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Brazilian Journal of Physics

On the delta function normalization of the wave functions of the aharonov-bohm scattering of a dirac particle

Vanilse S. Araujo, Francisco Antônio Bezerra Coutinho, J. Fernando Perez

Published Jun 1, 2002• 1 citation
Publisher

Abstract

In a previous paper, we found the most general boundary conditions for the Aharonov -Bohm scattering of a Dirac particle. We found the resulting wave functions but we did not worry about delta normalizing them. As is well know, in practice, it is not easy to evaluate the diverging integrals occurring in the process. The purpose of this paper is to evaluate those integrals and present the resulting delta normalized eigenfunctions.

DOI
10.1590/s0103-97332002000300027

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In a previous paper, we found the most general boundary conditions for the Aharonov -Bohm scattering of a Dirac particle. We found the resulting wave functions but we did not worry about delta normalizing them. As is well know, in practice, it is not easy to evaluate the diverging integrals occurring in the process. The purpose of this paper is to evaluate those integrals and present the resulting delta normalized eigenfunctions.
Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.
The Aharonov-Bohm scattering of massive spin-one particles is computed. Starting from the relativistic wave equation for a charged, massive spin-one particle, a reduction to two spatial dimensions is made, consistent with the standard Aharonov-Bohm (AB) geometry. It is found that the problem consists of the independent scattering of the three vector-meson spins. While the amplitude for the spin-0 mode is trivially identical to the AB amplitude, the same result is obtained for the spin-up and spin-down components, in sharp contrast to the corresponding result for the spin-1/2 AB amplitude. This can be traced to the fact that the Hamiltonian is unbounded as a consequence of the presence of a \ensuremath{\delta}-function term in the potential. In the Galilean limit it is found that the amplitude of the nonzero-spin components is identical to the spin-1/2 amplitude in that limit, in agreement with a general claim made earlier by one of the authors. The Galilean limit of the spin-one AB cross section is found to be modified and calculations for some pure-state polarizations are presented.
The scattering of relativistic spin-one-half particles in an Aharonov-Bohm 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 this 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=\ensuremath{\infty} limit.