Photon absorption and probability of atomic electron transition

[PhysicsAm55]

I have only taken one very intro quantum mechanics class and the furthest we got was solving the Schrödinger equation for the Coulumb potential of the hydrogen atom.
So we solved for the allowed energy levels and the wavefunction for the hydrogen atom. We also learned how using the wavefunction, the electron orbitals can be constructed as probability "clouds" around the nucleus.

Now the question I have is how does one deal with this hydrogen atom absorbing a photon in terms of the Schrödinger equation?

I was taught in modern physics that when an atom absorbs a photon, its energy is increased and the electron moves up to a higher energy level. But is it even correct to say that the atom absorbs the photon? I know some sources state and my professor stated a few times that the photon 'bumps' the electron up to a higher energy level but I assumed that this was a simplified and wrong way to look at the situation since it is treated like a classical collision (the photon actually colliding with the electron). I just wanted to mention this and ask here if I was right to dismiss this classical picture of the situation?

To try to answer my own question: I guess when the photon comes close enough to the hydrogen atom you must resolve the Schrödinger equation with a different potential energy function. This is different from anything I have learned because the potential would now depend on time so I would not know where to begin here, but let's say you solved the Schrödinger equation and find the wavefunction. Do you use this wavefunction to compute the probability of the photon being absorbed? and for the electron making an atomic transition? If so how would you go about this? Lastly how would you find the probability of Auger electrons after the excited electron returns to its ground state.

I thank you for your time and eagerly await your response!

 

[PhysicsAm55]

Does anyone have any recommended resources that could answer my questions?

 

[f95toli]

There is no simple answer to your question.
Yes, it is possible to calculate all those things, and yes the probability amplitude for a transition depends on the overlap of the wavefunctions of the orbitals involved in the transitions.
Look up e.g. Fermi's Golden Rule.

However, the details are quite complicated and require both a working knowledge of "standard" QM at the level of e.g. Sakurai, and some knowledge about quantum optics.
The best reference I can think of is Cohen-Tannoudji's book on atom-photon interactions.

 

[Bob_for_short]

The electron-photon interaction term (always present in nature) is first omitted in the Schroedinger equation. This gives the atomic wave functions (non perturbed). Then the interaction term is taken into account perturbatively. Often one uses the so called dipole approximation, so the matrix elements contain simply r instead of exact interaction potential.