Schrodinger's equation for an electron in a vector potential A has a one-to-one correspondence with the wave equation for an electric field in a moving medium. First the author points out a few errors in the literature concerning the Aharonov-Bohm (AB) effect then she explores the optical analogy further. The author finds that curl A plays a similar role to curl v where v is the velocity of the moving medium.
Schrodinger's equation for an electron in a vector potential A has a one-to-one correspondence with the wave equation for an electric field in a moving medium. First the author points out a few errors in the literature concerning the Aharonov-Bohm (AB) effect then she explores the optical analogy further. The author finds that curl A plays a similar role to curl v where v is the velocity of the moving medium.
The electromagnetic wave equations in a moving medium may be approximated by a form similar to that of the Schrödinger equation for a particle in an electromagnetic field, with the velocity v of the medium and the vorticity ∇×v playing the roles of the vector potential and magnetic field, re- spectively. A purely classical optical analogue of the Aharonov–Bohm effect follows by consider- ation of the interference pattern produced by two beams, each of which propagates in a region with zero vorticity, but such that the flux of the vorticity through the closed loop defined by the optical paths does not vanish. Fizeau’s experiment (1851) on the velocity of light in moving media may be regarded as an example of such a situation.
Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.
It is frequently stated that the electromagnetic vector potential acquires a fundamental role in quantum physics, whereas classically it only represents a convenient, but by no means necessary, way of representing the electromagnetic field. Here we argue that this is a historical accident due to the fact that the electromagnetic field was discovered before photons, while the electron itself was discovered first as a particle, before it became clear that it must also be treated as a wave and therefore as an excitation of the underlying electron field. We illustrate the fact that the vector potential ought to play a fundamental role classically using the Aharonov-Bohm effect. This effect is considered as the strongest argument for the role the vector potential plays in quantum physics, however, here we offer a fully classical account of it. This is a consequence of the fact that any account, be it classical or quantum, must involve the vector potential in order to preserve the local nature of the Aharonov-Bohm (as well as all the other) phases.