DOC.
38
QUANTUM
THEORY OF RADIATION
227
(12)
must
satisfy
the
equation identically
if
p
is
expressed as a
function
of
v
and
T
[16]
from Planck's
equation
(4).
§5.
Calculation of
R
Let
a
molecule of
the kind under consideration be
moving along
the
X-axis
of the
coordinate
system
K with the uniform
velocity
v.
We ask for the
mean
value of
momentum
per
time
unit,
which is transferred from the radiation
to
the molecule. In
order
to
be able
to
calculate
this,
we
have
to
judge
the
radiation
from
a
coordinate
system
K' that is
at rest
relative
to
the molecule. For
we
formulated
our hypotheses
about emission and
absorption only
for molecules
at rest.
The transformation
to
the
system
K' has been done several times in the
literature,
with
particular accuracy
in
Mosengeil's
Berlin dissertation.
However,
for the sake of
completeness
I
will
repeat
[17]
here these
simple
considerations.
The radiation is
isotropical
relative
to
K,
i.e.,
the radiation
of
frequency range
dv
[p. 56]
per
unit
volume attributed
to
an
infinitesimal solid
angle
dk about the direction of
the
beam
is
pdvdk/4x,
(13)
where
p
depends only upon
v
but
not upon
the
direction. To
this
particular
radiation
corresponds,
relative
to
the coordinate
system
K',
another
particular
radiation that is
also characterized
by
a
frequency range
dv'
and
a
solid
angle
dk'.
The volume
density
of this
particular
radiation is
p'(v'Q')
(13')
This defines p'. It is
direction-dependent
and
commonly
defined
by
an
angle
Q'
with the X'-axis
and
an angle
W'
between the
Y'-Z'-projection
and the Y'-axis. These
angles correspond
to
the
angles
Q
and
W
which
fix,
in
an analogous
manner,
the
direction
of
dk relative
to K.
Next,
it
is
clear that the
same
law
of
transformation
must obtain between
(13)
and
(13')
as
it does for the
squares
of the
amplitudes
A2
and A'2 of
a plane wave
of
corresponding
direction. In the
approximation we
want,
we
therefore have
p'(v',0')dv'dK'
p(v)dvdk
=
1
-
2
v-c
cos
Q
c
(14)
or