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arXiv:0911.5063

A proposal for investigating three-body forces in Aharonov-Bohm sytems

Y. Ben-Aryeh

Published Nov 26, 2009• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.0911.5063

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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.
The Aharonov–Bohm solenoid effect is related to the inequality of the forces of action and reaction of two subsystems in a three-system configuration. During the motion of the electron around the solenoid there are momentum exchanges between the solenoid and the electromagnetic field that are produced by the electron and the solenoid. The integral of the momentum of interference over the trajectory of the electron, which is equal to the integral of the energy of interference over the corresponding time, gives the Aharonov–Bohm phase shift.
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.
An experiment to observe the Aharonov-Bohm effect is discussed. A solenoid which consists of a large number of point magnetic dipoles is considered as the source of a vector potential, which acts on a charged particle, and such potential has an electromagnetic field of zero strength in the region of a nonzero vector potential. A detailed microscopic analysis of the change in the quantum state phase of the entire system, namely, a particle and a set of dipoles, reveals the origin of the apparent nonlocality of the action of the vector potential, and shows the locality of the phase change mechanism. An analysis of an experiment with a solenoid shielded by a superconducting shell is given.
Following semiclassical arguments by Vaidman [Phys. Rev. A 86, 040101(R) (2012)], we show that the phase shifts arising in the Aharonov-Bohm (AB) magnetic or electric effects can be treated as due to the electric force of a classical electron, respectively acting on quantized solenoid particles or quantized capacitor plates. This is in contrast to the usual approach that treats both effects as arising from non-field-producing potentials acting on the quantized electron. Moreover, we consider the problems of an interacting quantized electron and a quantized solenoid or a quantized capacitor to see what phase shift their joint wave function acquires. We show, in both cases, that the net phase shift is indeed the AB shift (for one might have expected twice the AB shift, given the above two mechanisms for each effect). The solution to the exact Schr\"odinger equation may be treated (approximately for the magnetic AB effect, which we show using a variational approach, exactly for the electric AB effect) as the product of two solutions of separate Schr\"odinger equations for each of the two quantized entities, but with an extra phase. The extra phase provides the negative of the AB shift, while the two separate Schr\"odinger equations without the extra phase each provide the AB phase shift so that the product wave function produces the net AB phase shift.