Question

Decay of excited electron from Schrödinger equation?

Answer 1

Schrödinger equation for an atom which is not coupled to a quantized electromagnetic field (or to a time dependent periodic external field) will not show decay processes (as the energy eigenstates are by definition, up to the phase factor, time independent states).

Answer 2

I'm currently studying for a quantum mechanics exam. I found some old exams which has the following question I'm trying to solve. Please tell me if my reasoning is wrong.

Can you see from the Schrödinger equation why an excited electron decays? If yes, explain how. If no, explain why not.

I find this question a bit annoying because there are multiple interpretations possible. The answer could be as simple as saying "Yes because the time-dependent Schrödinger equation allows for the wave function of the electron to change in time and the ground state is energetically much more favorable than an excited state".

But it could also be "no" since the derivation we saw for spontaneous emission (Quantum Mechanics by Bransden & Joachain, section 11.3, p. 527) is based on a thermodynamical argument by Einstein. But that derivation deals more with the how and the how fast rather than why.

I think the answer is "yes" but feel like there's more to it than the simple explanation that I gave. Moreover, the derivation of the decay rates of stimulated emission (which implies an excited electron falling back to the ground state) in the textbook is based on perturbation theory to find the decay and absorption rates of electrons in the presence of an external electromagnetic field. Maybe the actual meaning of the question was "show me how the Schrödinger equation is used to obtain time-dependent perturbation theory"?

Or maybe it's something entirely different.