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Isotropic 3-D harmonic oscillator

  1. Apr 22, 2007 #1
    1. The problem statement, all variables and given/known data
    Use Frobenius’ method to solve the problem of the isotropic three dimensional
    harmonic oscillator in polar coordinates. It is sufficient to
    find the energy levels and degeneracies, but it would be nice to plot the
    spectrum like we did for hydrogen. Be sure to introduce the natural
    length scale and energy scale and to incorporate the asymptotic forms
    at r = 0 and r = infinity of the radial wave function into your power series
    solution. You will find that the recursion relation skips a term, i. e.,
    relates c_j+2 to c_j . Thus two coefficients c_0 and c_1 are left undetermined.
    How do you handle this situation? (Note that you can check
    your answers against the Cartesian solution of the same problem.)


    2. Relevant equations
    See attachments. (Had to put them as images, I guess.) But it only let me put 3. a_0=hbar^2/(me^2), rho=r/(a_0), epsilon=-(E/R) where R=(me^4)/(2hbar^2)

    3. The attempt at a solution
    Writing down the equations is about as far as I've gotten. I think that the V(r) potential is right for the spherical (the last attached image) but I'm not sure. This stuff is just really too abstract for me.
     

    Attached Files:

  2. jcsd
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