8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf

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J o u r n a l

o f

T he P r a n k l in n s i i lu t e

D e v o t e d t o S c i e n c e a n d t h e M e c h a n i c r t s

Vol. 207 APRIL 1929 No. 4

EDITORS

NOTE: The following fourteen papers were read

at a Symp osium on Quan tu m Mechanics held unde r the aus-

pices of the American Physical Society in New York City

December 3~ I928. It is much regretted tha t the editors

have not been able to procure two other papers which were

read on that occasion.

P H Y S I C A L M E A N I N G O F W A V E M E C H A N I C S

BY

J . C . S L A T E R PH .D .

Harvar d University.

AN understanding of the physical meaning of wave

mechanics is essential if one is going to do useful work in

the subject; even purely math emat ical research in qu an tu m

theory is of small value unless it is carried out with the

proper unders tanding of the physics behind it. For that

reason it seemed well to start this discussion with a brief

tre atm ent of physical interpretation rather t han mathemati cal

details. Of course only a small portion of so large a subject

can be tak en up; and I have chosen to speak abou t the

statis tical side of wave mechanics both because this is one

of its most fun dament al sides and because it is a topic that

will fit in well with the other papers.

Wave mechan ics is an extension not of ord inary New-

tonian mechanics but of statistical mechanics; and this simple

observation is enough to explain many of its otherwise pUZ-

Note .--Th e Franklin Institute is not responsible for the statemen ts and opinions advan ced

by contributors to the JOURNAL.)

VOL. 207, NO. I24 o-- 3I 449

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4 5 o J . C . SLAVF~. IJ. V . I.

z l i n g f e a t u r e s . S i n c e o r d i n a r y s t a t i s t i c a l m e c h a n i c s is a s u b -

j e c t a b o u t w h i c h t h e r e is m u c h m i s u n d e r s t a n d i n g i t w i l l b e

w e l l t o s t a r t w i t h a l i t t l e d e s c r i p t i o n o f i t. B e i n g s t a t i s t ic a l i t

m u s t d e a l w i t h a g r e a t m a n y s im i l a r o b j e c t s o r e v e n t s ; b u t i t is

n e c e s s a r y a t t h e o u t s e t t o u n d e r s t a n d t h a t t h i s b y n o m e a n s

l im i t s u s t o p r o b l e m s w h e r e a s in t h e t h e o r y o f g a s es t h e r e a r e

a g r e a t m a n y i d e n t i c a l p h y s i c a l s y s t e m s . O n e c a n p e r f e c t l y

w e l l t r e a t s t a t i s t ic a l l y a p r o b l e m w i t h e v e n o n e si ng le d e g r e e o f

f r e e d o m a s f o r e x a m p l e an o s c il la t o r . T h e s t a t i s t i c s c o m e s

in i n t h e f a c t t h a t o n e w o r k s n o t w i t h s in g le o b s e r v a t i o n s

b u t w i t h e n s e m b l e s o f o b s e r v a t io n s . B y a n e n s e m b l e o n e

m e a n s a s e t o f r e p e t i t i o n s o f t h e s a m e e x p e r i m e n t - - j u s t s u c h

a s e t o f r e p e t i t i o n s a s o n e a c t u a l l y m a k e s in w o r k i n g a r e al

p h y s i c a l e x p e r i m e n t . O n e c a n s e t u p a s im p l e m a t h e m a t i c a l

p i c t u r e o f a n e n s e m b l e o f o b s e r v a t i o n s . S u p p o s e w e h a v e a

s p a c e w i t h a n u m b e r o f d i m e n s i o n s t o c o in c i d e w i t h t h e

n u m b e r o f m e a s u r e m e n t s w e m a k e ; f o r e x a m p l e if o u r

s y s t e m is a m e c h a n i c a l o n e w i t h n d e g r e e s o f f r e e d o m t h e r e

w i ll b e 2 n d i m e n s i o n s n fo r t h e c o 6 r d i n a t e s o f t h e s y s t e m

t h e o t h e r n f o r i ts v e l o c i t i e s o r m o m e n t a . E a c h t i m e w e

m a k e o u r m e a s u r e m e n t s w e c a n r e p r e s e n t t h e r e su l t b y a

