What Are the Eigenstates and Energies of the Quantum Harmonic Oscillator?

[atomicpedals]

Homework Statement

Consider the Hamiltonian

H=\frac{p^2}{2M}+\frac{1}{2}\omega^2r^2-\omega_z L_z

Determine its eigenstates and energies.

2. The attempt at a solution

I want to check my comprehension; by eigenstate they mean

\psi(r)
from the good old

H\psi(r)=E\psi(r)
and then the energies would then be solutions for E?

 

[atomicpedals]

To sort of answer my own question the eigenstates would more properly probably be
|\psi>
in
H|\psi>=E|\psi>

 

[vela]

It's a three-dimensional problem, so the wave function will be a function of r, θ, and φ.

 

[atomicpedals]

Ah, right. (and just as I was getting comfortable in 2D)

 

[dextercioby]

Convert L_z and p to spherical coordinates and separate variables just like in the H-atom case, or the isotropic 3D oscillator.