A quasi-exactly solvable model: two charges in a magnetic field, subject to a non-Coulomb mutual interaction

TL;DR

This paper extends quasi-exact solvability in quantum mechanics to a two-charge system in a magnetic field with non-Coulomb interactions, using algebraic methods to find partial solutions.

Contribution

It introduces a method to obtain quasi-exact solutions for two charges in a magnetic field with non-Coulomb potentials by algebraic reduction of the Hamiltonian.

Findings

Partial diagonalization of Hamiltonian achieved

Explicit solutions found for certain energy levels

Method applicable to various non-Coulomb potentials

Abstract

We extend the class of QM problems which permit for quasi-exact solutions. Specifically, we consider planar motion of two interacting charges in a constant uniform magnetic field. While Turbiner and Escobar-Ruiz (2013) addressed the case of the Coulomb interaction between the particles, we explore three other potentials. We do this by reducing the appropriate Hamiltonians to the second-order polynomials in the generators of the representation of $SL(2,C)$ group in the differential form. This allows us to perform partial diagonalisation of the Hamiltonian, and to reduce the search for the first few energies and the corresponding wave functions to an algebraic procedure.