s in g le p o i n t in t h i s s p a ce T h e n i f w e r e p e a t t h e e x p e r i m e n t

m a n y t i m e s w e g e t a s m a n y d i f fe r e n t p o i n t s a s t h e r e a r e

r e p e t i t io n s a n d t h i s s w a r m o f p o i n t s i s t h e m a t h e m a t i c a l

p i c t u r e o f t h e e n s e m b l e . I n m o s t c a se s w e w i s h t o s u p p o s e

t h e n u m b e r o f t r i a ls t o i n c re a s e w i t h o u t l im i t s o t h a t t h e

s w a r m a p p r o a c h e s a c o n t i n u o u s d i s t r i b u t i o n a n d w e c a n

d e f in e a f u n c t i o n r e p r e s e n t i n g t h e d e n s i t y o f p o i n t s i n o u r

s p a c e . T h i s d i s t r i b u t i o n f u n c t i o n t h e n w i ll a l so s e r v e a s a

m a t h e m a t i c a l r e p r e s e n t a t io n o f t h e e n s e m b l e .

T h e p a r t i c u l a r s o r t o f e n s e m b l e t o b e u s e d in a n y c a s e

d e p e n d s o n t h e p h y s i c a l c o n d i t io n s . W e m u s t ju d g e a s

c l o s e ly a s p o s s ib l e w h a t s o r t o f d i s t r i b u t i o n o f t h e q u a n t i t i e s

w e a r e m e a s u r i n g w i l l r e a l l y o c c u r i n o u r e x p e r i m e n t a n d s e t

u p a n e n s e m b l e a c c o r d i n g ly . H e r e a s i n e v e r y c a se t h e

p u r p o s e o f o u r m a t h e m a t i c s i s t o r u n s o t o s p e a k p a r a ll e l t o

t h e p h y s i c s ; o n e c a n im a g i n e t h e m w r i t t e n i n p a r a l le l c o l u m n s

w i t h a d e f i n i te c o r r e s p o n d e n c e b e t w e e n s u c h t h a t w h e r e v e r

w e c a n m a k e a p h y s i c a l o b s e r v a t i o n i t c a n b e m a t c h e d w i t h

s o m e f e a tu r e o f t h e m a t h e m a t i c s . T h e n i t is p l a in w h y w e

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Ap r., I 9 2 9. 1 P H YS ICA L M E A NIN G OF W AV E ~{E CHANICS . 4 5 I

n e e d e n s e m b l e s : t o m a t c h t h e p h y s i c a l p ro c e s s o f r e p e t i t i o n .

A n d t h e t a s k o f s e t t i n g u p t h e p r o p e r o n e m u s t d e f i n i t e l y b e

o n e o f f i n d i n g a p a r a l l e l t o t h e p h y s i c s .

F o r e x a m p l e , s u p p o s e t h a t o u r s y s t e m w e r e a l in e a r

o s c i l la t o r; t h a t w e fi rs t t o o k i t , w i t h o u t e n e r g y ; a t a n

a r b i t r a r y t i m e g a v e i t a d e f in i t e e n e r g y , t h e s a m e in e a c h

r e p e t i t i o n o f t h e e x p e r i m e n t ; a n d t h e n a t a l a t e r t i m e m e a s -

u r e d i t s c o 6 r d i n a t e a n d m o m e n t u m , p l o t t in g t h e m in a

t w o - d i m e n s i o n a l s p ac e . T h e n th e e n s e m b l e o f r e p e t i t io n s o f

t h is m e a s u r e m e n t w i l l be e a s y t o d e s c r i b e ; t h e p o i n t s w i ll a ll

c o r r e s p o n d t o t h e s a m e e n e r g y , a n d w i ll a ll l ie o n a c u r v e

in t h e t w o - d i m e n s i o n a l s p ac e w h o s e e q u a t i o n is e n e r g y

= c o n s t a n t a n e l l ip s e , s i n c e o u r s y s t e m is a n o s c i l l a t o r ) .

