Try to find angular momentum eigenstates for the 2D harmonic oscillator.

If the approach of solving the radial eqn in cylindical coord.'s doesn't work, it may be interesting to look at the matrix represantation of Lz in the basis of the cartesian eigenstates. You may be able to calculate the matrix elements using a+ and a- operators... (note that x and y commute, as does px and y, etc...)

You don't have to get all of them necessarily. Any result with three or more (eigenstates) is an OK start...



Questions to think about and discuss:

Does Lz commute with the Hamiltonian?
Can you find simultaneous eig states of H and Lz ?
What is the nature, origin, combinatorix of the degeneracies of this system?