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Journal of Mathematical Physics

Motion of a charged particle in a periodic Aharonov-Bohm potential

G. E. Drăgănescu

Published Apr 1, 2006• 5 citations
Publisher

Abstract

In this paper we introduce a periodic Aharonov-Bohm potential, produced by two parallel infinite thin layers. The motion of a quantum charged particle without spin, under the action of this Aharonov-Bohm potential is examined. Finally we introduce a periodic Aharonov-Bohm potential, produced by a system of parallel solenoids and the motion of the charged particle in this field is also examined.

DOI
10.1063/1.2189199

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In this paper we introduce a periodic Aharonov-Bohm potential, produced by two parallel infinite thin layers. The motion of a quantum charged particle without spin, under the action of this Aharonov-Bohm potential is examined. Finally we introduce a periodic Aharonov-Bohm potential, produced by a system of parallel solenoids and the motion of the charged particle in this field is also examined.
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.
The authors establish the exact solution of the Dirac equation for a charged particle moving in the Coulomb field plus Aharonov-Bohm field plus Dirac monopole field by using the equation in two-dimensional complex space. On the basis of the found solution, the relativistic analogy of the Aharonov-Bohm effect for the system of a diogen atom existing in an infinite cylindrical solenoid is studied. In this case the authors obtain some surprising results. One of them is that the Aharonov-Bohm effect is absent for certain states.
The locality principle fulfillment in the Aharonov-Bohm (AB) effect is analyzed from the point of view of a self-sufficient potential formalism based on so-called gradient hypothesis in electrodynamics. The "magnetic" kind of AB effect is examined (as the quantum charged particle moves to an infinitely long solenoid with a permanent current), and no locality principle violation recognized if the gradient hypothesis is used. A conclusion is made that AB effect is no longer a physical and electrodynamic "paradox".
The Aharonov-Bohm effect is considered by most authors as a quantum effect, but a generally accepted explanation does not seem to be available. The phenomenon is studied here under the assumption that hypothetical electric dipole distributions configured by moving charges in the solenoid act on the electrons as test particles. The relative motions of the interacting charged particles introduce relativistic time dilations. The massless dipoles are postulated as part of an impact model that has recently been proposed to account for the far-reaching electrostatic forces between charged particles described by Coulomb's law. The model provides a quantitative explanation of the Aharonov-Bohm effect.
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.