B u t , o n a c c o u n t o f t h e a r b i t r a r y t i m e w h e n t h e e n e r g y w a s

c o m m u n i c a t e d , t h e p h a s e w il l b e d i f f e r e n t e a c h t i m e t h e

m e a s u r e m e n t is m a d e , a n d t h e p o i n t s o f t h e s w a r m w i ll be

d i s t r i b u t e d in a u n i f o r m f a s h io n a b o u t t h e e ll ip se , a c c o r d i n g

t o a l a w w h i c h c o u ld b e r e a d i l y f o u n d . T h i s e n s e m b l e is

w e l l k n o w n i n s t a t i s t i c a l m e c h a n i c s ; i t is t h e m i c r o c a n o n i c a l

e n s e m b l e . I t is u s e fu l in t h e r m o d y n a m i c s , w h e n w e w i s h t o

w o r k w i t h s y s t e m s o f c o n s t a n t e n e r g y .

A s a n o t h e r e x a m p l e , w e m i g h t t a k e o u r s a m e o sc i l la to r ,

b u t e x p o s e i t t o d i f f e r e n t e x t e r n a l c i r c u m s t a n c e s . W e n o w

c o u p l e i t , w i t h s m a l l f o r c e s o f i n t e r a c t i o n , t o a s y s t e m w i t h

a g r e a t m a n y d e g r e e s o f f r e e d o m , a n d a d e f i n i t e e n e r g y . A t

a d e f i n i t e t im e , w e m e a s u r e i ts c o 6 r d i n a t e a n d m o m e n t u m .

I n t h i s c as e , t h e o s c i l l a to r c a n a c q u i r e a n y e n e r g y b y i n t e r -

a c t i o n , a n d s o m e t i m e s i t w i l l h a v e o n e v a lu e , s o m e t i m e s

a n o t h e r , b u t t h e r e l a t i v e c h a n c e s o f t h e d i f f e r e n t v a l u e s w i ll

b e d i s t r i b u t e d a c c o r d i n g t o a la w w h i c h c a n b e c a l c u la t e d .

A s b e fo r e , t h e p h a s e s w i ll b e a r b i t r a r y . T h e s w a r m o f

p a r t i c l e s i n t h e e n s e m b l e w i l l t h e n c o v e r t h e w h o l e o f t h i s

t w o - d i m e n s i o n a l s p ac e , w i t h a d e n s i t y t h a t c a n b e d e f in i t e ly

f o u n d . T h i s is t h e c a n o n i c a l e n s e m b l e . I t is p a r t i c u l a r l y

u s e f u l in t h e r m o d y n a m i c s , b e c a u s e i t r e p r e s e n t s w h a t o n e

m e a n s b y a s y s t e m a t c o n s t a n t t e m p e r a t u r e ; t h a t is , a

s y s t e m c a p a b l e o f i n t e r c h a n g i n g e n e r g y w i t h a m u c h l a r g e r

s y s t e m - - t e m p e r a t u r e b a t h - - o f f ix ed p r o p e r t ie s .

T h e t w o e n s e m b l e s w e h a v e m e n t i o n e d a r e th e m o s t

u s e fu l in t h e a p p l i c a t i o n o f s ta t i s t ic a l m e c h a n i c s t o t h e r m o -

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452 J . C . SLATER. lJ. I:. I.

dyna mic problems; but statistical mechan ics is a nmch wider

subject in its possibilities, and an indefinite number of other

ensembles may be useful. Suppose we still tak e our oscillator;

but let us now give it approximately a definite energy, with

approximately a definite phase, the precise values varying

from one repe tition of the experime nt to another. Then, if

we measure the co6rdinate and mo me nt um at a definite time,

all the points in our swarm will be clustered together in a

small region of the two-dimensional space. This is the sort

of ensemble that could be used to describe a measurement,

almost definite, th at was subject to small errors. The amoun t

of spreading of the swarm would be directly connected with

ideas of probable error.

Wave mechanics, being a form of statistical theory,

operates with ensembles. Th ey are represented mat hem ati -

cally by giving the distribution function representing the

swarm of points. Thus, the familiar 6~* gives the dens ity

of points in a space in which the various co6rdinates of the

sys tem are plotted. This is, it is true, different from the

space usually used in ordinary statistical mechanics, where

co6rdinates and mome nt a both are plotted. But the differ-

ence is not essential, from the present point of view, and it

is easy to show the relation of the two methods . The re are

some other differences; we can not, for example, restrict all

the points of the ensemble to too small a region, or we meet

the difficulty of the principle of indete rminate ness. But the

essential principles are the same. Jus t as in ordi nary sta-

tistical mechanics, we must here choose the ensemble, deter-

mined by , by considering the sort of statistical distributions

actua lly present in the repetitions of the experim ent being

performed.

