Suppose we have a system that has discrete energy levels (e.g. hydrogen atom, potential well) and the stationary solutions for the wave function are $\psi_n$. I would assume that there should be a way in which one can model the transition $2\rightarrow 1$ by using as initial condition the stationary state $\psi_2$ for the Schrodinger equation (temporal version). However I did not manage to find materials that cover this approach.
I assume that the Schrodinger eq. should not be changed (for a spontaneous transition there is no need of a photon to trigger it). From this, it should follow that a stationary wave function should evolve in another one of lower energy, or, if the transition is possible on multiple lower energy levels, I assume that the solution of the Schrodinger eq. is a superposition of the possible lower energy states.
So, in order to conclude the question, are there any materials on this approach that I did not manage to find? If so, I would appreciate suggestions on the topic.