Solving a differential equation similar to Legendre

[Physicslad78]

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 realized 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 :

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

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

 

[azntoon]

for your question. The issue you are facing is due to the fact that the differential equation you are trying to solve is non-separable, meaning that it cannot be written as two separate equations in terms of $\theta$ and $\phi$. This makes it difficult to use the properties of spherical harmonics to derive a recurrence relation between the coefficients in your solution. One way to approach this problem is to use the method of separation of variables. This involves assuming a solution of the form $\psi = \Theta(\theta) \Phi(\phi)$, and then substituting this into the original differential equation. This will lead to two separate equations in terms of $\theta$ and $\phi$, which can then be solved using the properties of spherical harmonics.Another approach that may be useful is to make use of the symmetry of the problem. Since the equation is invariant under rotations around the $z$-axis, it is possible to take advantage of this to reduce the number of independent variables. Specifically, you can make a substitution of the form $\phi = \alpha + \beta$ where $\alpha$ is an arbitrary constant and $\beta$ is the angle around the $z$-axis. This will reduce the equation to one in terms of just $\theta$ and $\alpha$, which can then be solved using separation of variables or other methods.I hope this helps!