DOC.
38
QUANTUM
THEORY OF RADIATION
225
PnBnm =
PmBnm.
We then
get,
as a
condition
of
dynamic equilibrium,
(6)
P
=
K
b:
em
-
enn
tn
(7)
kT
-
1
This is how the radiation
density depends
upon temperature according
to Planck's
law. It
now
follows
immediately
from Wien's
displacement
law that
necessarily
A
"
ZüL
=
av3
b:
(8)
and
em -
en =
hv,
(9)
where
a
and
h
are
universal constants. In
order to
obtain
the numerical value
of
constant
a
one
would have to have
an
exact
theory
of
electrodynamic
and
mechanical
processes.
For the time
being we
must
use
Rayleigh's limiting
case
of
high
temperatures,
for
which the
classical
theory applies
in the limit.
Equation
(9)
constitutes,
as
is
well
known,
the second
major
rule in Bohr's
theory
of
spectra,
and after its
completion by
Sommerfeld and
Epstein,
one
can
claim it
as
a
secured
part
of
our
science.
Implicitly,
it also contains the
photochemical
law
of
equivalence,
as
I
have shown.
[11]
[12]
§4.
Methods of
Calculating
the Motion
of
Molecules in
a
Field
of Radiation
We
now
turn to
the
investigation
of the motion which
our
molecules
execute under
the influence of radiation. In
doing
this
we use a
method which is well known from
[p. 54]
the
theory
of Brownian
movement,
and which
I
have used
repeatedly
for
calculations
of
movements
in
a
domain of radiation. In order
to
simplify
the
calculation,
we
carry
[13]
it
out
only
for the
case
where
movement
occurs
just
in
one
direction, i.e.,
in the
X-
direction
of
the
coordinate
system.
We also confine ourselves
to the calculation of the
mean
value of the kinetic
energy
of
the translational
motion; i.e.,
we
skip
the
proof
that the velocities
v
are
distributed
according
to
Maxwell's law. Let the
mass
M of
the molecule be
sufficiently large
that
higher powers
of
v/c
can
be
neglected against
the lower
ones.
The molecule
can
then
be
treated with
ordinary
mechanics.
Furthermore,
we
can
calculate without
any
real loss of
generality, as
if
the states
m