Degeneracy in Ring Hamiltonian

Classical Mechanics Level 3

A particle constrained to move on a ring can be described by one parameter, the angle $\phi$ on the ring with respect to some coordinates. In quantum mechanics, an otherwise free particle constrained to move on this ring has the Hamiltonian

$\hat{H} = -\frac{d^2}{d\phi^2}.$

Which of the following correctly describes the spectrum of this Hamiltonian?

There is no degeneracy in the spectrum. Every eigenvalue but one has a degeneracy of order 2. Every eigenvalue has a degeneracy of order 2. Exactly one eigenvalue has a degeneracy, which is of order 2.

1 solution

Matt DeCross
May 10, 2016

The eigenfunctions of this Hamiltonian are the sines and cosines; at each wavenumber $k$ corresponding to each energy eigenvalue there exist a sine and cosine solution, so the degeneracy is two. However, there also exists exactly one zero energy solution $\psi_0 = \frac{1}{\sqrt{2\pi}}$ , which has no degeneracy.

Degeneracy in Ring Hamiltonian

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