the actual content of quantum theoretical kinematics and mechanics (nasa)

8/19/2019 The Actual Content of Quantum Theoretical Kinematics and Mechanics (NASA)

1/35

NAS_ TECHNICAL MEMORANDUM NASA TM-77379

%

i

THE ACTUAL CONTENT OF QUANTUM

_, THEORETICAL KINEMATICS AND MECHANICS

,

W

l

"

Translation of Uber den anschaulichen Inhalt der

quantentheoretischen Kinematik und Mechanik", Zeit-

schrift fur Physik, v. 43, no. 3-4, pp. 172-198, 1927.

--i

18_- 170 _6

(lIISk-TB-773791 Tile ICTaAL COIJTEIJT OF

QU&IITUB THEOiigTZCaL KZlfEH&TICS &lid BKCiikJZCS

(18at

i

onal ae

r

onaut

i

cs and Space

&dminist_ation) 35 p iIC &O3/_F &OI CSCL 126 Uncla8

63/77 18109

• s

L

J

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

. WASHINGTON D.C, 20546 DECEMBER 1983

o ,, , m


2/35

_1 i I I I I I • I II

First, e

x

act definitions are supplied in this paper f

o

r

the terms: position, velocity, energy,

e

tc. (of the electron,

for instance), such that they are valid also in quantum mech-

anics; then we shall show that canonically conjugated variables

can be determined simultaneously only with a characteristic

uncertainty. This uncertainty is the intrinsic reason for the

occurrence of statistic_l relations in quantum mechanics. Their

mathemat

i

cal formulation is made possible by the Dirac-Jordan

theory. Beginning from the basic principles thus obtained, we

shall show how macroscopic processes can be understood from.the

viewpoint of quantum mechanics. Several imaginary experiments

a

re disc

u

sse

d

t

o

el

u

cidate t

h

e theory.

I i.

a. _,,_ _,_ioi_i_i_ u. I,,,,_.,_kRRI, l

_c

l

a

ss

l

J

_l_

_md

Unllmi

t

e

d

III II_ ..... U


3/35

W

THE ACTUAL CONTENT OF QUANTUrl THEORETICAL KINEMATICS AND

1

172

MECHANICS

By W Heisenberg, Institute for Theoretical Physics of the

University, Copenhagen, Denmark

SUMMARY First, exact definitions are sup-

plied in this paper for the terms: position,

velocity, energy, etc. (of the electron, for

nstance), such that they are valid also in

quantum mechanics; then we shall show that

canonically conjugated variables can be de-

term,ned simultaneously only with a charac-

teristic uncertainty _§I]. This uncertainty

is the intrinsic reason for the occurrence

of statistical relations in quantum mechan-

ics. Their mathematical formulation is made

possible by the Dirac-Jordan theory (§2). Be-

ginning from the basic principles thus oh-

rained, we shall show how macroscopic pro-

cesses can be understood from the viewpoint

of quantum mechanics (§3).Several imaginary

experiments are discussed to elucidate the

theory (§4

)

.

We believe to understand a theory intuitively, if in all sim-

ple cases we can qualitatively imagine the theory's experi-

mental consequences and if we hav

e

simultaneously realized

I that the appli

c

ation of the theory excludes internal

c

ontra-

di

c

tions• F

o

r i

n

st

a

n

c

e: we be

l

ieve t

o

und

e

r

s

t

a

nd Einstein's

concept of a finite three-dimensional space intuitively, be-

cause we can imagine the experimental consequences of this

concept wit

h

out contr

ad

ictions. Of course

,

t

h

ese

c

onsequen

c

es

contradict our customary intuitive space-tlme beliefs. But we

can convince ourselves that the possibility of applying this

customary view of space and time can not be deduced either

from our laws of thinking, or from experience. The intuitive

, i

• Numbers in the margin indicate foreign pagination

I


4/35

,

O

R

I

G

INAL PAGE I

g

OF P

OO

R QUALITY

interpretation of quantum mechanics is still full of internal

contradictions, which become apparent in the battle of opin-

ions on the theory of continuums and discontinuums, corpuscles

and waves. This alone tempts us to believe that an interpre-

tation of quantum mechanics is not going to be possible in the

customary terms of kinematic and mechanical concepts. Quantum

theory, after, derives from the attempt to break with those

customary concepts of kinematics and replace them with rela-

tions between concrete, experimentally derived values. Since

this appears to have succeeded, the mathematical structure of

quantum mechanics won't require revision, on the other hand.

By the same token, a revision of the space-time geometry for

small spaces and times will also not be necessary, since by a

choice of arbitrarily heavy masses the laws of quantum mechan-

ics can be made to approach the classic laws as closely as 117___3

desired, no matter how small the spaces and times. The fact

that a revision of the kinematic and mechanic concepts is re-

quired seems to follow immediately from the basic equations

of quantum mechanics. Given a mass _, it is readily understand-

able, in our customary understanding, to speak of the position

and of the velocity of the center of gravity of that mass m.

h

But in quantum mechanics, a relation

Pq

--

qP

:'f_-_

i

exists

between mass, position and velocity. We thus have good reasons

to suspect the uncritical application of the terms position

and velocity . If we admit that for very small spaces and

times discontinuities are somehow typical, then the failure

of the concepts precisely of position

a

nd velocity become

immediately plausible: if, for instance, we imagine the uni-

it_

_1

.

..q.z

dimensional motion of a mass point, then in a continuum theory

2


5/35

it will be possible to trace the trajectory curve x(t) for

the particle's trajectory (or rather, that of its center of

mass) (see Fig. I, above), with the tangent to the curve in-

dicating the velocity, in each _ase. In a discontinuum theo-

y, in contrast, instead of the curve we shall have a series

_ of points at finite distances (s_e Gig. 2, above). In this

case it is obviously pointless to talk of the velocity at a

certain position, since the velocity can be defined only by

means of two positions and consequently and inversely, two

different velocities corresponded to each point.

The question thus arises whether it might n

o

t be p

o

ssible, by

means of a more precise analysis of those kinematic and me-

chanical concepts, to clear up the contradictions currently

existing in an intuitive interpretation of quantum mechanics,

to thus achieve an intuitive understanding of the relations of

quantum mechanics.*

§ I The concepts: position, path, velocity, energy /17--4

In order to be able to follow the quantum-mechanical behavior

of any object, it is necessary to know the object's mass and

and the interactive forces with any fields or other objects.

Only then is it possible to set up the hamiiconian function

for the quantum-mechanical system. [The considerations below

* This pa

p

er was written as a consequen

c

e of the efforts and

wishes expressed clearly by other scientists, much earlier, be-

fore quantum mechanics was developed. I particularly remember

Bohr's papers on the basic tenets of quantum theory (for

instance, Z.f.Physlk 13, 117 (1923)) and Einstein's discus-

s

i

on

s on t

he

rel

a

tio

n

--S

e

twee

n

w

a

v

e

fi

e

l

d

s

and

light q

u

anta.

In more recent times, the problems here mentioned were dis-

cussed most clearly by W. Pauli, who also answered some of

the questions that arise ( Ouantentheorle , Handbuch d.Phys.

[ Quantum theory , Handbook of Physics] Vol. XXIII, subse-

quentl

y

cite

d

as l.c.

)

. Qu

a

ntum mech

a

nics has

c

h

a

nge

d

little

in the formulation Pauli gave to these problems. It is also

a special pleasure for me here to thank Mr. W. Paull for the

stimulation I derived from our oral and written discussions,

which have substantially contributed to this paper.

3

II l II I III . , ,.,,m,,,-._lh_


6/35

shall in general refer to non-relativistic quantum mech

a

nics,

since the laws of quantum-theory electrodynamics are not com-

pletely known yet.* No further statements regarding the ob-

Ject's gestalt are necessary: the totality of those inter-

active forces is best designated by the term gestalt .

