Schrödinger equation in multiply connected spaces and phase optics

Abstract

The magnetic vector potential$\vec A$ in a field free spaceR ₀ cannot be removed by gauge transformations in general, ifR ₀ is multiply connected.Aharonov andBohm ¹ have noticed recently that$\vec A$ therefore should have more physical meaning than only to give the magnetic field by differentiation. They could show that$\vec A$ inR ₀ may influence the phase ofSchrödinger'sψ-function in an observable manner. We want to point out here that this influence can be expressed in a simple, general form: “A closed magnetic field line operates uponψ like ae Φ/ħ-phase-shifter placed on any area bounded by the field line.” Surface like phase shifters are familiar in phase optics. There exists a narrow relationship between electron scattering at magnetic fields and some special problems of phase optics. An electron phase contrast microscope is discussed.