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Solving a differential equation similar to Legendre

  1. Jun 1, 2009 #1
    I am trying to solve the following differential equation:

    [tex](\frac{L^2}{6k^2}+\frac{w\sqrt{3}}{2}\sin^2\theta\ sin 2\phi)\psi=E\psi[/tex]

    where is the angular momentum given by:

    [tex]L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial\theta }\left(\sin\theta\frac{\partial}{\partial\theta}\right)-\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}.[/tex]

    goes from 0 to [itex] $\pi$ [/itex] while [itex] $\phi$ [/itex] goes from 0 to [itex]2 $\pi$ [/itex]. w and k are constants and E is the energy of the system.. This differential equation is non separable. However i have realised that

    [tex] \begin{equation*} \sin^2\theta\ sin 2\phi= i w\sqrt{\frac{2\pi}{15}} (Y_2^{-2}-Y_2^{2}). \end{equation*} [/tex]

    I plugged that in differential equation above and multiplied the whole equation by [itex]Y_{l}^{m}[/itex] and then used the integral properties of three spherical harmonics multiplied together (in terms of wigner 3 j symbols). This is to get a recursion relation between different coefficients found in the solution that I assumed :

    [tex] \begin{equation*}
    \psi=\sum_{l=1}^{n} A_{l,1} Y_l^1 .

    I am looking at the case when m=1 hence the substitution by 1 for m in above equation. My two questions are: (1) How will I treat the term above contatining i? I only took the part without i and got solutions but some of the eigenvalues are complex (it is an energy term) so it is not possible! (2) even inserting i would cause more problems as u still get complex values..can anyone tell me what might be going wrong?.


    Last edited by a moderator: Jun 1, 2009
  2. jcsd
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