DOC.

5

CONTRIBUTIONS TO

QUANTUM

THEORY

25

energy

value associated

with Z

is

e*.3

Equation 4a) expresses

Boltzmann's

principle

in the formulation

of Boltzmann-Planck.

Up

to now,

we

only

considered

changes

of

state under

constant

X.

The

question

hence arises whether

or

not

4a)

remains valid under

changes

of

state in the

system

when

X changes.

This

question

cannot

be

answered without

making special

hypotheses.

The

most

natural

hypothesis

which offers itself

is

Ehrenfest's

adiabatic

[18]

hypothesis,

which

can

be formulated thus:

With reversible adiabatic

changes

of

X

every quantum-theoretically possible

state

changes

over

into another

possible

state.

It is

a

consequence

of

this

hypothesis

that the number Z of

quantum-theoretically

possible

realizations does not

change during

adiabatic

processes.

Since the

same

is

[19]

true for

S, we

have to conclude from

Ehrenfest's

adiabatic

hypothesis (which

is

a

natural

generalization

of

Wien's

displacement law)

that the

Boltzmann

principle

in

[p.

827]

formula

4a)

has

general validity.

The

entropy

of

a

system

has, therefore,

for all

[20]

(thermodynamically defined)

states

of

a

system-provided

they are quantum-

theoretically

realizable in the

same

number

of

ways-the

same

value.

We ask ourselves

now

if

we

can

deduce

some expectation

with

respect

to

the

range

of

validity

of Nernst's

theorem. Let there be

a

physical system

at

absolute

zero

in two

thermodynamically

defined states

A1

and

A2.

We

can compare

the

entropy

values of these states if

we

can

find

the number Z

of

quantum-theoretically

possible

realizations

of

the

system.

We

can

consider the state

of

the

system

at

absolute

zero

as

quantum-theoretically

and

molecular-theoretically

(i.e.,

in its

micro-state)

as completely

described if the

positions

of

the

centers

of

gravity

of

the

individual

atoms

which constitute the

system

are

given

(the

atoms

imagined are

to

be

numbered).

Z then is the number which

says

how

many

of these micro-states

are possible

without the

system leaving

its

thermodynamically

defined state.

If

all

phases

of

the

system

are

chemically

homogeneous

and

crystallized

in

spatial

lattices such that it is determined in which

positions

the atoms

of

various kinds

are

situated,

then

I

can change

from

one

micro-state

to

another

one,

contained in

Z,

only

in

that

manner

that

I

exchange positions

of

atoms

of the

same

kind. States caused

by

the

exchanges

of

atoms

of

different kinds

are,

in

contrast,

not to

be counted.

If

the

system comprises

of

n1

molecules of the first

kind,

n2

of the

second, etc.,

Z has the

value

Z

=

nx\ n2\

...

From this

follows,

under consideration

of

4a),

that the

entropy

in all these states

has the

same

value.

Therefore,

the

validity

of

Nernst's

theorem

in

Planck's

3This

is

equivalent

to

a

transition from

a

"canonical" to

a

"micro-canonical" ensemble.