Aharonov-Bohm scattering of particles with spin

Abstract

The scattering of relativistic spin-one-half particles in an Aharonov-Bohrn potential is considered. It is shown that earlier approaches to this problem have neglected a crucial delta-function contribution to the potential. By formulating the problem with a source of finite radius which is then allowed to go to zero, it is established that it is the delta function alone that causes solutions that are singular at the origin to become relevant. The changes in the amplitude which arise from the inclusion of spin are seen to modify the cross section for the case of polarized beams. Finally, the calculated Aharonov-Bohm amplitude is shown to describe the scattering of particles with arbitrary spin in the c~limit. PACS numbers: 03.65.BzThe Aharonov-Bohm effect' has long been recognized for its crucial role in demonstrating the importance of potentials in quantum mechanics. More recently, interest in this topic has been stimulated by the considerable effort currently being expended on the study of (2+ 1)-dimensional models in both superconductivity and particle theory. With the increased application of the results of Ref. 1 to other problems (e.g. , Ref.2) it has become important that certain limitations on the original Aharonov-Bohm calculation be removed. This has been accomplished in a recent series of papers.A concern which has been increasingly addressed of late has to do with the question of how the inclusion of spin modifies the results of Ref. l. In particular, Alford and Wilczek have applied their calculations for Dirac particles to the interaction of cosmic strings with matter. They assert that the scattering amplitude is unaff'ected by the spin, a result which certainly follows if one accepts their requirement that the upper component of the two-component spinor be regular at the origin. On the other hand, Gerbert and Jackiw have suggested a more general boundary condition which introduces a new parameter into the calculation. This has been applied to the question considered in Ref. 5 with quite different results being obtained. This paper approaches the same problem by attempting to infer the behavior of the wave function at the origin in terms of the underlying physics.One begins by writing the Dirac equation for a particle of mass M which, in terms of two-component spinors P', 1S Etir [MP+Py" II]y where the matrices P and Py; are conveniently defined in terms of the Pauli spin matrices as p 03 py; = (o,sa2) and s is twice the spin value (+1 for spin "up" and -1 for spin "down"). The form (1) follows most simply by using the decoupling of the usual four-component Dirac equation in the absence of a third spatial coordinate into two uncoupled two-component equations for s =+ 1 and where the potential A, is related to the magnetic field in the usual way H VxA.(2) Since one is generally interested in the situation in which H is restricted to a flux tube of zero radius, it is conventional to write eH -(a/r )b (r ),with r being the two-dimensional radius vector. In the Coulomb gauge this yields for 8; the result ...

Scattering of spin-polarized electron in an Aharonov–Bohm potential

“…Solutions (25) and (27) differ from those obtained in [6]. This is because the magnetic field is equal to zero in the range r < R in the model used in [6]. In the range r > R the linearly-independent solutions for F l (r) are given by…”

“…(20), up to the replacement of ϕ by ϕ + π, coincides with the expression for the cross section of polarized beams given in [6]. The difference is due to the fact that the scattering angle here is measured from the positive x-axis, whereas in [6] it is measured from the negative x-axis.…”

“…In this appendix we shall demonstrate that the results of [4,6] for the scattering of relativistic spinone-half electrons in 2 + 1 dimensions and the corresponding results presented here for the scattering of nonrelativistic spin-1/2 electrons in 3 + 1 dimensions coincide. For this we follow the approach of [6] by introducing the particle spin in the two-component Dirac equation as a new parameter.…”

“…The scattering of spin-polarized fermions in an Aharonov-Bohm potential was first considered in 2+1 dimensions by Hagen in [6]. The particle spin in [6] is artificially introduced into the two-component Dirac equation as a new parameter (see, also, [7]).…”

“…The particle spin in [6] is artificially introduced into the two-component Dirac equation as a new parameter (see, also, [7]). The term including this new parameter appears in the form of an additional delta-function interaction of spin with magnetic field in the Dirac equation.…”