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arXiv:hep-th/9509003 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Scattering amplitude in the Aharonov-Bohm gauge field

Paola Giacconi A, Fabio Maltoni B, Roberto Soldati

Published Jan 15, 1996• 17 citations
arXivHTMLPDF

Abstract

A general expression for the scattering amplitude of nonrelativistic spinless particles in the Aharonov-Bohm gauge potential is obtained within the time-independent formalism. The result is valid also in the forward direction as well as for any choice of the boundary conditions on the wave function at the flux tube position. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevd.53.952

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Aharonov-Bohm scattering amplitude

C. R. Hägen

The two principal approaches to the quantum-mechanical calculation of scattering by flux tubes in two spatial dimensions are reviewed and corrections made where needed. It is noted that even when appropriate care is taken the standard phase-shift method fails to agree with the (corrected) Aharonov-Bohm result. The discrepancy is resolved by noting that the former approach requires an improper interchange of a limit and a summation. That interchange is justified only when the phase shifts go to zero sufficiently rapidly as the angular momentum becomes arbitrarily large, a condition which fails to hold in this particular scattering problem.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1990•63 citations
A general expression for the scattering amplitude of nonrelativistic spinless particles in the Aharonov-Bohm gauge potential is obtained within the time-independent formalism. The result is valid also in the forward direction as well as for any choice of the boundary conditions on the wave function at the flux tube position. \textcopyright{} 1996 The American Physical Society.
The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits.
The Born scattering amplitude of a non-relativistic spin one-half particle in an Aharonov-Bohm potential is calculated up to second order. It is demonstrated that perturbation theory works well for this model, in contrast with the case of scalar particles, thanks to the spin-magnetic moment interaction term. The first order amplitude is shown to coincide with the exact amplitude when expanded to the same order; the second order amplitude is finite and null. The polarized scattering cross section is found to be different from the unpolarized one only if the incident particle has a spin component perpendicular to the flux tube.
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 two principal approaches to the quantum-mechanical calculation of scattering by flux tubes in two spatial dimensions are reviewed and corrections made where needed. It is noted that even when appropriate care is taken the standard phase-shift method fails to agree with the (corrected) Aharonov-Bohm result. The discrepancy is resolved by noting that the former approach requires an improper interchange of a limit and a summation. That interchange is justified only when the phase shifts go to zero sufficiently rapidly as the angular momentum becomes arbitrarily large, a condition which fails to hold in this particular scattering problem.