Quantum confinement particle in a 2D quadrupole potential

Abstract

We analytically solve the Hamiltonian for a quantum particle confined in a cylindrical hard-wall well, subject to the action of a two-dimensional quadrupolar potential at the well center. The angular part of the wavefunction is expressed by Mathieu functions whose angular eigenenergies take negative values when the quadrupolar momentum is above a certain threshold. We show that in this case, the radial part of the eigenfunctions is expressed in terms of Bessel functions of an imaginary order which are imaginary-value functions whose phases are not well defined at the origin. However, the density of probability is well defined everywhere and the wave function satisfies hard-wall boundary conditions for any value of the parameters involved. We discuss an alternative criterion for determining the eigenenergies of the system based on the expected value of the symmetrized radial momentum.