# Solving a differential equation similar to Legendre

1. Jun 1, 2009

I am trying to solve the following differential equation:

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

where is the angular momentum given by:

$$L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial\theta }\left(\sin\theta\frac{\partial}{\partial\theta}\right)-\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}.$$

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

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

I plugged that in differential equation above and multiplied the whole equation by $Y_{l}^{m}$ 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 :

$$\begin{equation*} \psi=\sum_{l=1}^{n} A_{l,1} Y_l^1 . \end{equation*}$$

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?.

Thanks

Thanks

Last edited by a moderator: Jun 1, 2009