An ensemble, in statistical mechanics, takes the place of

a single set of mea su remen ts in ord ina ry mechan ics; and as

such it is bu t the beginning of the problem. Our real task is,

for example, to trace what happens to the system as time

goes on. Given an initial ensemble, we wish the final en-

semble. This is where mechan ics enters the problem; and

here a characteristic and impo rta nt difference in metho d

between classical mechanics and wave mechanics appears.

In classical statistical mechanics, we consider each separate

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A p r ., 1 9 ~ 9. 1 I l t Y S I C A I . M I ; A N I N ( ; O F \ V A \ [ ~ ; M E C I t A N I C S .

453

point of the original ensemble representing a sys tem with

definite initial conditions; find its behavior according to the

principles of Newton ian mechanics; derive in this way a

final point from each initial one and so a final ensemble from

the initial one. The result of course depends on the sort

of external forces to which the system is subjected. [;or

example if we have the microcanonical ensemble and expose

the syst em only to the internal forces natu ral to the oscillator

then it can be easily shown that although the individual

points of the swarm move the ensemble as a whole remains

unchanged. This would not: be true if an external force

were present. With a canonical ensemble we are most

interested in finding what happens when the system interacts

with the tem pera tu re bath. Here each point moves so as to

change not only its phase but also its energy with time;

yet the ensemble is so chosen that it remains unchanged as

a whole. Both these ensembles since th ey remain constant

under the action of forces are useful in the the ory of steady

states. The third ensemble we have mentioned in which all

the points were concent rat ed close together nat urally behaves

different ly; as these systems are exposed to forces all the

points of the ensemble travel together so th at after the lapse

of an in terva l of time the ensemble will consist of a concen-

trated set of points in some new parts of the space.

The method by which the wave mechanics investigates

the change of the ensemble with time is essentially different.

The problem briefly is this: given one ensemble find

ano the r; given one function find another. The wave me-

chanics achieves this directly without inter mediate steps.

A thing which converts one function into another is an

operator; hence the importance of operator theo ry in wave

mechanics. Functions are convenient ly considered in a space

of an infinite number of dimensions where each point repre-

sents a function; in this space an ope rator changes one

point into another. All the operators we are inte rest ed in

are linear operators in function space and this explains the

importance of linear transformations and matrix theory

which is bound up directly with them in wave mechancs.

As is well known it is not the dist ribu tion functions with

which we work direct ly but the wave func tion 4/. For

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4 5 4

J . C .

S I A T E R [ J 1 ; [

e x a m p l e , i n t h e p a r t i c u l a r c a s e o f t h e c h a n g e o f t h e e n s e m b l e

w i t h t i m e , w e f in d t h e t i m e r a t e o f c h a n g e o f ~b a s a c e r t a i n

o p e r a t o r a c t i n g o n ~ b :

h O~

H e .

27ri 0t

T h u s t h e p r o g r e ss o f t h e s y s t e m is s h o w n b y t h e p a t h o f a

p o i n t i n f u n c t i o n s p a c e w i t h t i m e , j u s t a s i n o r d i n a r y m e -

c h a n i c s i t is s h o w n b y t h e p a t h o f a p o i n t in o r d i n a r y s p a c e

w i t h t i m e . I f o u r p r o b l e m i s a s s i m p l e a s, g i v e n th e d i s t r i b u -

t i o n f u n c t i o n a t o n e t i m e , t o c a l c u l a t e i t a t a l a t e r t i m e ,

w e n e e d m e r e l y t a k e t h e p o i n t r e p r e s e n t i n g t h e i n it ia l ~b;

f o ll ow i ts m o t i o n b y S c h r 6 d i n g e r ' s e q u a t i o n u n t i l t h e l a t e r

t i m e ; a n d t h e f in a l p o i n t w il l g i v e t h e d i s t r i b u t i o n f u n c t i o n

o f t h e e n s e m b l e a t t h e f in a l t i m e .

