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

Fringing fields and criticisms of the Aharonov-Bohm effect

Harry J. Lipkin

Published Mar 15, 1981• 13 citations
Publisher

Abstract

A recent paper attributes the Aharonov-Bohm effect to the fringing field leaking out of a finite solenoid rather than to the magnetic flux in the excluded region. This argument is seen to be absurd for a very long solenoid where the effect is attributed to the tail of the electron wave function very far from the point of measurement, and is completely refuted by toroidal geometry with no fringing field.

DOI
10.1103/physrevd.23.1466

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A recent paper attributes the Aharonov-Bohm effect to the fringing field leaking out of a finite solenoid rather than to the magnetic flux in the excluded region. This argument is seen to be absurd for a very long solenoid where the effect is attributed to the tail of the electron wave function very far from the point of measurement, and is completely refuted by toroidal geometry with no fringing field.
Predictions made by Aharonov and Bohm concerning the effects of electromagnetic potentials on electrons in field-free regions of space, have been demonstrated in superconductors with the use of the flux produced by a very fine toroidal solenoid. The use of a toroidal geometry sidesteps the objections raised by Roy concerning the use of sources of flux with linear geometries. The use of superconductors excludes the possibility that stray electrons can pass through the flux-filled region of space to produce the observed phenomena.
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
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.
Three attempts for an explanation of the Aharonov–Bohm effect The Aharonov–Bohm (A–B) effect is an effect one finds in every quantum field theory book and this is so for a very good reason. The prediction and subsequent experimental verification of the effect have been crucial cornerstones in the history of physics because they suggested that the gauge potential, also known as the A μ field, might be interpreted as a real field, rather than just a mathematical artifact. Hence, ever since its discovery, physicists have taken it for granted that A μ does represent something at least as tangible as any matter field. However, when one examines these arguments more closely, one realizes that attributing the status of a really existent field to the A μ field is not as straightforward as was originally thought. But first things first: we begin with an account of the effect itself, and then attempt to give some explanation for it. The effect The setting for the A–B effect is very similar to the two-slit experiment, with just one difference: immediately beyond the two slits and in between them is a very fine and long solenoid, ideally infinitely long, producing a magnetic field that is confined entirely within the tube of the solenoid.
It is pointed out that accounts in the literature sometimes misinterpret the Aharonov-Bohm effect, involving the shift in interference pattern for electrons passing a long solenoid. The quantum description and experimental verification of the effect are reviewed, and it is emphasized that both theory and experiment indicate that there is no deflection of the average momentum of the electrons passing the solenoid. There is no average force on the electrons due to the solenoid. It is remarked that the Aharonov-Bohm effect involves the shift of double-slit interference fringes which is independent of the quantum unit of action ħ and yet allows a natural classical limit.
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.