Atom and Light interaction I want to qualitatively get a picture of what happens when a hydrogen atom interacts with electromagnetic waves.
If we consider that the hydrogen atom is in state $\phi_{100}$ (ground state) then the interaction can excite the atom to a higher state(let's say $\phi_{210}$) depending on the frequency so that the state after interaction could be described as an equal superposition:
$$|\psi\rangle=\frac{1}{\sqrt2}(\phi_{100}+\phi_{210})$$
Firstly, is the above a correct qualitative description?
Secondly, what would happen if light now interacts with the hydrogen atom in the superposition state $|\psi\rangle$ ?
My guess is the following: If the hbar frequency of the incoming light matches with the energy difference  between any higher energy level and any of the states $\phi_{100}$ or $\phi_{210}$
the atom would go in the superposition of these three states.
For instance, if $E_{\phi_{300}} - E_{\phi_{100}}=h\nu$ then the light which has 50% probabilty of finding the particle in state $\phi_{100}$ can excite the atom to $\phi_{300}$ so that effectively now we have a perturbed atom state as:
$$|\psi'\rangle=\frac{1}{\sqrt3}(\phi_{100}+\phi_{210}+\phi_{300})$$
Is all the above even close to correct or just baloney?
If baloney, how can one understand the interaction without much math?
 A: I would like to add to the @Hans Wurst answer the following:
There is also (depending on the frequency OR the intensity of the light) a probability of ionization: the electron is no longer bound to the atomic potential and moves freely in space (a spreading wavepacket). Here you have an example in 1D in were the intensity of the light is so strong that it produces ionization of part of the probability density:

As you can see there is probability density still remaining in the bound state $|\psi_{0}(x)|$. There are also probability density in its bound states but it's inappreciable.

A: Starting with a hydrogen atom in its ground state and a low-intensity monochromatic EM wave with $h\nu=E_2-E_1$, we have a driven two-level system, where one of the levels is the ground state $|100\rangle$, and another depends on the polarization of the incident EM wave.
Suppose the incident wave is linearly polarized, so that we have the target state with $n=2,$ $l=1,$ $m=0.$ Then this system will undergo Rabi oscillations with the state periodically becoming a superposition of $|100\rangle$ and $|210\rangle$ with growing relative contribution of $|210\rangle$, then a pure $|210\rangle$, then a superposition again with diminishing contribution of $|210\rangle$, then $|100\rangle$, and back to superposition repeating the cycle, and so on.
Your example of $(|100\rangle+|210\rangle)/\sqrt2$ is a very special case that might be crossed in the middle of the transition between the extremes of the Rabi cycle. But, depending on the initial conditions like phase of the incident light wave, you could instead get e.g. $(|100\rangle-|210\rangle)/\sqrt2,$ or maybe $(|100\rangle+i|210\rangle)/\sqrt2,$ etc. I.e., there may (and likely will) be a phase difference between constituent states, so that a mere "+" sign won't suffice. Generally, aside from phases, there will also be scale coefficients that describe relative contribution of each state in the superposition.
In the second part of your question your expectation is qualitatively correct, but the same concern applies: mere two "+" signs will likely not suffice, you'll need to also introduce relative phases between the superposed states, and also weights that describe how much of each state is in the superposition. And of course, all these complex-valued coefficients will evolve in time, and in a more complicated way than in the two-level case.
A: In my opinion, if you want to consider the interaction of an atom with electromagnetic waves, then you must take into account photon states. The system of an atom and photons has the tensor product of spaces
$$
\cal{H}_a\otimes \cal{H}_p
$$
as its state space. To describe the process of excitation of an atom by photons, it is natural to consider the following linear combinations of basis vectors:
$$
A(t)|nlm\rangle\otimes|1\rangle + B(t)|n'l'm'\rangle\otimes|0\rangle + \cdots
$$
I am not sure that the linear combinations of the atom's eigenstates
$$
A(t)|nlm\rangle + B(t)|n'l'm'\rangle
$$
are directly related to the processes of photon emission and absorption.
A: Your understanding is partly correct partly incorrect.
First question. The light matter interaction that takes the ground state to a superposition is not only frequency dependent but time dependent - a $\pi/2$ pulse. This can be found using semiclassical electrical field theory and quantum perturbation theory or full quantum theory. The resulting superposition is described correctly.
If light interacts with the electron in superposition, multiple things can happen. A laser beam of the correct frequency (as described) could cause an excitation, this interaction forces the electron to take a one of the superposition states with even 50/50 probability, and excitement therefore will take place with a 50/50 probability - destroying the superposition state in both cases. But another a $\pi/2$ pulse would set the electron in the $\psi_{210}$ state for sure.
To perturb the superposition to another superposition state (of 3 basis states) is surely possible, but would require another round of perturbation calculation to find frequency and pulse length. Not trivial at all, and not without math.
A: Maybe page 117 from http://www.damtp.cam.ac.uk/user/tong/aqm/topics4.pdf may help (Rabi oscillations and resonances)?
