Electron emitting a photon

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fluidistic
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From my understanding, when solving the hydrogen atom (getting the wavefunction of the electron) in introductory quantum mechanics, we assume that the Hamiltonian doesn't change with time (because in this case the solution of the S. equation is separable and therefore "easy" to obtain). In other words, we assume that the system proton+electron is stable so that we dismiss any emission of the electron. It means we do not seek to explain using the Schrödinger's equation how an electron emits and do its transition of "orbit".
I would to know if there is some mathematical model that exactly explain what happens to the electron when it get "hit by a photon" or when the atom is in excited unstable state and then emits a photon and comes back to the ground state.

When a charge is accelerated it must emit EM waves according to classical mechanics. Is there a similar description in quantum mechanics? If so, where can I read more about this?
 

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fluidistic
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Thank you very much.
 

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