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American Journal of Physics

Aharonov-Bohm effect in perturbation theory

Kay M. Purcell, Walter C. Henneberger

Published Dec 1, 1978• 16 citations
Publisher

Abstract

The Aharonov-Bohm effect is obtained in first-order perturbation theory. It is shown that the effect occurs only when the initial state is a superposition of eigenstates of Lz corresponding to eigenvalues having opposite sign. This corresponds to the two paths separated by the magnetic flux that are considered in usual discussions of the effect.

DOI
10.1119/1.11390

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The Aharonov-Bohm effect is obtained in first-order perturbation theory. It is shown that the effect occurs only when the initial state is a superposition of eigenstates of Lz corresponding to eigenvalues having opposite sign. This corresponds to the two paths separated by the magnetic flux that are considered in usual discussions of the effect.
The perturbation theory expansion of the Aharonov-Bohm scattering amplitude has previously been studied in the context of quantum mechanics for spin-0 and spin-1/2 particles as well as in Galilean covariant field theory. This problem is reconsidered in the framework of the model in which the flux line is considered to have a finite radius which is shrunk to zero at the end of the calculation. General agreement with earlier results is obtained but with the advantage of a treatment which unifies all the various subcases.
A fully quantum theory of the Aharonov-Bohm effect is presented. It is based on the interaction, mediated by virtual photon exchange, between the traveling electron and the atomic magnetic dipoles of an infinitely long permanent magnet. The calculation involves second-order time-dependent perturbation theory. As expected, the relative phase between the two states—corresponding to the two alternative paths—interfering on the screen agrees with that predicted in the usual theory. However our method may allow to get corrections, for instance due to the size and shape of the electron wave packet, the possible soft-photon emission or the excitation of the atoms in the magnet.
Results obtained for the nonrelativistic Aharonov-Bohm effect, together with contact interactions, are presented and discussed. These results are compared with those concerning the spectrum and the scattering amplitude for two interactions separately. We use the formalism of first quantization and refer, for technical tools, to the von Neumann-Krein theory for the construction of self-adjoint extensions of symmetric operators and to scattering theory. Despite the rotational invariance of both magnetic and contact interactions, their superposition can give rise to an anisotropy effect
We study a generalization of Aharonov-Bohm effect, the potential effect. The discussion is focused on field-free effects in simply connected region, which obviously can not have any local field-flux. Among the published discussions about this kind of effects, it is generally agreed that this kind of effects does not exist due to gauge invariance. However, there are also opinions that this effect is a trivial variation of Aharonov-Bohm effect and therefore there is no need to check its existence. To my knowledge, it has never been tested. My first goal here is to supply enough theoretical reason to motivate the experimental test of this effect. I start with an intuitive derivation, then I introduce a wave-front theory as a theoretical consideration. Logically, the existence of potential effect implies the existence of the AB effect, but not vice versa. The purpose of this paper is to provide a physical connection in the opposite direction.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.
The Aharonov-Bohm effect including spin-noncommutative effects is considered. At linear order in $\ensuremath{\theta}$, the magnetic field is gauge invariant although spatially strongly anisotropic. Despite this anisotropy, the Schr\"odinger-Pauli equation is separable through successive unitary transformations and the exact solution is found. The scattering amplitude is calculated and compared with the usual case. In the noncommutative Aharonov-Bohm case the differential cross section is independent of $\ensuremath{\theta}$.