B u t m o r e o f t e n w e h a v e a m o r e c o m p l i c a t e d p r o b l e m

w h e r e o u r f in a l d i s t r i b u t i o n f u n c t i o n is n o t t o b e m e r e l y a

f u n c t i o n o f t h e s a m e v a r i a b l e a s t h e i n i t i a l o n e , o n l y a t a

l a t e r t i m e , b u t is to b e a f u n c t i o n o f d i f f e r e n t v a r i a b l e s .

F o r e x a m p l e , w e m a y o b s e r v e t h e d i s t r i b u t i o n o f p o s i t i o n s o f

p a r t i c l e s to s t a r t w i t h ; o u r f in a l m e a s u r e m e n t m a y b e t h e

d i s t r i b u t i o n i n e n e r g y . I n a c a s e l ik e t h is , i t p r o v e s t h a t

t h e p r o c e s s o f f i n d in g t h e d i s t r i b u t i o n w i t h r e s p e c t t o a n e w

q u a n t i t y m e a n s r e a l ly f i n d i n g th e r e p r e s e n t a t i o n o f t h e s a m e

f u n c t i o n ~b w i t h r e s p e c t t o n e w c o 6 r d i n a t e s i n f u n c t i o n s p a c e .

I f , f o r e x a m p l e , t h e s y s t e m is a q u a n t i z e d o n e , a n d w e w i s h

t o f i n d t h e d i s t r i b u t i o n i n e n e r g y , w e r e f e r t o n e w c o -

o r d i n a t e s i n w h i c h t h e u n i t v e c t o r s a r e t h e o r t h o g o n a l

f u n c t i o n s f o u n d in S c h r 6 d i n g e r ' s t h e o r y . T h e d i s t r i b u t io n

f u n c t i o n , a s r e f e r r e d to e n e r g y , is a d i s c r e t e o n e : i t c o n s i s t s

o f a v a l u e crick* g i v i n g t h e p r o b a b i l i t y t h a t t h e e n e r g y is

t h a t o f t h e n t h s t a t i o n a r y s t a te . T h e s e t o f c n c , * s as a

f u n c t i o n o f n is a s t r u l y a d i s t r i b u t i o n c u r v e , d e s c r i b i n g a n

e n s e m b l e , a s is ~ b * a s a f u n c t i o n o f x , a n d o n e f o r m d e p e n d s

d i r e c t l y o n t h e o t h e r , a s w e c a n s e e f r o m t h e r e l a t i o n

= ~ n)cnu~,

w h e r e t h e u .,,'s a r e t h e o r t h o g o n a l f u n c t i o n s m e n t i o n e d a b o v e .

T h e g e n e r a l t h e o r y o f c h a n g i n g f r o m o n e s e t o f v a r i a b le s t o

a n o t h e r , i n th i s w a y , i s t h e t r a n s f o r m a t i o n t h e o r y o f D i r a c

8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf

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Apr., I929 .1 ] I tYSIC AI , M EA NIN G OF \~ :A VE ~ IECHANICS. 4 5 5

a n d J o r d a n a n d w e s ee t h a t i t is j u s t p a r t o f t h e g e n e r a l

s c h e m e o f w a v e m e c h a n i c s in f in d i n g a d i s t r i b u t i o n f u n c t i o n

f o r t h e f in a l e n s e m b l e f r o m t h a t o f t h e o r ig i n a l o n e .

I t is p l a i n f r o m w h a t h a s b e e n s a id t h a t t h e s t a t i s t ic a l

s id e o f w a v e m e c h a n i c s is a v e r y i m p o r t a n t sid e . T o g e t f ul l

a d v a n t a g e o f t h e m e t h o d s of t h e su b j e c t t h e s t a t i s t ic a l

f e a t u r e s h o u l d b e c o n s t a n t l y k e p t in m i n d . P r o b l e m s s h o u l d

b e fo l l ow e d t h r o u g h m a t h e m a t i c a l l y a s t h e y a re w o r k e d

p h y s i c a l ly ; b y l o o k in g a t t h e m in t h is b r o a d w a y m a n y

e r r o r s a n d u n c e r t a i n t i e s w i l l b e a v o i d e d .