The answer is, of course, yes!
Following Bohr, we can start from a simple assumption, that electron is both a wave and a particle, and derive a formula for Hydrogen atom electron energy levels involving lots of goodness: Plank's constant, vacuum permittivity, elementary charge, and pi along the way.
Now let's start:
When n=1, we have the first energy level, E=-13.6 eV, which is just the observed. At higher energy levels, we have a inverse proportional relation between energy and n squared, where n is the level number, with n=1 being the lowest level closest to the nucleus.
The energy of trapped electrons are lower than free electrons, because it requires work to overcome the attraction of the nucleus and become free. As n tends to infinity, E becomes 0, meaning that free electrons by themselves have zero energy, which is intuitive. Bound electrons enter a negative potential well.
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