Allowed energy levels of electrons

[Jimmy87]

What is the fundamental reason why the allowed energies of electrons are quantized? I did some research on the web before posting this and what I found seems to link this to the wave-like behaviour of electrons. That is, if electrons could hypothetically exist within the band gaps of the atom, then the electron wave would effectively cancel itself out (like standing waves which effectively give no waves between the various harmonics). Therefore, as it would cancel itself out it cannot exist there. Is this an accepted explanation? I thought that the allowed energy states also depend on the values of the various quantum numbers of the electrons?

 

[hilbert2]

When you solve the Schrödinger equation of a quantum system, you initially get a continuous spectrum of states as a solution. However, one has to impose boundary conditions, for example, the electron wavefunction can't grow without bound when one goes arbitrarily far from the nucleus of a hydrogen atom. The boundary conditions restrict the allowed spectrum of electronic states to a discrete (quantized) spectrum.