Einstein did not really derive the spontaneous emission rate from relativistic quantum mechanics. That did not exist at the time. Instead Einstein used a dirty trick; he peeked at the solution.
To see how, consider a system of identical atoms that can be in a low
energy state or in an excited energy state
. The fraction of atoms in the low energy state
is and the fraction in the excited energy state is
. Einstein assumed that the fraction
of excited atoms would evolve according to the equation
Then Einstein demanded that in an equilibrium situation, in which
is independent of time, the formula must agree with
Planck’s formula for the blackbody electromagnetic radiation
energy. The equilibrium version of the formula above gives the energy
density as
That must equal is a consequence of the symmetry property mentioned at the end of chapter 7.7.2. But it was not self-evident when Einstein wrote the paper; Einstein really invented stimulated emission here.
The valuable result for this book is the formula for the spontaneous emission rate . With given by (7.47), it determines the spontaneous emission rate. So it has been obtained without using relativistic quantum mechanics. (Or at least not explicitly; there simply are no nonrelativistic photons.)