Two Charged Particles in the Plane Under a Constant Perpendicular Magnetic Field

Abstract

We show that a system with two particles of equal masses and opposite electric charge on a plane subject to a constant magnetic field perpendicular to this plane is a completely integrable system. In the classical case, the separation of variables can be performed in oblate coordinates. In the quantum case, we obtain two independent Hill equations of fourth order. Some approximations lead to Mathieu equations, for which the solutions are well known.