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 Schrodinger equation using the harmonic oscillator potential. Use separation of variables.

 

[paranormal]

As a scientist, it is important to accurately and clearly communicate your findings and thoughts. Here is a possible response to the above content:

Based on the given information, it seems that we are looking to find the wave functions of a particle in a harmonic oscillator potential that are eigenstates of the angular momentum operator, specifically with eigenvalues of -1 h, 0, and 1 h. The goal is to determine if these states are also eigenstates of the L^2 operator and to write the wave functions using spherical coordinates while normalizing the radial and angular parts independently.

To incorporate the harmonic oscillator potential into the problem, we can use the Hamiltonian operator for the harmonic oscillator, which includes the potential energy term. This will give us a more accurate representation of the system and allow us to find the appropriate wave functions. Additionally, we can use the Schrödinger equation to solve for the wave functions and eigenvalues.

Using the given solution for the eigenstates of the Lz operator, we can calculate the corresponding eigenfunctions for the harmonic oscillator potential. From there, we can check if these states are also eigenstates of the L^2 operator by plugging them into the L^2 operator and seeing if they satisfy the eigenvalue equation.

Furthermore, to write the wave functions in spherical coordinates and normalize them independently, we can use the radial and angular parts of the wave function separately. The radial part can be obtained by solving the radial Schrödinger equation, while the angular part can be obtained by using the spherical harmonics. Normalization can then be achieved by ensuring that the squared magnitude of the wave function is equal to 1.

In conclusion, by incorporating the harmonic oscillator potential and using the appropriate equations and techniques, we can find the wave functions of the states of a particle in a harmonic oscillator potential that are eigenstates of the Lz operator and determine if they are also eigenstates of the L^2 operator. We can also write these wave functions using spherical coordinates and normalize them independently. Further analysis and calculations may be necessary to fully understand the properties and behavior of these states.