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erok81
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Homework Statement
I have a quantum harmonic oscillator with quantum numbers nx,ny,nz ≥ 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 Enx,ny,nz of a QHO in 3d with quantum numbers nx,ny,nz ≥ 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 ψnx,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...
[tex]\psi (x,y,z) = Ae^{\frac{1}{2}(x^2 +y^2 +z^2)}[/tex]
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...
[tex]U(\vec{r}) = \frac{1}{2}m \omega^{2}\vec{r}^2~=~\frac{1}{2}m \omega^{2}(x^2+y^2+z^2)[/tex]
Back to my Schrödinger...
[tex] \frac{- \hbar ^2}{2m} \nabla ^2 \psi (\vec{r}) +U(\vec{r}) \psi (\vec{r}) = E \psi (\vec{r}) [/tex]
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.
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