If we want to clearly understand what is meant by the word

position of the object - for instance, an electron - (rela-

tive co a given reference system}, th_n we must indicate the

d

e

finite e

xpe

riments by me

a

ns

o

f which we intend to determine

the position of the electron Otherwise the word is meaning-

less In principle, there is no shortage of experiments that

determination of the of the electron to

ermit a position

any desired precision, even. For instance: illuminate the e-

lectron and look at it under the microscope. The highest pre-

cision attainable here in the determination of the position is

substantially determined by the wavelength of the light used.

But let us build in principle, a r-ray microscope and by means

of it

d

etermine the position as precisely as desired. But in

this determination

a

sec

o

nd

a

ry

c

ircu

m

stance becomes essential:

the Compton effect. Any observation of the scattered light

c

oming from the ele

c

tron (into the eye, onto a photographic

plate, into

a

phot

oc

ell}

p

resupposes

a

photoelectric effect,

t

ha

t is, it c

an

also be interpreted

a

s

a

light qu

a

ntu

m

strik-

ing the

e

l

ec

tron, there being ref]ectedordiffracted to then

- deflected on

c

e ag

a

in by the mi

c

roscope's lense - finally

/

17__55

triggering t

he

ph

o

t

o

el

ec

tric effect. At t

h

e instant

o

f the

de

t

e

rmination of its position - i

.e.,

the instant at which

th

e

light qu

an

tu

m

is

d

iffr

a

cted b

y

the ele

c

tron - the electron

discontinuousl

y

c

ha

nges its im

p

ulse. T

ha

t c

ha

nge will be m

o

re

p

r

on

o

un

c

ed

, t

he

sm

a

l

le

r th

e wavelen

gth of

t

h

e l

ig

h

t

used,

i

.

e.

the more precise t

h

e posit

i

on

d

etermin

a

t

io

n is to be.

I

n the

iii J u • i

,

*

Ho

we

v

er, sig

n

ific

an

t

pro

g

r

e

s

s w

a

s

made ve

ry

rec

e

n

t

ly

t

hrou

g

h

t

h

e w

o

rk

o

f

P

.

V

lr

ac [P

r

oc. Roy

. S

oc

. (A

)

,

114

,

24

3 (

1927

)

and

s

ub

seq

u

e

n

t st

ud

ie

s.]

4

...........


7/35

_ ORIGINAL PAGE |g

' OF POOR qUALITY

instant at which the electron's position is known, therefore,

its impulse can become known only to the order of magnitude

corresponding to that discontinuous change. That is, the more

precisely the position is determined, the more imprecisely

will the impulse be known, and vice-versa. This provides us

with a direct, intuitive clarification of the relation

h . Let q be the precision to which the value

Pq --qP--__i

I

of _ is known (ql is approximately the average error of _),

or here, the wavelength of the light; Pl is the precision to

which the value of _ can be determined, or in this case, the

discontinuous change in _ during the Compton effect. Accord-

ing to the basic equations of the Compton effect, the rela-

tion between Pl and ql is then

P,_l _ _'.

, (l)

That relation (

I) a

b

ove

st

an

ds in

a

dire

c

t

ma

them

a

ti

ca

l

con

-

h

nection wit

h

the

c

ommutatio

n

relation Pq--q

P--

_;i s

ha

ll

be shown below. Here we shall point out that equation (I) is

the precise expression for the fact that we once sought to

des

c

ri

b

e by dividing t

h

e p

ha

se sp

a

ce into

c

ells of size

h

.

Other experiments can also be performed to determine the e-

lectron's position, such as impact tests. A very precise de-

termination of the position requires impacts with very fast

p

a

rtic

l

es

,

sin

c

e for slow electrons t

h

e

d

iffr

a

ction phenomen

a

- which according to Einstein are a consequence of the de

Broglle waves (see for instance the Ramsay effect) - preclude

a precise determination of the position. Thus, once again for

a precise position measurement the electron's impulse changes

disontlnuously and a simple estimate of the precision with

the equations of the de Broglie waves once again leads to e-

quation (1).

This

d

is

cu

ssi

on s

e

e

ms t

o

define

the co

n

c

e

pt

po

si

t

i

on o

f t

he

electron clearly enough and we only need to add a word about

5


8/35

the size of the electron. If two very fast particles strike

the electron sequentially in the very brief time interval At,

then the two positions of the electron defined by these two

particles lie very close together, separated by a distance AI.

From the laws observed for m-particles we conclude that AI can

be reduced to a magnitude of the order of 10-12 cm, provided

At is sufficiently sma

l

l

a

nd the p

a

rti

c

les sele

c

ted

a

re suf- /17--6

ficiently fast. That is the meaning, when we say that the e-

lectron is a particle whose radius is not greater than 10-12 cm.

L

et us m

ov

e

o

n to the

co

n

cep

t of the path of the ele

c

tron.

By

pa

th or tr

a

jectory we me

a

n a series of points in s

p

a

c

e (in

a given referen

c

e system) th

a

t the electron adopts as su

c

essive

p

ositions. Since we already know what position

a

t a

c

ert

a

in

time means, there &re no new difficulties, here. It is still

readily understood that the often used ex

p

ression, for instance,

the I-S orbit of the ele

c

tron in the hydrogen atom makes no

sense, from out

p

oint of view. Be

ca

use in order to me

a

sure this

IS orbit, we woul

d

ha

v

e to illumin

a

te the

a

tom with light su

c

h

that its wa

v

elength is considerably shorter than 10-8 cm. But

one light quantum of this kind of light would be sufficient to

completely throw the electron out of its orbit (for which

reason never more than a single point of this path could be

defined, in sp

a

ce)

a

nd hence the word path is not very sen-

sible or meaningful, here. This can be easily derived from the

experimental possibilities, even without

a

ny knowledge of the

new theories.

I

n cont

r

as

t

,

t

he

im

ag

i

na

r

y po

siti

o

n m

ea

s

u

r

e

ments c

an be p

er-

forme

d

f

o

r m

a

n

y a

toms i

n a

IS

s

t

a

te. (At

o

ms in

a g

iven st

a

ti

on

-

a

ry st

a

te, f

o

r inst

a

nce, can in

p

ri

n

ci

pl

e be is

ola

te

d

b

y

t

h

e

Stern-Gerl

a

c

h

e

xp

eriment.) T

hu

s, f

o

r

a g

i

v

e

n

state, f

o

r i

n

s-

t

an

ce 1S, of an atom, a probability function must exist for the

elect

r

on

'

s pos

i

t

i

o

n

s

,

such

t

ha

t i

t co

rr

esponds

,

on th

e

av

er

ag

e,

t

o

t

he class

i

cal t

r

a

j

ecto

r

y ove

r

all phases

, a

nd

t

ha

t

can be


9/35

est

a

blishe

d

by me

a

suremen

t

s to

a

ny desired pre_ision. Ac

c

ord-

ing to Born* this fun

c

tion is given by _is(q

)

$1s(q

)

, if

$is(q

)

is the Schroedinger w

a

ve function corresponding to the

st

a

te IS. I w

a

nt t

o

J

o

in Dir

a

c*

a

nd J

o

rd

a

n*, in view of sub-

sequent generalizations, in saying: the probability is given /177

by S(IS,q)_(IS,q), where S(IS,q) is that column of the trans-

formation matrix S(E,q) from E to _, which corresponds to E =

EIS (E = energies).

In the fact that in quantum theory for a given state - for

instance IS - only the probability function for the electron

position can be given, we may see a characteristic statistical

feature of quantum theory, as do Born and Jordan, quite in

contrast to the classical theory. On the other hand, if we

want to we can say with Dirac that the statistics came in via

our experiments. Because also in classical theory only the

pr

ob

a

bility o

f

a cer

ta

in ele

ct

ron position could be given, if

a

nd

a

s long

a

s we do not know the

a

tom's ph

a

ses. R

a

ther, the

difference between c

la

ssic

a

l aud qu

a

ntum mech

a

nics consists in

this: cl

a

ssic

ally

, we can

a

lw

ay

s

a

ssume the ph

a

ses to h

a

ve

been determ

i

ned in

a

previous e

x

periment

.

But in re

a

lity this

is impossible, bec

a

use ever

y

e

x

periment to determine the ph

a

se

wou

l

d either destroy or modify the

a

tom.

