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Homework Help: 2 dimensional harmonic oscillator.find the energy eigenvalues?

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Potential of a simple harmonic oscillator is[itex] \frac{1}{2}m\omega
    ^{2}(x^{2}+4y^{2})[/itex].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

    [tex]\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 [/tex]

    [tex]\psi (x,y)=X(x)Y(y)[/tex]

    [tex]\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[/tex]

    [tex]\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[/tex]

    need a hint about how to proceed.
    Thanks.
     
    Last edited: Feb 1, 2012
  2. jcsd
  3. Feb 1, 2012 #2
    Do you know how to do the one dimensional harmonic oscillator?
     
  4. Feb 1, 2012 #3
    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. :)
     
  5. Feb 1, 2012 #4
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