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arXiv:1112.4958

Complemetarity of Phases in Aharonov-Bohm Solenoid Effect

Y. Ben-Aryeh

Published Dec 21, 2011• 1 citation
arXivHTMLPDF

Abstract

In the present Note it is suggested that there should be a certain complementarity of phases between Aharonov-Bohm (AB) solenoid phase calculation on one part of the system and a phase calculation about another part of the physical system. Assuming a unique value for the function of the total system, after a complete circulation of the electron around the solenoid, the sum of these two phase changes should vanish. Such assumption leads to a compatibility relation between our previous calculations for the AB solenoid phase effect and that of the original calculation by AB.

DOI
10.48550/arxiv.1112.4958

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In the present Note it is suggested that there should be a certain complementarity of phases between Aharonov-Bohm (AB) solenoid phase calculation on one part of the system and a phase calculation about another part of the physical system. Assuming a unique value for the function of the total system, after a complete circulation of the electron around the solenoid, the sum of these two phase changes should vanish. Such assumption leads to a compatibility relation between our previous calculations for the AB solenoid phase effect and that of the original calculation by AB.
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.
We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities: the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semiclassical calculation [L. Vaidman, Phys. Rev. A 86, 040101 (2012)], later confirmed by a fully quantum calculation of Pearle and Rizzi [preceding paper, Phys. Rev. A 95, 052123 (2017)], where it is supposed that the electron's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions: a vector potential wave function, an electron wave function, and a solenoid wave function. We apply the variational principle for the exact Schr\"odinger equation to this approximate form of solution. This leads to three Schr\"odinger equations, one each for vector potential, electron, and solenoid, each with classical sources for the other two entities. However, each Schr\"odinger equation contains an additional real ccc-number term, the time derivative of an extra phase. We show that these extra phases are such that the phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.
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.
The authors have found the scattering cross sections on two solenoids with opposite magnetic fluxes and on the toroidal solenoid in the presence or absence of a magnetic field. From this they extract the diffraction pattern shift which is due to the presence of the magnetic flux inside the solenoids. Practical recommendations for the performance of the experiments are given. It is proved that the crucial experiment on the existence of the AB effect suggested earlier could indeed be realised in practice.
It is argued that much of the recent controversy over the Aharonov-Bohm effect has been fuelled by the widespread consideration of solenoids of infinite length: in this limit, the field and the vector potential have somewhat anomalous properties and it appears superficially that the Aharonov-Bohm phase shift can be changed by a gauge transformation. This incorrect impression is here removed by considering a solenoid of finite length and by carefully distinguishing the longitudinal and transverse parts of the vector potential. It is shown that the phase shift depends only on the transverse part of the vector potential and it cannot be changed by a gauge transformation. The nature of the difficulty that occurs in the limit of a solenoid of infinite length is examined in detail. The finite-length theory given here also provides guidelines for the design of experiments on the Aharonov-Bohm effect.