Deriving Energy and Wave Functions from 3D Schrödinger

[erok81]

Homework Statement

I have a quantum harmonic oscillator with quantum numbers n_x,n_y,n_z ≥ 0 and frequency ω₀. There are three parts to the problem. To me it seems they are out of order, but I'm kind of shaky deriving these.

a) Write down the energy level E_nx,_ny,_nz of a QHO in 3d with quantum numbers n_x,n_y,n_z ≥ 0 and frequency ω₀?

b) Write down the normalized wave function for the ground state ψ₀(x,y,z)?

c) What is the Schrödinger equation for the generic 3d wave function ψ_nx,_ny,_nz(x,y,z)?

Homework Equations

n/a

The Attempt at a Solution

To me I think one would solve this c, b, a. I can state a and b first by copying out of the book, but I'd rather derive them since that's the correct method. The only thing I cannot figure out what my wave function is to involve the energy levels.

My guess for the wave equation is...

$$\psi (x,y,z) = Ae^{\frac{1}{2}(x^2 +y^2 +z^2)}$$

But I am missing my energy levels n. So I know that is wrong.

My 3d Schrödinger equation equation in this case is...

Well first the potential is given by...

$$U(\vec{r}) = \frac{1}{2}m \omega^{2}\vec{r}^2~=~\frac{1}{2}m \omega^{2}(x^2+y^2+z^2)$$

Back to my Schrödinger...

$$\frac{- \hbar ^2}{2m} \nabla ^2 \psi (\vec{r}) +U(\vec{r}) \psi (\vec{r}) = E \psi (\vec{r})$$

So. Where I am confused is guessing my wave equation and then solving for the energy.

I thought I would have to do the order like I posted above, but the question is worded differently. Maybe I am more confused then I thought. :)

The steps I would do is write out my Schrödinger, use separation of variables, solve for my wave equation, then find the energy. But I'm not fully sure since I won't have my n values.

Hopefully this makes sense. I've been up for a while and am having some trouble explaining what I've done so far. :rofl:

 

[erok81]

So my wave equation isn't correct.

I believe it should be

$$\psi (x,y,z) = Ae^{\frac{\alpha}{2}(x^2 +y^2 +z^2)}$$

Where α = mω/ℏ

However the problem states to write down the normalized wave function for the ground state! I don't have any n terms in my wave equation and therefore have no way to relate it to the ground state.

If this were a box, I'd have sin terms and n terms, and I'd be set. However I see no way to incorporate n into this type of wave function.

Any ideas?

On a side note: should I have put this in advanced physics?