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 $(q,$ $\ensuremath{\Phi}/Z)$ and the other with $(\ensuremath{-}Zq,$ $\ensuremath{-}\ensuremath{\Phi})$ where Z is a pure number is studied in detail. The bound-state problem is solved exactly for arbitrary q and $\ensuremath{\Phi}$ when $Z>0.$ The scattering problem is exactly solved in parabolic coordinates in special cases when $q\ensuremath{\Phi}/2\ensuremath{\pi}\ensuremath{\Elzxh}c$ takes integers or half integers. In both cases the cross sections obtained are rather different from that for pure Coulomb scattering.