Bound states of a periodic potential well in one dimension

[Chuckstabler]

Hi,

I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically would the wave-functions for these bound energy eigenstates be periodic. I chose this potential because it looks sort of like a harmonic oscillator's potential about x = 0. So would the lowest bound state look like the harmonic oscillators ground state except periodic? Or would it look like the harmonic oscillators ground state while dying off and not being periodic?

 

[DrClaude]

The answer is Bloch states.

 

[Misha]

Periodic and bound don't come together easily. if potential is finite, the respective Schrödinger equation doesn't have bound states.

The chosen potential is too complex for diving into periodic systems. It's better to start from piece-wise constant potentials. They're already quite rich.