220

DOC.

8

ANALYSIS OF A RESONATOR'S MOTION

Doc. 8

Statistical

Investigation

of

a

Resonator's

Motion in

a

Radiation

Field

by

A.

Einstein and

L.

Hopf

[Annalen

der

Physik

33

(1910):

1105-1115]

§1.

Train

of

Thought

It

has

already

been

shown in

a variety

of

ways

and

it

is

now generally

accepted

that,

when

correctly applied

in

the

theory

of

radiation,

our

current views

on

the distribution

and

propagation

of

electromagnetic energy on

the

one

hand,

and

on

the

statistical

distribution

[1]

of

energy on

the other

hand,

can

lead

to

no

other but the

so-called

Rayleigh

(Jeans)

radiation

law. Since this law

is

in

complete

contradiction

with

experiment,

it

is

necessary

to

undertake

a

modification

of

the

foundations of

the

theories

used for its

derivation;

and

it has

often been

suspected

that the

application

of the

statistical

energy

distribution

laws

[2]

to

radiation

or

to

rapidly oscillating

motions

(resonators)

is not flawless.

The

following

investigation

shall

now

show

that

such

a

dubious

application

is not

required

at

all,

and

that

it suffices to

apply

the

equipartition

theorem for

energy solely

to

the

translational

motion of the

molecules and oscillators in

order

to

arrive at

the

Rayleigh

radiation

law.

The

applicability

of the

law to

translatory

motion

has

been

adequately proved

by

the

successes

of the kinetic

theory

of

gases;

we

may

therefore

conclude

that

only a

more

radical

and

more

profound

change

in

our

fundamental

conceptions

can

lead

to

a

law

of

[3]

radiation that

is

in

better

agreement

with

experiment.

We consider

a

mobile

electromagnetic

oscillator1

that

is, on

the

one

hand,

subjected

to

the

effects

of

a

radiation

field

and, on

the other

hand,

possesses a

mass

m

and enters

into interaction

with

the

molecules

present

in

the radiation-filled

space.

If the

above

interaction

were

the

only one

present,

then the

mean

square

value

of the

momentum

associated with

the oscillator's

translatory

motion

would

be

completely

determined

by

statistical mechanics.

In

our case

there

also exists

the interaction of the oscillator

with

the radiation

field.

For

a

statistical

equilibrium

to be

possible,

this

latter interaction

must

not

produce

any

change

in

that

mean

value.

In other

words:

The

mean

square

value

of

the

momentum associated with

the

translatory

motion that the oscillator

assumes

under

the influence of

the radiation alone must

be the

same as

that

which it would

assume,

in

accordance with

statistical

mechanics,

under the

mechanical influence

of the

molecules

alone. This

reduces the

problem

to the task of

determining

the

mean

square

value

(mv)2

of the

momentum

assumed

by

the oscillator under the

sole influence

of the radiation

[4]

field.

This

mean

value must be

the

same

at time

t

=

0

as

at time

t

=

t, so

that

we

have

1

For the

sake

of

simplicity we

will

assume

that the

oscillator oscillates

only

in

the

z-direction and

moves only

in

the

x-direction.