I

n

a

definite st

a

tion-

a

ry st

a

te of the

a

tom, the ph

a

ses

a

re indetermined in

* The st

a

tistical meaning of the de Brog

l

ie waves was first

formu

la

ted by A. Einstein

[S

itzungsber.d.preuss.Ak

a

d.d.

Wiss. 192

5

, p.

3)

. This st

a

tistic

al

element then p

la

ys

a

slgnif

T

_t ro

l

e for M.

B

orn, W. Helsenberg

a

nd P.

J

ord

a

n,

Ouan

tum

m

ec

han

ics

II

.

[Z.

f

.

P

hy

s.

35

,

557

(1926

)]

,

e

sp

e

-

ci

ally

ch

a

pter 4, §3,

and

P. J

o

r

da

n-_Z.f.Ph

y

s

. 37

, 3

7

6

(1926

)]

; it is

a

n

aly

ze

d

m

a

them

a

tic

ally

in

a

fun_

-a

ment

al

paper b

y

M.

Bo

rn

[

Z.f.

P

h

y

s.

38

,

80

3 (1926

)] a

n

d

use

d

f

o

r

t

h

e i

n

ter

p

ret

a

t

io

n

o

f the c

o

IIisl

o

n

ph

e

no

men

a

.

T

he f

o

u

n

d

a

-

ti

o

n f

o

r using the

p

r

o

b

a

bi

l

it

y

the

o

rem fr

o

m the transf

o

rm

a

-

ti

o

n the

o

r

y

f

o

r m

a

trices c

a

n be f

o

un

d

in: W. Helsen

b

erg

[

Z.

f. Phys. 40, 501 (1926)], P. Jordan [ibid. 40, 661 (1926)],

W. Paull-TAnm. in

Z

.f.Phys. 41, 81 (1927)]_-P. Virac [Proc.

R

oy

.

So

c.(A

)

113, 621 (1

9

2

6)]

,

P

.

Jo

r

da

n

[Z

.f

.P

h

y

s. 40, 80

9

(

1

926)

]

.

The

_

a

ti

s

tic

al

si

de of q

u

a

nt

u

m m

ec

h

an

ics i_

gen

-

e

r

al

i

s dis

c

u

sse

d by P. Jo

r

dan (

N

a

t

u

rwiss. 1

5

, 1

0

5

(

1

9

2

7)]

a

n

d

M.

B

orn

[

N

atu

rwlss. 1

5

, 2

38 (

1

9

27

)]

.


10/35

ORIGINALPAGE [8

OF POOR QUALITY

principle, which we may consider a direct clarification of the

known equations

h or 3w- w3

=

El-- fE = _ =D _=-;

(

]

: action variable,

w

: angular variable).

T

h

e word velocity of an obje

c

t is e

a

sily defined by measure-

merit, if it is a force-free motion. For instance, the object

can be illuminated with red light and then the particle's ve-

locity c

a

n be

d

etermi

n

e

d

by the Doppler effect of the sc

a

tter-

ed light. The determination of the velocity will be the more

precise, the longer the wavelength of the light used is, since

then the particle's velocity change per light quantum due to

Compton effect will be the smaller. The position determination

becomes correspondingly uncertain, as required by equation(1).

If the velocity of the electron in an atom is to be measured

a

t

a c

ert

a

in inst

a

nt, we shoul

d

h

av

e to m

a

ke the nu

c

le

a

r ch

a

rge

and the forct:sdue to the other electrons disappear, at that

inst

a

nt

,

so t

h

at the motion m

a

y

p

roceed for

c

e free,

a

fter t

ha

t

instant, to then perform the determination described above. As

was the case earlier, we once again can convince ourselves that

a function p(t) for a certain state of the atom - say, IS - can

not be defined. In contrast, there again will be a function for

1

17--8

the probability of _ I_ this state, which according to Dlrac

and Jordan will have the value S(1S,p)_(1S,p). Again, S(1S,p)

means the column of the transformation matrix S(E,p) of E Int_

pth

a

t c

o

rres

po

n

ds

t

o

E

=

E

I

S

.

Finally, let us point out the experiments that allow the meas-

urement of the energy or the value of the action variables J.

Such experiment_ are particularly important since only with

their aid will we be able to define what we mean, when we talk

about the discontinuous change of the energy

or

or J. The

8


11/35

Franck-Hertz collision experiments permit the tracing back of

the energ_ measurements on atoms to the energy measurements of

electrons movin

g

in a straight line, because of the validity

of the ener

g

y theorem in the quantum theory. In principle,

this measurement can be made as precise as desired, if only

we forego the simultaneous

d

etermination of the electron posi-

tion, i.e., of the phase (see above, the determination of _),

k

correspondin

g

to the rel

a

tion £

t

--

t

E----_-z3

•

The Stern-

Gerl

a

ch experiment permits the

d

etermin

a

tion of the magnetic

or

a

n average electric moment of the

a

tom, i.e., the measure-

ment of magnitu

d

es th

a

t

d

epend onl

y

the

a

ction v

a

ri

a

bles J.

T

he

ph

a

ses remain un

d

etermined in principle. If it is not sensible

to t

a

lk of the frequenc

y

of

a

light wa

v

e

a

t a given inst

a

nt, it

is not possible either to spe

a

k of the ener

gy

of

a

n

a

tom

a

t

a

p

a

rticular inst

a

nt. In the Stern-Gerlach e

xp

eriment this cor-

respon

d

s to the situ

a

tion th

a

t the

p

recision of the ener

gy

me

a

surement will be the sm

a

ller, the shorter the time interv

a

l

d

urin

g

which the

a

tom is un

d

er the influence of the

d

eflecting

forcem.

B

ec

a

use

a

n upper limit for the deflecting force is

given by the f

a

ct th

a

t the

p

otenti

a

l energy of that

d

eflecting

force insi

d

e the be

a

m of r

ay

s c

a

n v

a

ry onl

y

by qu

a

ntities th

a

t

a

r

e

consider

a

bly sm

a

ller th

a

n the ener

g

y

d

ifferences of the

st

a

ti

o

n

a

ry st

a

tes, if

a d

etermin

a

ti

o

n of the st

a

tion

a

ry st

a

tes'

ener

g

y is t

o

be

p

ossible. If

E I

is the qu

a

ntit

y o

f energy th

a

t

s

a

tisfies th

a

t c

o

n

d

iti

o

n (

E

I

a

t the s

a

me time is a me

a

sure

o

f

the

p

recisi

o

n

o

f th

a

t ener

g

y me

a

surement), then

E

1/

d

is the

m

ax

im

u

m

val

u

e

f

o

r t

he d

efl

ec

t

i

n

g fo

r

c

e

,

if

d i

s t

he

wi

d

th

o

f

t

h

e ra

y

beam (measurable

by

mea

n

s

of

t

h

e wi

d

th of the s

l

it

used. The angular deflection of the atom beam Is then £1tl/dP,

w

he

r

e

t

I

is t

he p

er

iod o

f time

du

r

ing

w

hich

t

h

e at

o

m

s a

re

und

er

.

the effect

of

the deflecting force, _ the impulse

of

the atoms /179

in the direction of the beam. This deflection must be at least

of the same order of magnitude as the naturaZ beam broadening

caused by diffraction in the slit, in order for a measurement

u Cf. also W. Pault, 1.c.p.61

9


12/35

. OF POORQOAL

F

rV

to be possible. The angular deflection due to diffraction is

approximately _/d, where R is the de Broglie wavelength, i.e.,

a _ dp cr since _--_.

_

t

, _ h. (_)

This equation corresponds to equation (1) and it shows that a

precise energy determination can be attained only through a

corresponding uncertainty in the time.

