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arXiv:1104.1449

Derivation of the Aharanov Bohm phase shift using classical forces

H. Fearn, Nguyen, Khai

Published Apr 7, 2011• 0 citations
arXivHTMLPDF

Abstract

In 1959 Aharonov and Bohm suggested that an electron passing around a long solenoid would pick up a phase shift dependent on the magnetic field of the solenoid, even though the electrons themselves pass through a region of space which has a zero magnetic field. It has long been held that this result is purely quantum and is derived in many well known quantum mechanics text books using the Schrodinger equation and vector potential. Here the same phase shift is derived from a purely classical force, but relativistic transformations are taken into account. The force is in the direction of motion of the electron (or opposite) leading to a phase advance (or lag) and we obtain precisely the phase shift thought previously to be purely quantum. The only quantum result used here is the de Broglie wavelength of the particle, in order to get two slit like interference and the phase shift. We employ a stack of dipoles as the solenoid and note the same force on the electron in two different frames of reference. We shall consider the solenoid stationary and the electron moving, and then consider the electron rest frame and consider the solenoid moving in the opposite direction.

DOI
10.48550/arxiv.1104.1449

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In 1959 Aharonov and Bohm suggested that an electron passing around a long solenoid would pick up a phase shift dependent on the magnetic field of the solenoid, even though the electrons themselves pass through a region of space which has a zero magnetic field. It has long been held that this result is purely quantum and is derived in many well known quantum mechanics text books using the Schrodinger equation and vector potential. Here the same phase shift is derived from a purely classical force, but relativistic transformations are taken into account. The force is in the direction of motion of the electron (or opposite) leading to a phase advance (or lag) and we obtain precisely the phase shift thought previously to be purely quantum. The only quantum result used here is the de Broglie wavelength of the particle, in order to get two slit like interference and the phase shift. We employ a stack of dipoles as the solenoid and note the same force on the electron in two different frames of reference. We shall consider the solenoid stationary and the electron moving, and then consider the electron rest frame and consider the solenoid moving in the opposite direction.
In 1984 Aharonov and Casher proposed that a beam of neutral particles with magnetic dipole moments passing around opposite sides of a line charge will undergo a relative quantum phase shift. This is currently under experimental investigation. Aharonov and Casher claimed that the magnetic dipole particles should undergo the phase shift despite experiencing no classical force. They suggested that this new effect has a ``duality'' with the solenoid Aharonov-Bohm effect where charged particles passing around a magnetic solenoid experience a phase shift despite, it is claimed, experiencing no classical force. Here it is pointed out that a magnetic dipole particle passing a line charge does indeed experience a classical electromagnetic force in the usual electric-current model for a magnetic dipole. This force will produce a relative lag between dipoles passing on opposite sides of the line charge, and the classical lag then leads to a quantum phase shift in exact agreement with that calculated by Aharonov and Casher. Thus actually the proposed Aharonov-Casher effect has a transparent explanation as a classical lag effect. It is emphasized that the solenoid Aharonov-Bohm effect is currently the only phase-shift effect which cannot be explained in an obvious fashion as arising from a classical lag and it is again proposed that also this shift may actually involve a classical electromagnetic lag effect. A natural experimental consequence suggested by the lag point of view is the breakdown of the interference pattern when the lag becomes comparable to the wave-packet coherence length.
A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces.
In classical electrodynamics the vacuum is defined as a region where there are no electric or magnetic fields. In such a region, a charged particle (such as an electron) will feel no effect — the Lorentz force is zero. The space ex-ternal to a perfect (i.e., infinite) solenoid can be considered an electromagnetic vacuum, since ~E and ~B vanish there. While a non-zero vector potential ~A does exist outside the solenoid, it can exert no influence on the particle, and as such cannot be directly detected or quantified classically. However, in 1959 Aharonov and Bohm predicted that a vector field would exert a purely quantum-mechanical effect on the phase of the particle’s wave function, which in principle should be detectable. The predicted phase shift was not observed experimentally until 1986, when Tonomura brilliantly verified the effect using a microscopic solenoid. This paper provides a simplified explanation of the Aharonov-Bohm prediction using a path-integral approach that is suitable for the advanced undergraduate. 1.
The magnetic Aharonov-Bohm effect shows that charged particles may be affected by the vector potential in regions without any electric or magnetic fields [1]. The Aharonov-Bohm effect was experimentally confirmed [2-3] and has been found in many situations [4-6]. A common explanation is based on quantum mechanics, which states that the wavefunctions associated with the charges will accumulate a phase shift due to the vector potential. However, consensus about its nature and interpretation has not been achieved [7-14]. We here propose a simple but reasonable interpretation based on the theory for electromagnetic radiation and couplings [15]. The energy associated with a pulse radiator is divided into a Coulomb-velocity energy and a radiative energy, together with a macroscopic Schott energy accounting for the energy exchange between them. All these energies are expressed with terms including the potentials, so are the mutual coupling energies. There exists a force acting on the moving charges even though the fields completely vanish. This force makes the charges pass through the magnetic solenoid in different velocity with different path length, causing a phase shift the same as that obtained with quantum mechanics. The theory is originally aimed for providing an interpretation for electromagnetic radiation and mutual coupling. It is derived directly from the Maxwell theory with no modification but only substitution and reorganization.
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.
In the well-known Aharonov-Bohm and Ehrenberg-Siday effects, the diffraction envelope of an electron is not affected by Lorentz forces in a region of zero magnetic field; instead, a phase difference—created by the magnetic vector potential—shifts the constructive- and destructive-interference fringe positions when electron wavefunctions are overlapped. This shift in fringe positions has been previously derived as resulting from potential-induced phase differences, without being clear on the physical mechanism behind it. In this paper, we show that the de Broglie wavelength of the electron is changed locally by its interaction with the vector potential. The vector potential thus acts as a quantum “phase plate,” changing the phase difference between interfering electron wavefunctions in a non-dispersive, gauge-invariant manner.