Our expression for the decay rate of an initial state
into some particular final state
Taking out the harmonic time dependence (to the delta function) as before, we have the
matrix element of the perturbing potential.
We must sum (or integrate) over final states. The states are distinguishable so we add the decay rates, not the amplitudes. We will integrate over photon energies and directions, with the aid of the delta function. We will sum over photon polarizations. We will sum over the final atomic states when that is applicable. All of this is quite doable. Our first step is to understand the number of states of photons as Plank (and even Rayleigh) did to get the Black Body formulas.
Jim Branson 2013-04-22