You could derive the spontaneous emission rate (Equation 9.56) with-out the detour throu...

You could derive the spontaneous emission rate (Equation 9.56) with-out the detour through Einstein's A and B coefficients if you-knew the ground-stale energy density of the electromagnetic field, , for then it would simply be a as f stimulated emission (Equation 9.47). To do this honestly would require quantum electrodynamics, but if you are prepared to believe that the ground state consists of one photon in each mode, then the derivation is very simple:

(a) Replace Equation 5.111 by

(b) Use your result, together with Equation 9.47, to obtain the spontaneous emission rate. Compare Equation 9.56.