Eigenstate for a 3D harmonic oscillator

[JordanGo]

Homework Statement

A 3D harmonic oscillator has the following potential:

$V(x,y,z) = \frac{1}{2}m( \varpi_{x}^2x^2 + \varpi_{y}^2y^2 + \varpi_{z}^2z^2)$

Find the energy eigenstates and energy eigenvalues for this system.

The Attempt at a Solution

I found the energy eigenvalue to be:

$E = E_{x} + E_{y} + E_{z}$

$E = \hbar((n_{x}+\frac{1}{2})\varpi_{x} + (n_{y}+\frac{1}{2})\varpi_{y} + (n_{z}+\frac{1}{2})\varpi_{z})$

Now I know that the eigenstate is:

$\Psi = \Psi_{x} \times \Psi_{y} \times \Psi_{z}$

But I don't know how to find ψ_x, ψ_y or ψ_z.

Can someone help me?

 

[PhysicsGente]

Hey JordanGo.
Try writing down your 3-D Schrödinger equation and use separation of variables.

 

[vela]

How did you find the energy eigenvalues? It seems to me if you can figure those out correctly, it's pretty straightforward to see how to get the eigenstates. Show your work so far.