I've been thinking recently about Fermi's Golden Rule. Sparing the details of how to get Fermi's golden rule (you'll find like five derivations if you google it), the final result is, $$\frac{\partial P}{\partial t}=\frac{2\pi}{\hbar}|\langle f|H'|i\rangle|^2\rho(E_f)$$which says that the probability per unit time, to first order in the perturbation from $H_1$, is a constant. Hence, this quantity tells you the transition rate. Notice, however, that it depends on the density of final states. Notice, if I'm not mistaken, that this is not for a specific state, but this is more accurately the density of all states above the original states (let's say we're in the ground state and ignore stimulated emission). I understand how to calculate the density of states for a classical, and hence continuous system. It's just an integral in phase-space.
Things become more complicated in a molecule, however, because the states are quantized. So, how do I go about doing this calculation of the density of discrete states? A really great answer would sketch out how one might do this calculation for a quantum harmonic oscillator.
I've seen a couple things online which do this kind of calculation for the motion of a particle in an infinite potential well on a solid. You basically just use the relationship between energy and the wavenumber, which is easy because you know the solutions to the Schrodinger equation for all states ahead of time.