The energy e in the stationary Schrodinger equation H\psi = E\psi does not change with time, and the Hamiltonian h does not contain time. This is the case with square potential well and harmonic oscillator selected in general textbooks.

But the general Schrodinger equation depends on time:

d

i hbar - \psi = H(t) \psi

Trembling insanity (abbreviation for Delirium Tremens)

This can describe the motion in any potential field, whether it is conservative or not. This is very common, for example, to describe the movement of electrons in a light field. Optical pulse is a time-dependent non-conservative force field. If the light field is weak, it can be treated by time-dependent perturbation, typically Fermi's golden law, which increases the perturbation term of the light field on the basis of the two-level system. The idea of perturbation is only an approximation, and its purpose is to get the wave function under unknown Hamiltonian by adding perturbation term according to the known solution. However, if the light field is very strong, the size of the interaction with the light field can be compared with the zero-order interaction (for example, the time-invariant binding potential of electrons), and there is a problem in perturbation treatment. At this time, the time-dependent Schrodinger equation is generally solved directly numerically.

So the conclusion is that Schrodinger equation holds in any force field and has nothing to do with energy conservation (time-dependent allowable energy change); Perturbation is only an approximate method to solve Schrodinger equation, which has limitations. The general Schrodinger equation can be solved directly without the help of perturbation theory.