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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Aharonov-Bohm effect as a scattering event

Yakir Aharonov, C. K. Au, E. C. Lerner, J.-Q. Liang

Published May 15, 1984• 58 citations
Publisher

Abstract

The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric $(m=0)$ partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.

DOI
10.1103/physrevd.29.2396

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American Journal of Physics•Nov 1, 1978•43 citations
The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric (m=0)(m=0)(m=0) partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.
A method of partial-wave analysis is used to extend the treatment of Aharonov and Bohm (1959) for the scattering from an unscreened solenoid of infinitesimal radius to include screening and non-zero radius. The method yields a scattering amplitude which bears a formal resemblance to that appropriate to modified Coulomb scattering. The scattering amplitude and the momentum transfer cross section retain a dependence on the enclosed flux as the screening barrier becomes infinite, resulting in a component of force, periodic in the enclosed flux, being exerted on the barrier. The resulting implications concerning the locality of electromagnetic interactions are discussed. The force is shown to persist when the conditions of strict impenetrability are relaxed and the possibility of directly observing this quantum force is examined.
The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.
The Aharonov--Bohm effect in a model system described by the generalized Schrödinger equation is considered. The scattering cross section is calculated in the standard formulation: an electron beam impinges on a long impenetrable solenoid encompassing a magnetic field. It is shown that the incident wave is scattered regardless of whether the magnetic flux through the solenoid is an integer of flux quanta whereby the scattering cross section becomes a continuously nonzero function of the total magnetic flux. The problem may find its application in investigations of the soliton-magnon interaction in the low-dimensional magnetism.