Schrodinger Equation for Non-Pure Dipole Potential in 2D Systems

TL;DR

This paper analytically investigates the Schrödinger equation for a 2D non-pure dipole potential, deriving eigenenergies and eigenfunctions, and analyzing how the dipole moment affects bound states and energy reality conditions.

Contribution

It provides an analytical solution for the 2D Schrödinger equation with a non-pure dipole potential, including eigenenergy expressions and the impact of dipole strength on bound states.

Findings

Eigenenergies and eigenfunctions derived analytically.

Maximum dipole moment for bound states identified.

s states are absent when dipole term is present.

Abstract

In this work, we analytically study the Schr\"odinger equation for the (non-pure) dipolar ion potential V (r) = q/r + Dcos{\theta}/r 2 , in the case of 2D systems using the separation of variables and the Mathieu equations for the angular part. We give the expressions of eigenenergies and eigenfunctions and study their dependence on the dipole moment D. Imposing the condition of reality on the energies E n,m implies that the dipole moment must not exceed a maximum value otherwise the corresponding bound state disappears. We also find that the s states (m = 0) can no longer exist in the system as soon as the dipole term is present.