Bound states of a periodic potential well in one dimension

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SUMMARY

The discussion focuses on the bound states of a one-dimensional periodic potential well defined by V(x) = -A*(cos(w*x)-1). The primary conclusion is that while the wave-functions for bound energy eigenstates in such potentials may exhibit periodic characteristics, they do not correspond to true bound states due to the nature of the potential. The discussion emphasizes that for finite potentials, the Schrödinger equation does not yield bound states, and suggests starting with piece-wise constant potentials for a more manageable analysis.

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Chuckstabler
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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?
 
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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.
 

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