224
DOC.
38
QUANTUM
THEORY OF RADIATION
a
momentum
(em - en)/c
in the direction in which
the beam
propagates.
The
transferred momentum has the
same magnitude
but
opposite
direction
if
the
process
of
induced radiation effects
a
transition
Zm
-
Zn.
In
case
the
resonator is simulta-
neously exposed
to several
beams,
we assume
that the entire
energy
em -
en
of
an
elementary process
is taken from
(or
given to) only one
beam
of
rays; thus,
the
transfer
of
the
momentum
(em
-
en)/c
upon
the molecule
occurs
also in this
case.
If
the
loss
of
energy occurs through spontaneous
emission,
the Planck
resonator
does, as
a
whole,
not receive
a
transfer of
momentum because under the classical
theory
the emission
of
radiation
occurs
in the
form
of
a spherical wave.
But
we
noted
already
that
we can only
obtain
a quantum theory
that
is
free of contradictions if
we
assume
that
the
process
of
emission
of
radiation is also
a
directed
one.
Then
every process
of
emission of radiation
(Zm
-
Zn)
transfers
a
momentum
of
magni-
tude
(em en)/c
upon
the molecule.
For
an
isotropic
molecule
we
have to
assume
equal
probability
for all directions
of
emission. We arrive
at
the
same
statement
for
nonisotropic
molecules
if
the orientation
changes
over
time
according
to
the laws of
chance.
By
the
way,
such
assumption
also has
to
be made for the statistical laws
(B)
and
(B')
of induced
radiation;
otherwise the
constants
Bmn
and
Bnm
could become
dependent upon direction,
which
can
be avoided
by assuming isotropy or pseudo-
isotropy
of the molecule
(by
forming mean
values
over time).
§3.
Derivation of Planck's
Law
of
Radiation
[p. 53]
We
now
ask which effective radiation
density
p
must
prevail
in order to
ensure
that
the
energy exchange
between radiation and
molecules,
according
to the statistical
laws
(A), (B),
and
(B'),
does
not
disturb the distribution
of
molecules
given by
equation
(5).
For
this,
it is
necessary
and sufficient
that,
per
time
unit,
on
average
as
many elementary processes
of
type (B) occur
as
of
types (A)
and
(B')
together.
This
condition
provides-due
to
(5),
(A),
(B),
and
(B')
for the
elementary processes
that
correspond
to the index combination
(m,n)-the
following equation:
pne
kTBnmp
=pme
kT
(KP
+
Amn).
If
furthermore
p
shall
go
to
infinity
with
increasing
T-and
we
shall
assume
this
to
be the
case-then
the constants
Bmn
and
Bnm
must
satisfy
the relation