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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 Schrodinger 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 Schrodinger 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 lets say you solved the Schrodinger 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!