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.