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

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 $(R,\varphi )$ and relative $(\rho ,\phi )$ 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 $b\le 1$, (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.

20 pages

References in corpus (3)

• An infinite family of superintegrable deformations of the Coulomb potential
• Particular Integrability and (Quasi)-exact-solvability
• Two charges on plane in a magnetic field: special trajectories

Cited by in corpus (10)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Particular Integrability and (Quasi)-exact-solvability
• An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
• Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
• Ground state of an exciton in a three-dimensional parabolic quantum dot: convergent perturbative calculation
• Two charges on plane in a magnetic field I. "Quasi-equal" charges and neutral quantum system at rest cases
• Two charges on plane in a magnetic field: III. $H{e}^{+}$ ion
• Three charges on a plane in a magnetic field: Special trajectories
• A quasi-exactly solvable model: two charges in a magnetic field, subject to a non-Coulomb mutual interaction
• Trajectories of two identical particles on a plane in a constant magnetic field subject to non-Coulomb potentials