2 dimensional harmonic oscillator.find the energy eigenvalues?

[humanist rho]

Homework Statement

Potential of a simple harmonic oscillator is$\frac{1}{2}m\omega<br /> ^{2}(x^{2}+4y^{2})$.Find the energy eigenvalues?

Homework Equations

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

The Attempt at a Solution

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

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

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

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

need a hint about how to proceed.
Thanks.

 

[genericusrnme]

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

 

[humanist rho]

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

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. :)

 

[genericusrnme]

You should probably try and learn the one dimensional oscillator first then
Once you know the 1-d case you can solve this easily
You might find this useful http://www.physics.csbsju.edu/QM/Index.html