Aharonov-Bohm effect revisited

The Aharonov-Bohm (AB) effect shows that electromagnetic potentials can influence an electron in a field-free region. The single-slit and double-slit electron diffraction patterns are explicitly calculated here by the Feynman path integral method for different configurations of the magnetic field in order to compare the effect of the vector potential with the effect of the magnetic field. When an electron passes through a magnetic field behind the slits, the whole diffrection pattern is shifted due to the Lorentz force. When an electron passes through two slits with magnetic flux confined to a small cylinder between them, the double-slit diffraction pattern is shifted within the single-slit diffraction envelope. The asymmetric diffraction pattern corresponds to a nonzero average displacement and momentum of the electron even though the field exerts no force on the electron. This behavior can be understood on the basis of a quantum-mechanical interference effect. The classical limit of the electron diffraction patterns is taken to obtain the classical particle distributions. The effect of the potential vanishes in the classical limit, while the effect of the magnetic field persists because of the Lorentz force.