D.42 Derivation of the Einstein A coefficients
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
where is the ambient electromagnetic field energy
density, the frequency of the photon emitted in a
transition from the high to the low energy state, and the and
values are constants. This assumption agrees with the expression
(7.46) given in chapter 7.8.
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
Equating this to Planck’s blackbody spectrum as derived in chapter
6.8 (6.11) gives
The atoms can be modeled as distinguishable particles. Therefore the
ratio can be found from the
Maxwell-Boltzmann formula of chapter 6.14; that gives the
ratio as , or
in terms of the photon frequency. It then
follows that for the two expressions for to be equal,
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