2 dimensional harmonic oscillator.find the energy eigenvalues?

[humanist rho]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

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

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

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

need a hint about how to proceed.
Thanks.

 

[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

 

[asteriatic]

To find the energy eigenvalues for a 2-dimensional harmonic oscillator, we can use the separation of variables method. This involves separating the wavefunction into two parts, one for the x-direction and one for the y-direction. From there, we can use the Schrodinger equation to solve for the energy eigenvalues.

In the attempt at a solution, you correctly separated the wavefunction into X(x) and Y(y). However, the next step would be to plug these functions into the Schrodinger equation and solve for the energy eigenvalues. This would involve using the fact that the operator for the x-direction is $-\frac{\hslash^2}{2m}\frac{\partial^2}{\partial x^2}$ and the operator for the y-direction is $-\frac{\hslash^2}{2m}\frac{\partial^2}{\partial y^2}$.

From there, you can solve for the energy eigenvalues for each direction, and then combine them to get the overall energy eigenvalues for the 2-dimensional harmonic oscillator. Keep in mind that the energy eigenvalues will be in terms of the quantum numbers n and m, for the x- and y-directions respectively.