Charged particle in the field an electric quadrupole in two dimensions

TL;DR

This paper provides an exact analytical solution to the Schrödinger equation for a charged particle in a 2D electric quadrupole field, covering bound and scattering states with implications for particle binding.

Contribution

It introduces a series solution using special functions and orthogonal polynomials for the 2D quadrupole problem, including conditions for particle binding.

Findings

Solution valid for all energies including bound and scattering states

Wavefunction expansion coefficients satisfy three-term recursion relations

Particle binding depends on quadrupole moment exceeding a critical value

Abstract

We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. This solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. The charged particle could become bound to the quadrupole only if its moment exceeds a certain critical value.