Question

Why does a quantum harmonic oscillator have a discrete energy spectrum, and why does a free particle have a continuum?

Answer 1

I think it mostly has to do with boundary conditions - in the harmonic oscillator your wavefunction is defined explicitly by boundary conditions, making it able to only take certain discrete energy values so as to fit them. A free particle has *no boundary conditions placed on it, so its energy is not defined by either the geometry of its enclosure (it has none) or the value of some potential field its passing through (there also is none). Does that help?

Answer 2

Which boundary conditions are we asking for the QAO and how do they relate to the potential?

Thinking about an arbitrary potential, we can define areas where as the total energy of the system is below 0 where we would have discrete eigenvalues, then again if we have other areas where the energy can get positive values, our system there behaves as a free particle.

I want to connect these two notions.