§ 2 The Dirac-Jordan theory

We would like to summarize the results of the previous section

and generalize them in tLis statement: All concepts used in

classical theory to describe a mechanical system can also be

defined exactly for atomic processes, in analogy to the classic

concepts. But purely from experimentation, the experiments that

serve for such definitions carry an inherent uncertainty, if we

expect from them the simultaneous determination of two canoni-

cally conjugated variables. The degree of this uncertainty is

given by equation (I), widened to include any canonically con-

jugated varlab]es. It is reasonable to _ere compare the quantum

theory wlth the special theory of relativity. According to the

theory of relativity, the term slmultaneous _'can only be de-

fined by experiments in which the propagation veloclty of light

plays an essential role. if there were a sharper definition

of simultaneity - for instance, signals that propasate infl-

nltely rapidly - then the theory of relativlty would be Impos-

sl

b

le.

Bu

t since s

u

c

h

signals

do no

t e

x

ist -

b

ec

a

use t

h

e

v

el

o

-

city of light already appears in the defi

n

ltlo

n

of simultane-

ity - room is available for the postulate of a constant velo-

city of light and therefore th_a Fostulate is not contradicted

by the appropriate use of the terms, position, veloclty, time*.

T

h

e

s

i

tu

a

t

i

on I

s sim

ila

r

in r

e

g

a

rd to

t

h

e _efi

ntt

l

on o

f t

h

e

10


13/35

ORIGINALPAGE_

OF POOR QUALITY

concepts electron position and velocity , in quantum theory.

All the experiments we could use to define these terms neces-

sarily contain the uncertainty expressed by equation (I), even

though they permit

a

n ex

ac

t definition of the individu

a

l

c

on-

cepts £ and _. If experiments existed that allowed a more

precise definition of _ and _ than that

c

orresponding to e-

quation (I), th

e

n the quantum theory would be impossible. This

/

IBO

uncertainty - which is fixed by equation (I) - now provides the

space for the relations that find thel; terse expression in

the commutation relations of quantum mechanics,

k

Pv--qP --'-2xi

T

his equ

a

tion be

c

omes possible wit

h

out h

a

vin

g

to

c

h

a

n

g

e t

he

physical meaning of the variables E and _.

For those physical phenomena for which a quantum theory formu-

lation is still unknown (for instance, electrodynamics), equa-

tion (1) represents a demand that may be helpful in finding the

ne

w

la

ws. F

o

r

quan

t

u

m m

echan

i

c

s

, equa

ti

o

n

(1) can be de

ri

ved

from the Dirac-Jordan formulation, by means of a minor general-

iz

a

ti

on.

If f

o

r

a ce

rt

a

i

n value

n

o

f

an a

r

b

itr

a

r

y pa

r

a

m

e

t

e

r w

e

can determine the position _ of the electron at q' with a pre-

cision ql' then we can express this fact by means of a proba-

bility am_lltude $(n,q) that wlll be noticeably different from

ze

r

o only

i

n an a

rea

o

f

app

r

ox

im

a

te

d

ime

ns

i

on ql a

r

ound q'.

W

e

c_n thus say, more specifically

i

.

e

.,

We

t

hu

s

have

f

or t

he

probabili

t

y a

m

pli

t

ud

e

cor

res

pondtn8

t

o p:

s($_) : _s($ e)s_.t)de. (4)

Zn asreement wlth Jordan, we can say for :S(q,p) :hat

,(,,

.

)=

1'1


14/35

, ORIG,,._ = Pj

• OF POGR 4UALITY

In that

c

ase, ac

c

ording t

o (4),

S(

q

,p

)

will b

e

noti

ce

ably dif

-

ferent fr

o

m zero only for values of p for which 2_

(

p-p'

)

ql

/

h

is not substantially larger than I. More especially, in the

case of

(

3

)

we shall have:

S(

_. j

,)

pr

o

p

J e

_ 'v,'

,t

q

,

i.e.,

S{_

,

p

)

pr

o

p

¢

=l,= +h-¢'_-I _

that is

S_pr

op

e p

t

*

_iqt -- ....

(6)

/181

Thus, as

s

umption

(3)

for S(n,q

)

corresp

o

nds t

o

the e

xp

eriment-

al fact that the value p' of _ and the v

a

lue

q

' of _ were me

s

s-

ured

[

with the pre

c

i

s

ion restriction

(

6

)]

.

Th

e

purely

ma

the

ma

tical

c

h

a

r

ac

t

e

risti

c o

f th

e

D

i

r

ac-

J

o

r

d

an

formulation

o

f qu

a

ntum m

ec

hanics is that th

e

relations b

e

tween

p

,¢,

E ,

e

t

c.

,

ca

n be written

as equ

ations

b

etween

ve

ry g

e

n

-

eral matrices, such that any vari

a

ble indicated by quantum

theory a

p

pears as the diagonal matrix. The feasibility of such

a notation seem reasonable if we visualize the matrices

a

s

tensors

(

for instance, moments of inerti

a)

in multidimension

a

l

spaces,

a

mong which mathematical rel

a

tions exist. The

a

xes of

th

e

coordinate s

y

st

e

m in whi

c

h t

h

ese m

a

t

he

matic

a

l r

e

l

a

ti

o

ns

are expresse

d

can alw

a

ys be pl

ac

e

d a

long the m

a

in

ax

is of o

n

e

of these tensors. It is

a

fter

a

ll alw

a

ys possible to ch

a

r

a

cter-

ize the m

a

them

a

ti

ca

l rel

a

tion between two tensors A

a

n

d B

by

me

a

ns

o

f tr

a

nsf

o

rm

a

tion formul

a

e that will c

o

n

v

ert

a

system of

coordinates oriente

d a

long the m

a

in

ax

is of A, into one ori ....

e

n

ted a

l

on

g

th

e m

a

i

n ax

is

o

f

B.

T

he

latt

e

r f

o

rm

u

l

a

tion

co

rtes-

p

o

n

d

s to $c

h

roe

d

inger's the

o

ry. In c

o

ntr

a

st

,

Dir

a

c's not

a

tion

of the q-numbers m

u

st be c

o

nsi

d

ere

d

the tru

l

y Inv

a

rl

a

nt

12


15/35

formulation of quantum mechanics, independent of all coordi-

nate sy

s

tems.

I

f we wanted to derive physical results from

that mathematical model, then we must assign numerical values

to the quantum mechanics variables, i.e., the matrices (or

tensors in multidimensional space). This is to be understood

s meaning that in that multidimensional space a certain di-

rection is arbitrarily chosen

(

that is, establi

s

hed by the

kind of experiment performed

)

, and then the value of the

matrix is asked for (for instance, the value of the moment of

inertia, in that picturel, in the direction chosen. This ques-

tion has unequivocal meaning only if the direction chosen co-

incides with one of the matrix' main axes: in that case there

will be an exact answer to the question. If the direction

chosen deviates but little from one of the matrix' main direc-

tions, we can still talk with a certain imprecision, given by

the relative inclination, with a certain probable error, of

the value of the matrix in the direction chosen. We can thus

state: it is possible to assign a number to every quantum

theory variable, or matrix, which provides its value , with a

certain probable error. The probable error depends on the sys-

tem of coordinates. For each quantum mechanics variable there

/

182

exists one system of coordinates for which the probable error

vanishes, for that variable. Thus, a given experiment can

never provide precise information on all quantum mechanics

variables: rather, it divides the physical variables into

known and unknown {or: more or less precisely known vari-

ables

)

, in a m

a

nner ch

a

racteristic for that experiment. The

results of two experiments c

a

n be derived precisely from e

a

c

h

other only when the two experiments divide t

h

e physic

a

l v

a

ri-

a

bles in the s

a

me m

a

nner into

k

nown

a

nd unknown (i.e., if

the tensors in that multidimension

a

l sp

a

ce

a

lre

a

dy used for

visu

a

liz

a

tion are viewed from the s

a

me direction, in both

experiments.

) I

f two e

x

periments cause two different distribu-

tions into known

a

nd unknown v

a

ri

a

bles, then the rel

a

tion

of the results of those experiments c

a

n be given

a

ppropri

a

tely

only st

a

tistic

a

lly.

