What physical reason is there for a mass on a spring to have discrete energy levels? And why are those energy levels equally spaced, i.e. why is $E \ \alpha \ f$?
Personal background and clarification:
I'm a 2nd year college student. I've taken a QM class, so I know why things like "particle-in-a-box" are quantized (or at least, I can accept that they are).
Now I'm in thermo, and we keep talking about oscillators having discrete, equally spaced energy levels. From the comments, I gather that there's no classical explanation for a mass on a spring to have discrete energy levels.
My question then becomes, starting from "QM is right", how do we get to "oscillators have discrete, equally spaced energy levels? Might it have to do with how the potential well for an oscillator is quadratic (as opposed to that of a particle in a box, which is square)?