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Transcript of 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
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8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
1/7
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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8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
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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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8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
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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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8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
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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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8/10/2019 1929_Slater. Physical meaning of wave mechanics [J. Franklin Inst.]1.pdf
6/7
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
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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 .