1

3


16/35

' ORiGiNAL _A_ _

OF PO

O

R QUALITY

L

e

t us perform an imaginary experim

e

nt, to m

o

re pre

c

isely dis-

cuss these statistical relations We shall start by sending a

Stern-Gerlach beam of atoms through a field F I that is so in-

homogeneous in the beam direction, that it causes noticeably

numerous transitions due to a shaking effect . The atom beam

is then allowed to run unimpeded, but then a second field shall

begin, F2, as inhomogeneous as F I. We shall assume that it is

possible to measure the number of atoms in the different sta-

tionary states, between F I and F2 and also beyond F2, by means

of an eventually applied magnetic field. Let us assume the

atoms' radiative forces to be zero. If we know that an atom was

in the energy state En before passing through F I, then we can

express this experimental fact by assigning a wave function to

the atom - for instance, in p-space - with a certain energy Ep

and the indetermined phase Sn

After passing through field FI, the function will have become*

_

.

' _ _

:,

_. __)

S(E., _)--,. _]c.,. _(E.,, _)¢ h _.7)

Jl

Let us assume that here the 8m are arbitrarily fixed, such /183

that the Cnm is unequivocally determined by F]. The matrix

Cnm transforms the energy value before passing through F I to

that after passing through F]. If behind F] we perform a de-

termination of the stationary states - for instance, by means

of

a

n inhomog

e

neous m

a

gneti

c

field - then we sh

a

ll find, with

a probability of Cnm_nm that the atom has passed from the

st

a

te _ t

o

t

h

e st

a

t

e

_. If w

e de

t

e

rmine experi

m

e

n

t

a

lly th

a

t

the atom has actually acquired the state m, then in the sub-

sequent calculations we shall have to assign it the function

* See P. Dirac, Proc.Roy.Soc. (A)112, 661 (1926) and M. Born,

Z. f. Phys• 40, 167 (1926).

14


17/35

[,

o

ORIGINAL P

A

C

T _

OF POOR QUALITY

Sm with an indeterminate phase, instead of the function

_

c

_,.Sm . Through the experimental determination state m

we select, from among t

h

e different possibilities (Cnm) , a

certain _ and simultaneously destroy, as we shall explain

below, whatever remained of phase relations in the variables

Cnm. When the beam passes through F2, we repeat the same pro-

cedure used for F I. Let dnm be the coefficients of the trans-

formation matrix that converts the energies before F2 to those

after F2. If no determination of the state is performed bet-

ween F I and F2, then the eigen-function is transformed accord-

ing to the following pattern:

s

(

E.,p

)

r

-

__.,..s(_

.

,p

)

_-_

_

. _,_

.

._.

,

S(E,, _,).

(

8

)

m m I

Le

t _

=

g._--e

.

_ . If t

he

stati

o

nary st

a

t

e o

f t

he a

tom

is determined, after F2, we shall find the state _ with a pro-

bability of enlenl . If, in contrast, we determined state m

between F I and F2, then the probability for _ behind F2 is

given by dml_ml . Repeating the entire experiment several time

s

(determining the state, each time, between F I and F2) we shall

then observe the state _, behind F2, with the relative frequency

Z.L-

--_,,

c

..

c_

.

d,_a,.t

.

This e

x

pression does not agree with

m

enl_nl. For this reason Jordan (l.c.) mentions an interference

of the probabilities . I, for one, would not agree with this.

Because the two experiments leading to enlenl or Znl, respec-

tively, are really physically different. In one case the atom

suffers no disturbance between F I and F21 in the other it is

disturbed by the equipment that makes the determination of the

stationary states possible. The consequence of this equipment

is that the phase of the atom changes by quantities that are

un

c

ontr

o

llable in prin

c

iple, J

u

st

a

s th

e

impuls

e

w

a

s

cha

ng

ed

/18__4

in the determination of the electron's position (cf. § I). The

magnetic field for the determination of the state between F I

and F2 will change the eigen-values E and during the observa-

tion of the atom beam (I am thinking of something like a Wilson

track) the atoms will be slowed down in different degrees,

15


18/35

statistically, and in an uncontrollable manner. As a conse-

quence, the final transformation matrix enl (from the energy

values before F I to those after leaving F2) is no longer given

by 3_

,

_ , and instead each term of the sum will have, in

addition, an unknown phase factor. Hence, all we can expect

-

s for the average value of enlenl, over all eventual phase

changes, to be equal to Znl. A simple calculation shows this

to be the case.

Thus, following certain statistical rules, we can draw conclu-

sion3, based on one experiment, regarding the results possible

for _nother. The other experiment selects, by itself and from

among all the possibilities, one particular one, thus limiting

the possibilities for all subsequent experiments. This inter-

pretation of the equation for the transformation matrix S, or

Schroedinger's wave equation, is possible only because the sum

oe all solutions is also a solution. Here we can see the deeper

meaning of the linearity of Schroeding, r's equations and hence

t ey can be understood only as waves in the phase space; for

ttis same reason we would consider any attempt to replace

these equations - for instance, in the relativistic case (for

several electrons) - by non-linear equatio

n

s as doomed to fail.

§ 3 The transition from micro to macromechanics

I believe the analyses performed in the preceding sections of

the terms electron position , velocity , energy , etc., have

sufficiently clarified the concepts of quantum theory kinemat-

ics and mechanics, so that an intuitive understanding of the

_croscopic processes must also be possible, from the point of

view of quantum mechanics. The transition from micro to macro

mechanics _as already been dealt with by Schroedinger*, but I

* E. Scnroedinger, Naturwiss. 14, 664 (1926)

16


19/35

do not believe that Schroedinger's considerations address the

essence of the problem, for the following reasons: according

to Schroedinger, in highly excited states a sum of the eigen-

vibrations will yield a not overly large wave packet, that in

its turn, under periodic changes of its size, performs the

periodic motions of the classical electron . The following /185

objections can be raised here: If the wave packet had such

properties as described here, then the radiation emitted by

the atom could be developed into a Fourier series in which the

frequencies of the harmonic vibrations are integer multiples

of the fundamental fr_4uency. Instead, the frequencies of the

spectral lines emitted by the atom are never integer multiples

of

a

fundamental frequency, according to quantum mechanics -

with the exception of the special case of the harmonic oscil-

lator. Thus Schroedinger's consideration is applicable only to

the harmonic oscillator considered by him, while in all other

cases in the course of time the wave packet spreads over all

space surrounding the atom. The higher the atom's excitation

state, the slower will be the scattering of the wave packet•

But it will occur, if one waits long enough. The argument used _

above for the radiation emitted by an atom can be used, for the '

time being, against all attempts of a direct transition from

quantum to classical mechanics, for high quantum numbers. For b

this reason, it used to be attempted to circumvent that argu- |

ment by pointing to the natural beam width of the stationary I

states; certainly improperly, since in the first place this I

,°

way out is already blocked for the hydrogen atom, because of lnsufficient radiation at higher states; in the second place,

the transition from quantum to classical mechanics must be un- _

erstandable without borrowing from electrodynamics. Bohr* has

.

repeatedly pointed out these known difficulties, in the past, ;

that make a direct connection between quantum and classical

theory difficult. If we explained them here again in such ,_

* N

.

B

oh

r, B

a

sic

Po

stul

a

tes

o

f Q

ua

nt

um

The

o

ry, l

.

c.

1

7 _


20/35

detail, it is because apparently they have been forgotten.

