Calculation of the Aharonov-Bohm wave function

Abstract

A calculation of the Aharonov-Bohm wave function is presented. The result is a series of confluent hypergeometric functions which is finite at the forward direction.

Aharonov-Bohm scattering on a cone

“…These calculations are rather lengthy, although straightforward, and can be reproduced after following the explanations given in [9,18]. Using these results we can consider the limit a(αφ)ρ → ∞, which corresponds to long distances from the scattering center away from the classical scattering directions θ ± .…”

“…with A(r) given by (2) and the metric g ij corresponding to the line element (1). Based on previous results [9,18] we expect the wave function to exhibit two privileged scattering directions that depend only on the mass density of the flux line, and superimposed on each of these directions AB-like phases induced by the magnetic flux . As the separate cases of planar gravitational scattering and AB scattering are known, all we have to do is combine the propagators analyzed in the above mentioned references.…”

“…and we can consider one of the two integrals, say I 1 , and treat the other one by analogy. Following the lines proposed in [9,18] the integral I 1 can be expanded as a hypergeometric series that, as we shall see, has a finite discontinuity at the classical scattering directions:…”

“…The propagator (3) reduces to the AB propagator if α = 1 and to the Deser-Jackiw propagator[7] of planar gravitational scattering if ν = 0. As in[9,18] the propagator (3) leads to an integral than can be calculated by a saddle-point approximation for all scattered directions except the classical ones. The integral in question is I(ρ, φ, α, ν) = ∞ −∞ dy exp iρ cosh y + 1 α {ν}y e −i{ν} π α 1 − e −iφ+ 1 α (y−iπ) − e i{ν} π α 1 − e −iφ+ 1 α (y+iπ) .…”

The Aharonov–Bohm problem revisited

“…(A12) takes the usual Maxwell form. If, on the other hand, κ and λ are set to zero, the equations show a symmetry under duality between J, A and K, G: 13) and an invariance under both an electric and a magnetic gauge transformation: A → A + grad φ, G → G + grad χ (A.14)…”