions in the past

3 The Indivisibility of Quantum Processes Let us now consider some of the experimental evidence that indicates the need for introducing the concept of discontinuous or indivisible processes into the quantum theory The first important piece of evidence comes from the photo­electric effect We have already seen, for example, that while all efforts

to explain the photoelectric effect as a process of gradual transfer of energy from radiation field to matter have failed, the assumption that the trans-

i.4] FURTHER DEVELOPMENTS OF EARLY QUANTUM THEORY 2 7

fer of energy i s a discontinuous process that takes place i n jumps of size

6.E = hv is in agreement with all the experiments dealing with this phe­nomenon Moreover, the same assumption is also required by Planck's hypothesis that the energy of the radiation oscillators is restricted to discrete values That is, if the transfer of energy took place gradually,

it would be necessary to consider states in which radiation oscillators had part of a quantum and, according to Planck's hypothesis, no such state

is possible

As we shall see later, there are many other experiments which demand the interpretation that the transfer of energy is a discontinuous process For the present, we shall offer an experiment by Lawrence and Beams, * who tried to break up a light quantum into two parts by means of a very fast shutter, utilizing a Kerr cell that could be activated in 10-9 sec If the light wave were continuous, as described by classical theory then, with the intensities of light used, it would have taken much longer than 10-9

sec for a full quantum of energy to come through Thus, we should expect that the shutter would break up the quanta into smaller quanta They found, however, that none of the quanta was ever broken up

If we combine Planck's hypothesis with the fact that no one has ever been able to perform an experiment in which a part of a quantum has been detected, we are led to the conclusion that a quantum is an indivis­ ible unit of energy We may also see, from the failure of all attempts to follow the energy gradually, that the transfer of a quantum from one system to another is an indivisible process The indivisibility of the quantum of energy, and the indivisibility of the process of transfer go together ; they are necessary for each other's logical self-consistency We should conclude, therefore, that in the transfer of a quantum, the system cannot be regarded as passing through a succession of intermediate states, in which the energy is exchanged in a continuous fashion Instead, the quantum process must be regarded as discontinuous and as an indi­visible unit The transfer of a quantum is one of the basic events in the universe and cannot be described in terms of other processes It may

be called an elementary process, just as a proton or an electron is called an

e!ementary particle, because it does not seem to be made up of other particles

4 Probability and Incomplete Determinism in Quantum Laws The indivisibility of quantum processes is totally at variance with classical physics, which describes all processes in a continuous fashion, each change being caused by the state of the system just before the change took place Since classical laws presuppose the existence of continuous processes to which they apply, it is clear that discontinuous quantum jumps cannot

be predicted by our classical laws Our problem is, then, to find the new laws governing quantum transfers

* Ruark and Urey, p 83

28 PHYSICAL FORMULATION OF THE QUANTUM THEORY [2.4

We come now to the second important difference between classical and quantum laws It is an experimental fact, exemplified in the photo­electric effect and in a wide range of other experiments not yet studied here, that no law has been discovered which predicts exactly where and when an individual quantum will be transferred Instead, only the prob­ability of such a process may be predicted For example, if only one quantum is directed at a metal surface, it is impossible to predict whether

it will be absorbed and, if it is absorbed, exactly where and when But

if a beam contains many quanta, jt is possible from the intensity of the light used, to predict the mean number absorbed in any given region Thus, in this case, quantum laws appear to control ouly the probability of

an event and cannot predict its occurrence with certainty Wt> sha.U see that this behavior is not restricted to the photoelectric effect but is com� mon to all quantum processes

Thus, we can see that quantum laws are very different from their classical counterparts, which always imply that the behavior of the sys­tem is completely determined by exact causal laws F.or example, all material particles obey Newton's equations of motion, mx = F Once the initial position and velocity of each particle are given, the future motion is determined exactly by the differential equations of motion Thus, the trajectory of an electron is determined by three quantities : (1) The position at any instant of time

(2) The velocity at that time

(3) The value of the force F at all times

For an electrical particle, the force F is determined by the electric and magnetic fields But these can be calculated exactly with the aid of Maxwell's equations and the initial values of electric and magnetic fields everywhere Hence, according to classical physics, the motion of a charged particle (also of any other kind of particle) can be determined precisely for all time, once certain initial conditions are known The same can be said about changes of the electromagnetic field Classical theory may therefore be called completely deterministic

Applying these general ideas, one concludes from classical theory that,

in a light beam of a given intensity, electrons gain energy at a continuous rate, which is calculable from the light intensity and from the initial conditions of the electrons On the other hand, experiments show that the process of energy transfer is discontinuous and apparently not gov­erned exactly by deterministic laws, at least not by the deterministic laws

of classical mechanics Instead, so far as we can find out from experi­ment, only the probability of the process is determined

At this point, it is worthwhile to go more deeply into the connection between the appearance of probability and the indivisibility of a quantum process First, there is the previously mentioned fact that many classical

2.5] FURTHER DEVELOPMENTS OF EARLY QUANTUM THEORY 29

laws (including Newton's equations of motion) which are essential for the operation of classical determinism must, by their very nature, refer to gradual and continuous processes Hence, if only because this kind of law has no meaning in discontinuous processes, it cannot apply directly

to quantum transfers Some classical laws, however, do not require us

to follow particles through a continuous path in space time, for example, conservation of energy, momentum, or angular momentum Even in an impulsive collision in which we cannot follow the motion continuously, these laws apply for the collision as a whole Such laws do have meaning even in discontinuous processes It is an experimental fact that these laws can all be taken over directly into the quantum theory For exam­ple, it has been shown experimentally that energy is always conserved in the photoelectric effect Many other experiments also yield this result Hence, not all classical deterministic laws must be abandoned, but only these requiring a description in terms of continuous processes

5 Unlikelihood of Completely Deterministic Laws on a Deeper Level

One might wonder whether the appearance of probability in quantum processes is not a result of our ignorance of the correct variables to use in describing the system In classical physics, probabilities often appear for just this reason For example, in thermodynamics we measure the pressure, temperature, and volume of a given system In very small regions of space, especially near the crit:ca� point, we find that these quantities no longer obey an equation of state 3xactly, but instead exhibit large random fluctuations about a mean value that is predicted by the equation of state Hence, the deterministic laws of thermodynamics break down and are replaced by laws of probability This is because the thermodynamic variables are no longer appropriate for the problem and must be replaced by the position and velocity of each molecule, which are, from the viewpoint of thermodynamics, hidden variables The thermodynamic quantities are, then, merely averages of hidden variables that cannot be observed by thermodynamic methods alone

To find the underlying causal laws, we must accept a description in terms

of the individual molecules

The idea immediately suggests itself that probability in quantum processes arises in a similar way Perhaps there are hidden variables that really control the exact time and place of a transfer of a quantum, and we simply haven't found them yet Although this possibility cannot

be absolutely ruled out, we can show that this is unlikely The first point, of course, is that no experiment has yet shown the slightest trace of

retical arguments which make it unlikely that such hidden variables exist These will be discussed later (Chap 22, Sec 19) For the present, we shall merely assert, as a general principle, that only the probability of a quantum jump can be determined by the physical state of the system