Solving two coupled PDE from a quantum problem

[Physteo]

Hello Guys.
I have to solve two coupled PDE coming from a quantum physical problem, which possesses a cylindrical symmetry. They look like this
$\left\{-\frac{i\hbar}{2M}\left( D^2_\rho + \frac{1}{\rho}D_\rho + D^2_z - \frac{L_z^2}{\hbar^2 \rho^2} \right) + \frac{M\omega^2_\rho \rho^2}{2} + \frac{M\omega^2_z z^2}{2} + g\psi^2_0 - \mu \right\} f^+_\nu = \epsilon_\nu f^-_\nu$
$\left\{-\frac{i\hbar}{2M}\left( D^2_\rho + \frac{1}{\rho}D_\rho + D^2_z - \frac{L_z^2}{\hbar^2 \rho^2} \right) + \frac{M\omega^2_\rho \rho^2}{2} + \frac{M\omega^2_z z^2}{2} + 3g\psi^2_0 - \mu \right\} f^-_\nu = \epsilon_\nu f^+_\nu$

where
$f^{+,-}_\nu = f^{+,-}_\nu(\rho, z, \phi)$

$D_x \equiv \frac{\partial}{\partial x}$

$\psi_0^2 = \frac{\mu}{g} \left(1 - \frac{\rho^2}{R^2_\rho} - \frac{z^2}{R^2_z}\right)$

$L_z$ is the projection of the angular momentum operator, which operates like this on his eigenfunctions:
$L_z (\frac{\exp{(im\phi)}}{\sqrt(2\pi)}) = \hbar m \frac{\exp{(im\phi)}}{\sqrt(2\pi)}$

and $\hbar, M, \omega_\rho, \omega_z, R_\rho, R_z, \epsilon_\nu$ and $\mu$ are constants.

Actually, I have to find the energies $\epsilon$.
I would like to use the fact that I know how the operator Lz acts, but also, I know how the operator on the z coordinate acts. I would like to end up with only a radial equation on ρ.
To be clearer, I rewrite the equations like this (grouping operators acting on different variables, and plugging the expression for $\psi_0$):

$\left\{-\frac{i\hbar}{2M}\left( D^2_\rho + \frac{1}{\rho}D_\rho \right) + \frac{M\omega^2_\rho \rho^2}{2} -\frac{\rho^2}{R_\rho} \right\} f^+_\nu +\left\{ -\frac{i\hbar}{2M} D^2_z + \frac{M\omega^2_z z^2}{2} -\frac{z^2}{R_z}\right\}f^+_\nu -\frac{i\hbar}{2M} \left\{-\frac{L_z^2}{\hbar^2 \rho^2} \right\}f^+_\nu= \epsilon_\nu f^-_\nu$

And here comes my question. I know how Lz acts on his eigenfunctions, but I also know how the operator on z acts on his eigenfunctions ( because it is the 1D quantum oscillator).
Then, am I allowed to write the solution in the following way?
$f^{+,-}_\nu = \sum_{m} \sum_{n} f^{+,-}_{n,m,\nu}(\rho) \psi_n(z) \frac{\exp{(im\phi)}}{\sqrt(2\pi)}$

where
$\psi_n(z)$ are the eigenfunctions of the harmonic oscillator?

If this is possible, the equation will be simplified in a 1D equation in ρ, replacing the harmonic operator in z, and the operator Lz with their eigenvalues.
Thanks for the attention.

 

[jvicens]

Thank you for sharing your problem with us. It seems that you are dealing with a complex and interesting quantum physical problem. I am always intrigued by such problems and would be happy to offer some suggestions.

Firstly, it is great that you have identified the cylindrical symmetry of your problem. This will definitely help in simplifying the equations and finding a solution. Regarding your question about writing the solution in terms of the eigenfunctions of the harmonic oscillator, I would say that it is possible. However, it is important to check if the eigenfunctions of the harmonic oscillator are also eigenfunctions of the operator in the z coordinate in your problem. If they are, then you can definitely write the solution in that form and simplify the equations further.

Another suggestion would be to look for any symmetries in the problem that can help in simplifying the equations. For example, if there is any rotational symmetry, you can use it to reduce the number of variables in your equations. In general, symmetries can greatly simplify the problem and make it more manageable.

Additionally, it would be helpful to look for any known analytical or numerical methods that can be applied to your problem. There might be specific techniques or algorithms that can help in solving coupled PDEs with cylindrical symmetry. It is always a good idea to explore different approaches and see which one works best for your problem.

I hope these suggestions are helpful to you. Good luck with solving your problem!