# 2 dimensional harmonic oscillator.find the energy eigenvalues?

1. Feb 1, 2012

### humanist rho

1. The problem statement, all variables and given/known data
Potential of a simple harmonic oscillator is$\frac{1}{2}m\omega ^{2}(x^{2}+4y^{2})$.Find the energy eigenvalues?

2. Relevant equations

Seperation of variables,i think. But i got stuck in the midway.

3. The attempt at a solution

$$\frac{-\hslash ^{2}}{2m}\left( \frac{\partial ^{2}\psi }{\partial x^{2}}+% \frac{\partial ^{2}\psi }{\partial y^{2}}\right) +\frac{1}{2}m\omega ^{2}(x^{2}+2y^{2})\psi =E\psi$$

$$\psi (x,y)=X(x)Y(y)$$

$$\frac{\partial ^{2}X}{\partial x^{2}}-\frac{m^{2}}{\hslash ^{2}}\omega ^{2}x^{2}X+\frac{2m}{\hslash ^{2}}E_{1}X=0$$

$$\frac{\partial ^{2}Y}{\partial x^{2}}-\frac{2m^{2}}{\hslash ^{2}}\omega ^{2}y^{2}Y+\frac{2m}{\hslash ^{2}}E_{2}Y=0$$

need a hint about how to proceed.
Thanks.

Last edited: Feb 1, 2012
2. Feb 1, 2012

### genericusrnme

Do you know how to do the one dimensional harmonic oscillator?

3. Feb 1, 2012

### humanist rho

Actually no.I was absent in the class,failed to understand it and later found abstract operator form more comfortable.

But i'll try
Thanks for the hint. :)

4. Feb 1, 2012