I believe the genesis of the classical orbit can be precise-

ly formulated thus: the orbit only comes into being by o

u

r

observing it. Let us assume an atom in its thousandth excita-

tion state. The dimensions of the orbit are relatively large

here, already, so that it is sufficient, in the sense of § I,

to determine the electron's position with a light of relative-

ly long wavelength. If the determination of the electron's

position is not to be too uncertain, then one consequence of

Compton recoil will be that after the collision, the atom will

be in some state between, say, the 950th and the 1050th. At

the same time, the electron's impulse can be derived - to a

precision given by equation (I) - from the Doppler effect. The

experimental fact so obtained can be characterized by means of /186

a wave packet - or better, probability packet - in q-space, by

a variable given by the wavelength of the light used, essen-

tially composed of eigenfunctions between the 950th and the

1050th eigen-function, and through the corresponding packet in

p-space. After a certain time, a new position determination is

performed, to the same precision. According to § 2, its result

can be expressed only statistically; possible positions are all

those within the now already spread wave packet, with a calcu-

lable probability. This would in no way be different in clas-

sical theory, since in classical theory the result of the sec-

ond position could also be given only statistically, due to

the uncertainty in the first determination; In addition, the

system's orbits would also spread in classical theory similarly

to the wave packet. However, the laws of statistics themselves

are different, in quantum mechanics and classical theory. The

second position determination selected a _ from among all those

possible, thus limiting the possibilities for all subsequent

determinations. After the second position determination, the

results for later measurements can be calculated only by again

assigning to the electron a smaller wave packet of dimension

18


21/35

ORIG.,_AL =_'4

OF POOR OUALI'P[

_ (wavelength of the light used for the observation). Thus,

each position determination reduces the wave packet again to

its original dimension i. The values of the variables p

and q are known to a certain precision, during all experi-

ments. Since within these limits of precision the values of

p and q follow the classical equations of motion, we can

conclude, directly from the laws of q

u

ant

u

m mechanics,

dH #H

P=

-

q=

.

But as we mentioned, the orbit can only be calcu]%ted statis-

tically from the initial conditions, which we may consider a

consequence uncertainty existing in principle, in the initial

conditions. The laws of statistics are different for quantum

mech

a

nics and classical theory. Under certain conditions, this

can lead to gross macroscopic differences between classical and

quantum theory. Before discussing an example of this, I want

to show by means of

a

simple mechanical system - the force-free

motion of a mass point - how the transition to the classical

theory discussed above is to be formulated mathematically. The /18__/7

equations of motion are (for unidimensional motion)

1 , I ,

4=;i p p=o. (1o)

Since time can be treated as a parameter (as a c-number ) if

there are no external, time-dependent forces, then the solu-

tion to this equation is:

1 t

q ----._p, + q, ; p -- p,, (11)

where p, and _ represent impulse and position at time t=O.

At time t=O [see equations (3) to (6)], let qo = q' be meas-

ured with precision q1' Po = p' with precision p;. If from

the values of _ and _ we are to derive the value of q

at time _, then according to Dirac and Jordan we must find

that transformation function, that transforms all matrices

19


22/35

' ORIGINAL

PAGE

_

J

OF POOR QUALITY

: in which q

o

a

pp

ears

a

s

a

di

a

g

o

n

a

l m

a

tri

x,

into m

a

tri

c

es in

which

q

appears as the diagonal m

a

trix

. I

n the _atrl

x

p

a

t-

tern in which

qo

appears as the diagonal matrix,

p,

can be

k d

replaced by the

o

perator _ . Acc

o

rding t

o

Dirac

[

l

.

c

.

_ equation (11)

]

we then have for the transformation amplitude

sought,

S(qo,q) ,

the differential equation

li k

0

I

,

,

,

_ _-_q,_e

o

j s

(

q.,e)

=

e

s(

q

.,

_)

(

1_)

,,

,

, __

,

,).-

S

(

q

e,

e

)

_

const.

e ..... _.

-

t.....

(I

S

) .

Thus S_ is independent of qo' i.e

.

, if at time t : 0, qo is

known e

xac

tly

,

t

he

n at

a

n

y

time t > 0 all v

a

lu

e

s of q

a

r

e e-

qually likely, i.e., the proba

b

ility t

ha

t _ lle

s

within

a

fi-

nite range, is generally zero. T

h

is i

s q

uite

c

le

a

r

,

intuiti

v

e-

ly

.

B

eca

u

se

th

e e

x

ac

t d

e

termination of

q

o le

ad

s t

o a

n infi

-

itely l

a

rge

C

ompton recoil. The same would of course be true

of

a

ny me

c

h

a

ni

ca

l system.

H

owever, if at time t = 0 , qo i_

known only to

a p

re

c

ision q

l a

nd Po to pre

c

ision

P

I' t

hen [c

f.

equation

(3)]

S(,/,_,)= COat.e-- _ f--Tp _'--_,

and the probability function for _ will have be calculated /18_8

from the equation

We

ob

t

a

i

n

t

Bdm f I t ,%

If we introduce the abbreviation

20


23/35

oRIGINALpAGE_

• O

F PoOR QUAL

I

TY

then the expon

e

nt in (141 becomes

-- , (, ;,))+""I

The term in q,2 can be included in the constant factor (inde-

pendent of g); by integration we obtain

, l,'

,,

)r

l

q

t= 1

(

_,,j

)

-- eou.t.e

,

(16

(,

_

;

,,_.

,

,)

(

,

-

¢onst. e

-

" sqL'

(I

From which follows

(,--,._,.)'

S

(

e_

.

_J]._

(

_,__

-- eonst.

e e_t

(i

+P_

-

.

(IT)

Thus, at time t the electron is at position (tlm)p' + q' to

a precision _lyT_-_ . The wave packet or better, the

probability p_c_:et has become larger by a factor of }:I_.

ccording to (15), 13 is proportional to the time t, inversely

proportional to the mass - this is immediately plausible - and

inversely proportional to q2I. Too great a precision in qo has a

greater uncertainty in Po as a consequence and hence al.qo

leads to an increased uncertainty in _[. The parameter n, which

we introduced above for formal reasons, could be eliminated in

all equations, here, since it does not enter in the calcula-

tions

As an example that the difference between the classical laws

f statistics and those from quantum theory can lead to gross

m

ac

r

o

s

cop

i

c d

i

f

f

e

r

ence

s i

n the

r

es

u

lts

fr

o

m

bo

t

h theo

ri

es,

u

n

-

d

er

ce

rt

a

i

n cond

iti

ons

, sh

all be b

rief

ly d

i

sc

u

ssed

f

o

r

the

r

e

f

lect

i

on o

f

an

e

lec

tr

on

f

lo

w

by a

gr

at

i

n

g

. I

f t

he la

tti

ce


24/35

m

constant is of the order of magnitude of the de Broglie wave- /189

length of the elec

t

ron, then the reflection will occur in

c

ertain

d

is

c

rete

d

irections in

s

p

a

ce,

as

d

o

es the

li

ght at a

gr

a

ting. Here,

c

l

a

s

s

ical the

o

ry

y

iel

d

s m

ac

r

o

s

co

pi

c

ally some-

thing gr

o

ssly different.

A

nd yet,

we

can not find a c

o

ntra

d

ic-

a

g

a

inst classic

a

l theory in the

o

rbit

o

f

a

single electr

on

.

We could do it, if some

h

ow w

e

could dir

e

ct t

h

e electron t

o

a

certain l

o

cation

on a

grating line an

d

t

h

er

e

est

a

blish that the

ref

l

ection did n

o

t

occ

ur

c

l

a

s

s

i

c

al

l

y

.

But if w

e

w

a

nt t

o

deter-

mine the electron's p

o

sition s

o

pre

c

isely th

a

t we c

o

uld say at

w

h

i

c

h l

oca

ti

o

n

o

n

a

gr

a

ting line it w

o

uld imp

a

ct, then the elec-

tron would acquire such a velocity, due to this determination,

that the de Broglie wavelength of the electron would be reduced

to the point that in this approximation, the electron would be

a

-tu

all

y ref

l

e

c

ted in th

e d

ir

ec

tion pre

sc

ribed

b

y

c

l

as

si

cal

th

eory

, without

con

tr

ad

icting the l

a

ws

o

f

qua

ntum th

eo

ry

.

§ 4

D

iscussion of s

om

e s

p

eci

a

l

,

im

a

gi

na

r

y

e

xp

e

rim

e

n

ts

According to the intuitive interpretation of quantum theory at-

tempted here, t

h

e points in time at w

h

ich transitions - t

h

e

quantum Jumps - occur should be experimentally

d

etermin

a

ble

i

n a co

ncr

e

t

e

m

a

n

n

er, s

u

c

h a

s

ene

rgi

e

s

o

f st

a

ti

o

n

a

r

y

st

a

t

e

s,

f

o

r inst

a

nce. The

p

recisi

o

n t

o

which such

a po

int in time c

an

be de

term

in

e

d

is give

n by equa

ti

o

n

(2) a

s

h

lA

E

I

,

if AE

i

s the

change

in

energy accompanying the transition. We are thinking

of an experiment such as the following: Let an atom, in state

2 at tim

e t=O

, r

eturn

t

o

i

t

s n

o

rma

l sta

te I b

y

e

m

i

tt

in

g

ra

d

i

a

-

t

i

on

.

