Aharonov-Bohm scattering of a localized wave packet: Analysis of the forward direction

Abstract

The Aharonov-Bohm scattering of a localized wave packet is considered. A careful analysis of the forward direction points out new results: according to the time-dependent solution obtained by means of the asymptotic representation for the propagator (kernel), a phenomenon of auto-interference occurs along the forward direction, where, also, the probability density current is evaluated and found finite.

Calculation of the Aharonov-Bohm wave function

“…Before proceeding to the evaluation of the integral, we demonstrate that it also appears in the AB propagator. It has been shown in [8] that this propagator can be written as a Bessel series:…”

“…One of the few genuine time-dependent analyses of the AB effect has been given by Stelitano in a recent work [8]. This author decomposes the propagator in positive, negative and zero angular momentum components, and calculates each term independently in the asymptotic limit, following a procedure similar to the original method of Aharonov and Bohm [1].…”

Scattering amplitude in the Aharonov-Bohm gauge field

“…[9] to develop the time dependent approach which, however, appears to be more involved than the time independent one, at least in our opinion. Furthermore, in that paper an explicit expression is not given for the forward scattering amplitude, but only arguments concerning the scattering of some wave packets.…”

“…(2.22) could also be reinterpreted in the framework of the time dependent approach, as proposed in ref. [9].…”

“…here the wave functions φ AB k (r, ϕ) do not correspond to eigenfunctions of the free Hamiltonian H 0 = p 2 /2m, owing to the long range nature of the AB potential; they are given instead by the phase factors [1], [9] for the exact solution of eq. (2.6), or from a contour integral representation of the Bessel functions [4], [5].…”