D O C . 1 2 9 T H E O R Y O F R A D I A T I O N E Q U I L I B R I U M 1 2 7
(6)
holds between the number of molecules n in state Z and the number of molecules
n* in state Z*. In order that nothing change in this distribution by an exchange of
radiation of the kind envisaged, the condition
(7)
must be satisfied in accordance with (2) and (5).
From (6) and (7) it follows that
, (8)
that is, Planck’s radiation formula, insofar as the coefficients a and b always just
satisfy the relation
. (9)
§2. Extension of This Consideration to the Case That the Molecules Are Freely
Mobile. First, a comment that is useful for understanding what follows and can also
be found in Pauli’s work. For the derivation of §1 it is not essential that the mole-
cule be only capable of discrete states or energy values. For, if the state density is
a continuous function in phase space, we substitute a priori states Z, Z* by infinitely
small state domains of equal probability, between which a radiation transition that
preserves the relation is possible. Then equations of the given form
may hold, such as equations (2), (3), and (4). Because equation (6) also applies,
nothing essential changes in our consideration.
If, furthermore, the observed molecule is anisotropic with respect to the radia-
tion influence, then the elementary process being considered, or its probability, will
also depend on the molecule’s orientation and the direction and polarization of the
beam with which the molecule interacts in this elementary process. The consider-
ations of §1 will again apply to the elementary processes that are specialized ac-
cording to these aspects as well as to their “inverse.” One circumstance must still
be considered, however, that we did not need to take into account until now.
We are not permitted to regard the transition simply as the temporal
inversion of the process . In this case, the latter process would not only
have to emit the quantum in the opposite direction from the one at its absorption
during the former process, so both processes would not balance each other out with
respect to their influence on the statistical equilibrium. In certain cases, for ex., the
presence of a constant magnetic field and a hydrogen atom in the sense of Bohr’s
n*
n
----- - e
ε* ε –
κT
–--------------
e
–hν
κT
---------
= =
nbρ n*( a bρ) + =
ρ
a
b
-- -
eκT
hν
-------
1 –
---------------- -=
a
b
-- -
8πhν3
c3
--------------- =
ε* ε – hν =
[p. 304]
Z* Z →
Z Z* →