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2D quantum harmonic oscillator in cylindrical coordinates (radial part

  1. Jul 27, 2014 #1
    Dear kind helpers,

    actually I am not 100% sure whether this is the right place to post, as it is not a homework in the sense of an exercise sheet. But I think it could be because it feels pretty basic and that I should be able to solve it. Though I really searched for a solution but could not find one.
    What made me hesitate to post here, is that I am not sure, whether there is an analytic solution to my problem at all.
    So I hope to have found the right place to post. Please excuse if not.

    My problem is the following: I've got an effective 2D harmonic oscillator potential (no degree of freedom in z)
    [itex]V(r)=\frac{1}{2}m\omega^2(x^2+y^2)[/itex]
    so the Hamiltonian is
    [itex]\mathcal{H}=\frac{1}{2m}\left[\left(\frac{\hbar}{i}\nabla\right)^2+m^2\omega^2(x^2+y^2)\right][/itex]
    (where I allowed me to use [itex]\nabla=\left(\frac{d}{dx},\frac{d}{dy}\right)[/itex]
    I could, of course, solve this easily in cartesian coordinates as two one dimensional oscillators with the energy eigenvalues
    [itex]E=\left(n_x+n_y+\hbar\omega\right)[/itex]
    with the corresponding wavefunctions, which I could, with the aid of Griffiths, calculate pretty easily.

    The important point is however, the rotational symmetrie around the z-Axis to underline the degeneracy of the eigenstates.
    That is where cylindrical coordinates become obvious.
    with [itex]r^2=x^2+y^2[/itex] the Hamiltonian should become:
    [itex]\mathcal{H}=\frac{1}{2m}\left[\hbar^2\left(\frac{d^2}{d r^2}+\frac{1}{r}\frac{d}{d r}+\frac{1}{r^2}\frac{d^2}{d \phi^2}\right)+m^2\omega^2 r^2\right][/itex]

    plugging it in to the Schrödinger equation and doing a separation of variables ([itex]\Psi=R\Phi[/itex])
    the angular part is easy:
    [itex]\Phi=\Phi_0 e^{im\phi} \text{where}\ m=\pm 0,\pm 1,\dots[/itex]

    back into the Schrödinger equation leaves me with the radial part:
    [itex]r^2R''+rR'+\left(r^2E-m^2-\omega^2r^4\right)R=0[/itex]

    But I cannot find a solution to this Problem. But I have to add one last thing: I would rather not want to use a powerseries approach for some reason. That is why I didn't try yet. I hope someone nows a better solution?
    Without the harmonic potential part, Bessel functions would solve it. For a similar Problem in three dimensions spherical harmonics. In cartesian coordinates (without seperating the angular part) hermite polynomials. For some reason, Laguerre polynomials seem appealing, but I could not realy say way.
    But for this problem? Did I make a mistake? Am I mistaken, that the solution should depend only on one quantum number?

    I hope I made it clear what my problem is, and that someone here nows an answer, or at least if this is analytically solvable or not. And if someone has, citations would be nice of course.

    Best regards

    D0m2
     
    Last edited: Jul 27, 2014
  2. jcsd
  3. Jul 27, 2014 #2

    vanhees71

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    Science Advisor
    2016 Award

    Welcome to the physics forums!

    You find an excellent treatment of your problem in

    W. Pauli, Wave Mechanics (Vol. 5 of the Pauli Lectures on Physics)
     
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