Angular momentum operator eigenvalues in HO potential.

[Zaknife]

Homework Statement

Find wave functions of the states of a particle in a harmonic oscillator potential
that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually, write the wave functions using spherical coordinates and normalize independently
their radial and angular parts.

Homework Equations

The Attempt at a Solution

I know that solution of eigen-problem for $L_{z}$ operator is:
$$psi(\theta, \phi)= P(\theta) e^{im\phi}$$
But i don't know how to include harmonic oscillator potential into this problem. I already proved that eigenstates of Lz are also eigenstates of L^2. Thanks for any advice !

 

[BruceW]

that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually, write the wave functions using spherical coordinates and normalize independently their radial and angular parts.

This suggests that the question is about about a 3d isotropic harmonic oscillator. The solutions might looks similar to the hydrogenic atom, but be careful. Were you set this as homework? Its pretty difficult to do something like this from first principles...

 

[Zaknife]

Yes it's my homework actually. So the suggestion is to solve the radial equation, for the HO potential $$\frac{1}{2}\omega mr^{2}$$. For l=-1,0,1 and then "combine" it with the angular part of wavefunction for different m values ?

 

[vela]

You need to solve the Schrödinger equation using the harmonic oscillator potential. Use separation of variables.