Coaxial Cylinder-Aharonov-Bohm effect

[aurelio]

Hi Everyone,

I'm trying to find the eigenfunctions and eigenvalues for a quantum spinless particle within two concentric cylinders of radius \rho_a and \rho_b
, where \rho_a is less than\rho_b, with Dirichlet boundary conditions at these two radii. It's clear to me what the solution is for the case of \rho_a being equal to zero. The radial solution is the Bessel Function of order zero but how can I find the solution when \rho_a is finite.
IF I know what is the spectrum for this problem then I think I can determine what is the spectrum when there is a constant magnetic field B along the z direction only within the inner cylinder.
Please help

 

[AndyW]

!

Hi there,

The problem you are describing is known as the quantum particle in a cylindrical potential with Dirichlet boundary conditions. The solution for this problem is given by the Bessel functions of the first kind, J_n(\rho), where n is an integer. The eigenfunctions for this problem are given by the product of the Bessel function and the azimuthal angle, \theta, which is a solution to the angular part of the Schrödinger equation.

When \rho_a is finite, the solution can be found by applying the boundary conditions at \rho_a and \rho_b. This leads to a system of equations that can be solved to determine the eigenvalues and eigenfunctions. The eigenvalues, or energy levels, are given by the zeros of the Bessel function, and the corresponding eigenfunctions are given by the product of the Bessel function and the appropriate solution for \theta.

To find the spectrum when there is a constant magnetic field B along the z direction, you can use the Landau levels approach. This approach considers the effect of the magnetic field on the energy levels of the particle, leading to a quantization of the energy levels. The energy levels are given by E_n = \hbar \omega_c (n + 1/2), where \omega_c is the cyclotron frequency and n is the Landau level number. The corresponding eigenfunctions are given by the product of the Bessel function and the appropriate solution for \theta, with an additional phase factor to account for the magnetic field.

I hope this helps with your problem. Let me know if you have any further questions or need clarification on any of the steps. Good luck with your research!