Electrons move to higher energy levels and drop back (electron migration). When they drop levels, photons are emitted and the wavelength/frequency emitted is based on how many levels are dropped. My question is, how does this work with elements with only one shell, such as hydrogen? Also, what determines how many energy levels are dropped?
Actually, hydrogen can have as many electron levels as you want. However, they only exist at specific energy levels, with the potential energy given as:
Where $R_h$ is the Rydberg constant, and $n$ is any integer. When electrons are exited to a higher energy lever, they will fall back after some time and emit photons (not protons, by the way). So how many levels do they drop? Well, in the case of hydrogen, it will eventually go back to the basic energy level, but this is not guaranteed to happen in one jump. This may or may not take multiple jumps, that depends on probability. For a simple atom like hydrogen, those probabilities may be possible to calculate analytically. For more complex atoms however, this becomes an extremely difficult task, usually approached by simulations.