The Aharonov–Bohm effect in scattering theory

• Aharonov–Bohm effect as a scattering event.
• Impenetrable magnetic vortex of nonzero transverse size.
• Scattering cross section is independent of a self-adjoint extension employed.
• Classical phenomenon of elastic reflection and quantum phenomenon of diffraction.
• Aharonov–Bohm effect as a fringe shift in the diffraction pattern.

The Aharonov–Bohm effect is considered as a scattering event with nonrelativistic charged particles of the wavelength which is less than the transverse size of an impenetrable magnetic vortex. The quasiclassical WKB method is shown to be efficient in solving this scattering problem. We find that the scattering cross section consists of two terms, one describing the classical phenomenon of elastic reflection and another one describing the quantum phenomenon of diffraction; the Aharonov–Bohm effect is manifested as a fringe shift in the diffraction pattern. Both the classical and the quantum phenomena are independent of the choice of a boundary condition at the vortex edge, providing that probability is conserved. We show that a propagation of charged particles can be controlled by altering the flux of a magnetic vortex placed on their way.