Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

TL;DR

This paper investigates the quantum mechanics of two Coulomb charges in a magnetic field, revealing hidden algebraic structures, integrability properties, and polynomial eigenfunctions in specific cases, advancing understanding of such quantum systems.

Contribution

It identifies hidden $sl(2)$ algebra and quasi-exact solvability in two-charge systems under particular conditions, providing explicit polynomial eigenfunctions and analyzing their properties.

Findings

Eigenfunctions are factorizable with explicit $ ho$-dependence.

Hidden $sl(2)$ algebra appears at discrete magnetic field values.

Finite polynomial eigenfunctions are explicitly constructed.

Abstract

The quantum mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field $B$ perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically-important particular cases, namely that of two particles of equal Larmor frequencies, ${e_c} \propto \frac{e_1}{m_1}-\frac{e_2}{m_2}=0$ (e.g. two electrons) and one of a neutral system (e.g. the electron - positron pair, Hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS $(R, \phi)$ and relative $(\rho, \varphi)$ coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit $\rho$-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in $\rho$-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden $sl(2)$ algebra; at some discrete values of dimensionless magnetic fields $b \leq 1$, (iii) particular integral(s) occur, (iv) the hidden $sl(2)$ algebra emerges in finite-dimensional representation, thus, the system becomes {\it quasi-exactly-solvable} and (v) a finite number of polynomial eigenfunctions in $\rho$ appear. Nine families of eigenfunctions are presented explicitly.