W

e could

t

h

e

n ass

ign

to the a

t

o

m

,

i

n analo

g

y

t

o

e

qua

ti

on

(7), the eigenfunctton

i i gl i i el i i

m

See W. Pauli,

1.c., p.12

,?.2


25/35

' ORIGINALPAGE

OFP

OO

RQUALr

rf

A _1 e ' _(

E

,,p)

e

- -T'- (18)

s

(

t

,p) = _

.

,_(_,_e + - -

if we assume that the radiation damping wlll express itself in

t

he e

ig

e

n-f

u

n

c

tion b

y

m

ean

s of

a

f

ac

tor of t

h

e form

e-a

t

(

the

true dep

e

nden

c

e m

a

y not b

e

t

ha

t simpl

e). Let

us s

e

n

d

this

a

tom

t

h

rou

g

h

a

n in

h

omog

e

n

e

ous m

a

gn

e

ti

c

fi

eld,

to me

a

sur

e

its

e

nerg

y,

a

s is

c

ustom

a

ry in t

h

e Stern-Ger

lach e

xp

e

rim

e

nt

, e

x

c

ept t

ha

t

t

h

e inhomog

e

n

e

ous fi

e

l

d

s

ha

ll f

o

ll

o

w t

l

_e

a

tom be

a

m for

a

g

o

o

d

por

t

io

n

of

the pa

t

h. The c

orresponding

acc

e

le

r

ati

o

n c

ou

ld

be

me

a

sured by

d

ivi

d

ing t

he

entire p

a

t

h

fo

ll

ow

ed

by t

he a

t

o

m

bea

m

in t

h

e m

agne

t

ic

fi

e

l

d,

into s

ma

ll p

a

rti

a

l p

a

t

h

s,

a

t t

he

en

d

of

eac

h

of whi

c

h w

e

m

ea

su

r

e t

h

e

b

e

a

m's

d

ef

lec

ti

on

.

De

p

e

n

d

i

n

g on 1

19

--0

t

he at

om b

ea

m's

v

e

loc

ity

,

t

h

e

d

i

v

ision into

par

ti

al pa

t

h

s wi

ll

f

o

r t

he a

lso t

o d

i

v

ision into time

correspon

d, a

tom

,

p

a

rtl

a

l

int

e

rv

al

s At

. Acc

or

d

i

n

g to §

I,

e

q

u

a

tion

(2), to

t

he

inter

va

l

At

cor

respo

n

ds

a p

r

ec

isi

on

i

n

t

h

e

e

n

e

r

g

y of

h/A

t

. The

pr

oba

bll-

i

ty of me

a

suring

a ce

rt

a

in

e

n

e

rgy

ca

n

b

e

d

lre

c

t

l

y

de

riv

e

d from

S(p,E) and

is

h

e

nc

e

calcu

l

a

t

ed i

n t

he

l

.

,t

e

r

va

l from

n

at to

(

n

+1)

At by me

a

ns of

+ I)

4e I

md

&J

mAt_ (a + I)_/ &

m4t

If

a

t

t

ime

(

n

+1)A

t w

e

m

ak

e t

he d

et

e

rm

i

n

a

tion,

st

a

t

e 2 ,

t

hen

for

all

s

ub

se

q

uent eve

n

ts we m

a

y

no lo

nger

a

ssign t

o

t

h

e

a

t

o

m

t

he e

lg

en

-fu

nc

tion

(18], bu

t

on

e

d

er

iv

e

d

from

(18)

if we re

-

plac

e

t w

ith

t

-

(

n

+1)A

t

. I

f, i

n con

t

r

ast

, we d

e

t

ermi

n

e

stat

e

I

, t

hen

from

then on

we

mu

st ass

ign

t

o

t

h

e

a

t

om

t

he

elgen-

f

unc

ti

on

Thus, in

a se

r

ies

of

I

n

te

rv

a

ls &t

we

would

first

observe

"

st

a

te

2 e, then continuously estate 1. e To hake a differentiation of

the two states possible, At must not fall below h/AE. Thus, the

23


26/35

tr

a

nsition-poin

t

in

t

ime can be

d

etermi

n

ed with th

a

t precision.

W

e conc

eive

o

f t

he

e

x

periment a

bov

e

en

tirely i

n

t

h

e s

e

nse

o

f

the old _nterpretation of qu

a

ntum t

h

eory, as

ex

pl

a

ined by

Pl

a

nck, Einstein

a

nd Bohr when we speak of a discontinuous

c

h

an

ge

o

f e

n

ergy

.

Since su

ch

an e

x

periment

c

an be

pe

rforme

d

,

in principle,

a

greement

a

s to its results must be possible,

I

n Bohr's b

a

si

c

post

u

l

a

te of the q

ua

ntum t

h

eor

y

,

t

he e

n

ergy

of an

a

tom, as we

ll a

s th

e valu

es

o

f the

ac

tion

var

iabl

e

s

J,

h

a

s the privilege over other it

e

ms to be

d

etermine

d

(such

a

s

the position of the ele

c

tron, et

c

.

)

th

a

t its numeric

a

l v

a

lue

can always be given. This privileged position held by energy

over

oth

e

r qu

a

ntum m

ec

hani

c

s m

a

gnitudes is ow

e

d strictly to

he circumstance th

a

t in

a

closed system

,

it represents

a

n

integral of the equ

a

tion of motion (for the energy m

a

tri

x

we

h

a

ve E

=

const.

)

. In contr

a

st

,

in open s

y

stems the energ

y

h

a

s no preference over other qu

a

ntum mech

a

nics v

a

ri

a

bles. In /

1

9

1

p

a

rticul

a

r

,

it will be possible to con

c

eive of e

x

periments

,

in which the

a

tom's ph

a

ses w

a

re precisel

y

me

a

sur

a

b

l

e

a

nd

for w

h

ich then the energy will rem

a

in

,

in principle, Indeter-

mi

n

e

d, cor

r

e

s

pondin

g

to a

re

la

ti

o

n Jw-wJ.-

:-

_s

-

i ,

o

r

J1

w

l

_ h. Suc

h a

n e

xp

eriment is provi

ded b

y res

o

n

a

nce

f

l

uorescence, for inst

an

ce. If

an a

tom is irr

a

diate

d

w

l

t

h an

etgen-frequency of say, v12 : (E 2 - E1)/h, then the atom will

vibrate in

ph

ase wlt

h

t

h

e e

x

terna

l

ra

d

iation, i

n

w

hlch ca

se

i

n p

ri

nc

i

pl

e

I

t

i

s

s

e

ns

el

e

ss t

o a

sk

,

i

n

w

hich

st

a

te - E

I o

r

E2

-

the

a

to

m i

s

v

lb

r

atlns. Th

e

phas

e re

lat

i

on b

e

t

we

en

at

o

m

and external radiation can be determined, for instance, by

means of the phase relations among many atoms (Woods experi-

me

n

t

). IF one does no

t

w

a

n

t t

o use

e

xp

er

i

me

n

t

s Involvin

g

ra

-

diation, the phase relation can also be measured by perf

o

rm-

lng precise position measurements In the sense oF J 1 For the

electr

on,

at

di

ff

e

r

en

t t

i

mes, r

el

at

lv

e t

o

t

he

p

h

a

se o

f t

he

ltsht

u

se

d

f

o

r

Illu

m

in

at

ion (

f

o

r m

an

y at

oms).

T

o e

a

ch

at

o

m

we could then assign a wave function such as

24


27/35

Z

ORIGINALPAGE

• OF POOR QUAL_P(

s(e. 0 --'=c,_, (J:,,,_); _ + I/T-- ,'7v,,(_,, _)e- ,, (l_) .

