The A coefficient corresponds to spontaneous transitions m —» n, which occur with
a probability that is independent of the spectral density p of the radiation present.
The B terms refer to induced emission and absorption. In Eqs. 21.7 and 21.8, p
is a function of v and T, where 'we shall assume that a molecule can go from the
state En to the state Em by absorption of radiation with a definite frequency v, and
[similarly] for emission' [E9]. Microscopic reversibility implies that dWmri =
dWnm. Using Eq. 21.6, we therefore havewhere pm is a weight factor. Consider a pair of levels Em, Ea, Em > Ea. Einstein's
new hypothesis is that the total number dW of transitions in the gas per time
interval dt is given by(21.7)
(21.8)(21.6)
400 THE QUANTUM THEORYDenote by Em the energy levels of a molecule and by Nm the equilibrium number
of molecules in the level Em. Then(Note that the second term on the right-hand side corresponds to induced emission.
Thus, if there were no induced emission we would obtain Wien's law.) Einstein
remarked that 'the constants A and B could be computed directly if we were to
possess an electrodynamics and mechanics modified in the sense of the quantum
hypothesis' [E9]. That, of course, was not yet the case. He therefore continued his
argument in the following way. For fixed Em — £„ and T -* oo, we should get
the Rayleigh-Einstein-Jeans law (Eq. 19.17). This implies that(21.9)
(21.10)
(21.11)
whencewhere «„„, = A^/B^. Then he concluded his derivation by appealing to the uni-
versality of p and to Wien's displacement law, Eq. 19.4: 'a^, and Em — Ea cannot
depend on particular properties of the molecule but only on the active frequency
v, as follows from the fact that p must be a universal function of v and T. Further,
it follows from Wien's displacement law that «„„, and Em — En are proportional
to the third and first powers of v, respectively. Thus one has
(21.12)where h denotes a constant' [E9].
The content of Eq. 21.12 is far more profound than a definition of the symbol