EARLY WORK

ON QUANTUM

HYPOTHESIS

137

is

a new

constant

(later

called

Planck's

constant).

To derive this

formula,[21]

Planck

cal-

culated the

entropy

of

the

oscillators,

using

what Einstein later called

"the Boltzmann

principle":[22]

S

=

k.log W,

where

S is

the

entropy

of

a macroscopic

state

of

the

system,

the

probability

of

which

is W.

Following

Boltzmann,

Planck took

W proportional

to the

number

of

"complexions,"

or possible microconfigurations

of

the

system corresponding

to

its

state. He calculated this number

by

dividing

the total

energy

of

the state into

a

finite

number

of

elements of

equal magnitude,

and

counting

the

number of

possible

ways

of

distributing

these

energy

elements

among

the individual oscillators.

If

the size

of

the

en-

ergy

elements

is

set

equal

to

hv,

where

v

is

the

frequency

of

the

oscillators,

an expression

for the

entropy

of

an

oscillator results that leads to

eq. (2).

Although

Einstein

expressed misgivings

about

Planck's

approach

in

1901,[23]

he did

not

mention Planck

or black-body

radiation

in

his

papers

until

1904.[24] A

study

of

the

foun-

dations

of

statistical

physics,

which he undertook between

1902

and

1904,

provided

Ein-

stein with the tools he needed to

analyze

Planck's

derivation and to

explore

its

conse-

quences.[25]

At least three elements

of Einstein's

"general

molecular

theory

of

heat"

("allgemeine

molekulare Theorie der

Wärme")[26] were

central to his

subsequent

work

on

the

quantum

hypothesis:

the introduction

of

the canonical

ensemble;

the

interpretation

of

probability

in

Boltzmann's

principle;

and the

study

of

energy

fluctuations in thermal

equi-

librium.

In

an analysis

of

the canonical

ensemble,

Einstein

proved

that the

equipartition

theorem

holds for

any system

in

thermal

equilibrium.[27]

In

1905

he showed that, when

applied

to

an

ensemble

of

charged

harmonic oscillators

in

equilbrium

with thermal

radiation,

the

equipartition

theorem

leads,

via

eq. (1),

to

a black-body

distribution

law

now

known

as

the

Rayleigh-Jeans law:[28]

877V2

|rT,

Pv

=

--

kT.

(3)

Despite

its

rigorous

foundation in classical

physics, eq. (3) agrees

with the observed

en–

[21]

See

Planck

1900e,

and 1901a.

[22]

Einstein

1905i

(Doc. 14),

p.

140.

[23]

See Einstein to Mileva

Maric,

4

April

1901 (Vol.

1,

Doc.

96).

In

explanation

of

his

misgivings,

he cited his

objection

to

Planck's

use

of

one type

of

oscillator,

with fixed

period

and

damping

constant

(see

Einstein to

Maric,

10

April

1901,

Vol.

1,

Doc.

97).

[24]

See Einstein 1904

(Doc. 5), p.

354.

[25]

See

Einstein

1902b

(Doc. 3),

Einstein

1903

(Doc. 4),

and Einstein 1904

(Doc. 5).

For

a

discussion

of

this

work,

see

the editorial

note,

"Einstein

on

the Foundations

of

Statistical

Physics,"

pp.

41-55.

[26]

See the title

of

Einstein 1904

(Doc. 5).

[27]

See Einstein

1902b,

pp.

427-428.

[28]

See Einstein 1905i

(Doc. 14),

§

1.

Ein-

stein rederived

this

result in Einstein 1907a

(Doc.

38),

and Einstein 1909b

(Doc. 56).

An

expression

equivalent

to

eq.

(3)

was

obtained in

1905

by Rayleigh

and Jeans

by

other methods

(see

Rayleigh

1905a, 1905b,

and

Jeans

1905a,

which

appeared

after the

receipt

of Einstein's

paper).

In

1900

Rayleigh

obtained

the

propor-

tionality

between T and the

energy

of

a

vibration

mode

by applying

the

equipartition

theorem

to

the normal modes

of

the radiation field

(Ray-

leigh

1900).