Spectrum with Hamiltonian that has a periodic potential

Question

Hi

If I have a Hamiltonian \(H=-\frac{d^2}{d \theta^2}+ V\) where the potential is a periodic function with periodic \(2 \pi \) and I know the ground state solution and ground state eigenvalue. Is there any canonical way how I could get all the other eigenvalues and eigenfunctions? The only I am aware of would be to introduce ladder operators, right? But is it possible to construct these kind of operators for arbitrary Hamiltonians? Or maybe there is a better method available for this kind of Hamiltonian?

Answer 1

Check out Bloch's theorem which tells you how to handle periodic potentials. This is a standard problem in condensed matter physics where one is dealing with electrons moving in a lattice and hence any standard book on Condensed Matter Physics will have something on this. The ladder operator method will only work for special one-dimensional potentials of the form $ V(x) = W(x)^2 \pm  W'(x)$ that appear in the context of supersymmetric quantum mechanics. Right?

Comments

Bloch's theorem does not help much I think. It only helps for potentials that are periodic with respect to translation, this potential is rotational symmetric (therefore the 2 pi), hence you do not gain much from this. Your second argument looks interesting. If you have this W how do you get your ladder operator?

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If you have translational invariance in one-dimension with translational period $2\pi$ -- it is identical to being on a circle with circumference $2\pi$. Think about it! (There are many reviews on susy QM -- just do an internet search.)

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true, but how should it help me finding the ladder operators and spectrum? Bloch's theorem tells me then that I can split it up into \(e^{ik 2\pi}u(\theta), \) where u itself is 2pi periodic. So how does this help finding the full spectrum of a hamiltonian like \(H = -\frac{d^2}{d \theta^2}+ \sin(\theta)+sin^2(\theta) + \sin^{50}(\theta) \)?

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I agree with your statement about Bloch's theorem. Who said that there must exist ladder operators for your example? Can you give me a reference? I doubt if there is one for a generic potential.  Anyway, here is a paper that deals with ladder operators but I doubt if your example will show up there.

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@Lefschetz: Could you be more specific about the Hilbert space of your problem? It does make a difference whether the particle moves on an infinite line in a periodic potential, or whether it moves on a circle. (The latter restricts the boundary conditions of the wave function to \(ψ(θ+2π)=ψ(θ)\)while the former does not.)