Here

c2 dep

e

nds on

t

h

e _

n

te

n

sity

and

B

on the pha

s

e of

t

he

illumi

na

tin

g

li

gh

t. T

hu

s, t

he

pr

obab

ilit

y

_

o

f

a ce

rt

a

i

n

p

o

sl-

ti

o

n is

s(q,o + (,-4),,,

The periodic *,erm in (20) can be experimentally separated

from the non-periodical, since the position determi_ _.ion can

be

p

e

rf

o

rm

ed a

t

d

iff

e

r

en

t

pha

s

e

s

o

f t

he

i

llu

mi

nat

i

ng l

ig

h

t

.

In a known imaginary experiment proposed by Bob,-, Lhe atoms of

a Stern-Gerlach atom beam are initially excited to resonance

fl

uo

r

e

s

c

en

ce, a

t

a ce

rt

a

i

n

l

oca

ti

on, b

y m

ean

s

o

f

l

ig

h

t irr

ad

i

a-

tion

.

After

a c

ert

a

in le

n

gt

h,

t

h

e

a

toms

pa

ss t

h

r

o

u

sh an

I

n

h

o

m

o

-

ge

n

e

o

us m

a

g

n

etic fiel

d;

t

h

e r

ad

i

at

i

on

emitte

d

b

y th

e

a

t

o

ms c

an

b

e

ob

ser

v

e

d o

ver the e

n

tire le

ng

t

h o

f t

h

eir

pa

th, bef

o

re

and

be

h

in

d

t

h

e m

a

g

n

etic fle

ld. B

ef

o

re t

h

e

a

t

o

ms e

n

ter t

h

e m

a

g

n

etic

f

i

e

ld

, t

h

ey e

xh

i

b

it

no

rm

al

res

o

n

an

ce flu

o

resce

n

ce, i.e.,

I

n

analo

gy t

o

t

he d

_s

pe

rs

lon

t

h

e

o

r

y,

we m

u

st

a

ss

u

m

e

t

h

at

all a

t

o

ms

emit i

n pha

s

e

wlt

h

t

he

i

nc

i

d

e

n

t

,

s

phe

r

ical l

ig

h

t w

ave

s.

A

t

f

i

rs

t, thl

s

lat

t

e

r i

nte

r

p

re

tat

i

o

n s

t

an

d

s

i

n

co

nf

l

i

c

t wlth w

ha

t

a rough application of the light quanta theory or the baslc /1_

rules of quantum theory indicate: from it one would conc].udo

that that only a few atoms would be ra sed to an upper state

by the absorption of a light quantum and hence, that _11 of

the resonance radiation would come from Intensively radiating

exci

te

d c

e

n

t

e

r

s. T

h

us, I

t

used

t

o be te

m

p

t

in

g t

o say:

t

h

e

con-

cept ot ltght quanta can be called upon here only for the

energy tmpulse balance; in reality all atoms radiate In lower

states as a weak and coherent spherical wave. Once the atoms

have passed through the magnetic field, there can hardly b_


28/35

any doubt left that the atom beam has split into two beams

of which one corresponds to atoms in the higher state and the

other, to atoms in the lower state. If the atoms in the lower

state were radiating, this would be a gross infringement of

the energy theorem, because all of the excitation energy is

contained in the fraction with the higher state. Rather, there

can be no doubt that behind the magnetic field, only the atom

beam with the upper states is emitting light - and non-coherent

light, at that - from the few intensively radiating atoms in

the upper state. As Bohr showed, this imaginary experiment makes

particularly clear how careful we must be with the application

of the concept stationary state . From the conception of the

quantum theory developed here, it is easy to discuss Bohr'S ex-

periment without any difficulty. In the outer radiation field

the phases of the atoms are determined and hence there is no

sense in talking of the energy of the atom. Even after the atom

has left the radiation field we can not say that it is in a

certain stationary state, if we are asking for coherence charac-

teristics of the radiation. But experiments can be performed to

test in which state the atom is; the result of this experiment

car only be given statistically. Such an experiment is actual-

ly performed by the inhomogeneous magnetic field. Behind the

magnetic field, the energies of the atoms are determined and

hence their phases are undetermined. The radiation is incoher-

ent and emitted only by atoms in the upper state. The magnetic

field determined the energies and hence destroys the phase re-

lations. Bohr's imaginary experiment provides a beautiful

clarification of the fact that the energy of the atom is also,

in reality, not a number, but a matrix. The law of conserva-

tion applies to the matrix energy and hence also to the value

of the energy, as precisely as it is measured, in each case.

Analytically, the cancellation of the phase relations can be

/

19--3

followed approximately thus: let Q be the coordinates of the

atom's center of mass; we can then assign to the atom (instead

of (19)) the eigen-function

26


29/35

OR,GINAL PAG_ ?_

• OF POOR QUALITY

s(Q,Os(q, t) -- s(_, _,0 ('_D

wher

e

S(

Q

,t) is a function that [as S(n,q) in (1611 is differ-

ent from zero in only a small area around a point in Q-space,

and propagates with the velocity of the atoms in the direction

of the beam. The probability of a relative amplitude q for

some values Q is given by the integral of

S(Q,q,t)S(O,q,t) over Q, i.e., via (20).

The eigen-function (21)

,

however, will change in the magnetic

field in a calculable manner, and because of the differing de-

flection of the atoms in the upper and the lower state, will

have become, behind the magnetic field,

S

(Q

,

_

,t) =

%s

,(0, t)

,

/,

,(,_;

,v)e

h

_=l£,t

-

_ _/i

-- ,

'_ S

,

(Q, t) ea (El

,

q

)

¢ 1

(22)

S1(Q,q,t) and S2(Q,t) will be functions in Q-space differing

from zero only in a small area surrounding the point. But this

point is different for S1_%nd for S 2. Hence SIS 2 is zero

e

very-

where. Hence, the probabilzty of a relative amplitude R and a

definite value 0 is

The periodic term in (201 has disappeared and with it, the pos-

sibility of measuring a phase relation. The result of the sta-

tistical position determination will always be the same, regard-

l

e

ss of the phas

e

of the incident light for which it was deter-

mined. We may assume that experiments with radiation whose theo-

ry has not yet been fully elaborated will yield the same re-

sults regarding the phase relations of atoms to the incident

light.

Finally, let us examine the relation between equation (2),

E1t I =h, and a problem complex discussed by Ehrenfest* and two '

ot

h

er

r

esea

r

chers b

y m

eans o

f B

o

h

r

's

co

rr

espo

n

de

n

ce p

ri

ncip

le,

2

7


30/35

i

in two important papers**. Eflrenfest and Tolman speak of weak

quantization when a quantifiea periodic motion is subdivided,

by quantum jumps or other disturbances, into time intervals

/

I__9_

that can not be considered long in relation to the system's

period. Supposedly, in this case there are not only the exact

energy values from quantum theory, but also - with a lower a

priori probability that can be qualitatively indicated - energy

values that do not differ too much from the quantum theory-based

values. In quantum mechanics, such a behavior is to be inter-

pretated as follows: since the energy is really changed, due to

other disturbances or to quantum jumps, each energy measurement

has to be performed in the interval between two disturbances,

if it is to be unequivocal. This provides an upper limit to t I

in the sense of § I. Thus the energy value Eo of a quantified

state is also measured only with a precision E I = t

/

t I. Here,

the question whether the system really adopts energy values

E that differ from Eo-with the correspondingly smaller statis-

tical weight - or whether their experimental determination is

due only to the uncertainty of the measurement, is pointless,

in principle. If t I is smaller than the system's period, then

there is no longer any sense in talking of discrete stationary

states or discrete energy values.

In a similar context, Ehrenfest and Breit (l.c.) point out the

following paradox: let us imagine a rotator - for instance

,

in

the shape of a gear wheel - fitted with a mechanism that after

f revolutions just reverses the direction of rotation. Let us

further assume that the gear wheel a

the actual content of quantum theoretical kinematics and mechanics (nasa)

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