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Frontiers of Physics

A possible interplay between electron beams and magnetic fluxes in the Aharonov-Bohm effect

Ruifeng Wang

Published Apr 17, 2015• 10 citations
Publisher

Abstract

Most studies on the magnetic Aharonov-Bohm (A-B) effect focus on the action exerted by the magnetic flux on the electron beam, but neglect the back-action exerted by the electron beam on the magnetic flux. This paper focuses on the latter, which is the electromotive force Δ U across the solenoid induced by the time-dependent magnetic field of the electron beam. Based on the back-action analysis, we observe that the magnetic A-B effect arises owing to the interaction energy between the magnetic field of the electron beam and the magnetic field of the solenoid. We also demonstrate that the interpretation attributing the magnetic A-B effect to the vector potential violates the uncertainty principle.

DOI
10.1007/s11467-015-0470-4

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Most studies on the magnetic Aharonov-Bohm (A-B) effect focus on the action exerted by the magnetic flux on the electron beam, but neglect the back-action exerted by the electron beam on the magnetic flux. This paper focuses on the latter, which is the electromotive force Δ U across the solenoid induced by the time-dependent magnetic field of the electron beam. Based on the back-action analysis, we observe that the magnetic A-B effect arises owing to the interaction energy between the magnetic field of the electron beam and the magnetic field of the solenoid. We also demonstrate that the interpretation attributing the magnetic A-B effect to the vector potential violates the uncertainty principle.
The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
Whether the time-dependent Aharonov-Bohm (AB) effect even exists or not has been the subject of long-standing debate. There are two factors complicating the problem. First, in the closed spacetime line integral of the vector potential that is thought to give the AB-phase shift, how to treat the time-varying vector potential is highly nontrivial. Second, the time-varying magnetic flux generates induced electric field even outside the solenoid. In the present paper, motivated by a recent work by Gao, we re-investigate the role of the induced electric field with the utmost care. This analysis reveals a highly nontrivial effect of the induced electric field, which turns out to be useful for verifying the very existence of the time-dependent AB-effect.
Although there is no force on the electron in Aharonov-Bohm solenoid effect, the electron exerts a force on the solenoid related to the inequality of action and reaction forces of two subsystems in three-system configuration. The AB phase which is related to the force exerted by the electron on the solenoid . The momentum changes of the mechanical oscillator are equal in magnitude and opposite in sign to the changes in the momentum of the em fields. It is proposed to investigate momentum changes of "micro" bodies producing magnetic fields in AB systems which will clarify the nature of these effects. The problem of magnetic fields shielded from the electron wave packet is also discussed.
We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.
We present a review and discussion of the physical meaning of the vector potential in electromagnetism by means of a classical experiment in which a long solenoid, connected to a sinusoidal voltage supply, produces an electric field and a vector potential in a circuit exterior to the solenoid itself. Some basic considerations are recalled concerning the meaning of action-at-a-distance and local description, in terms of 'real fields' of the induced electromotive force on the circuit outside the solenoid. From this viewpoint the roles played by the magnetic field, the electric field and the vector potential to account for the physical effects are reviewed. Finally, the connections between these considerations and the magnetic Aharonov–Bohm effect are recalled in the light of recent theoretical work aimed at explaining the Aharonov–Bohm phase shift in terms of the magnetic field without resorting to the vector potential.