Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Abstract

We propose a simple quantum mechanical equation for $n$ particles in two dimensions, each particle carrying electric charge and magnetic flux. Such particles appear in (2+1)-dimensional Chern-Simons field theories as charged vortex soliton solutions, where the ratio of charge to flux is a constant independent of the specific solution. As an approximation, the charge-flux interaction is described here by the Aharonov-Bohm potential, and the charge-charge interaction by the Coulomb one. The equation for two particles, one with charge and flux ($q, Φ/Z$) and the other with ($-Zq, -Φ$) where $Z$ is a pure number is studied in detail. The bound state problem is solved exactly for arbitrary $q$ and $Φ$ when $Z>0$. The scattering problem is exactly solved in parabolic coordinates in special cases when $qΦ/2π\hbar c$ takes integers or half integers. In both cases the cross sections obtained are rather different from that for pure Coulomb scattering.