Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue

Abstract

We study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux @. Defining the quantum flux parameter by a = q@/h, we show that for the case

Scattering amplitude in the Aharonov-Bohm gauge field

“…As already noticed, this point of view gives rise to the sum of a phase factor φ AB k = (2π) −1/2 exp{−i(kr cos ϕ + αϕ)} and a scattered wave ψ sc = (2πr) −1/2 f AB (k, ϕ) exp{ikr}, the amplitude corresponding to the well known AB scattering amplitude [1]. However, as already noticed, the above mentioned asymptotic form holds true for ϕ = lπ, l ∈ Z; in other words, the above decomposition of the exact wave function, in the large-r limit, strictly speaking loses its meaning in the backward and forward directions [5].…”

“…To our knowledge this point has been firstly discussed in a very clear and rigorous way in ref. [5]. In this paper it is shown that, strictly speaking, the celebrated formula for the differential cross section holds true outside two opposite narrow cones, with common vertices at the point-like flux tube position, which contain the backward and forward directions respectively.…”

“…In order to define the scattering amplitude, there basically are two attitudes in the Literature. In the first one [1], [5] the amplitude is obtained from the asymptotic behaviour of the exact regular scattering solution of eq. (2.6).…”

“…On the other hand, the alternative method described in ref. [5] to treat the problem, which makes use of a contour integral representation of the Bessel functions, is such that the separation between incident and scattered waves is no longer possible in the backward and forward directions; sic stantibus rebus a meaningful definition and eventual evaluation of the backward and forward scattering amplitudes are still lacking. There have been proposals [4], [6] to introduce some regularizations, in order to give a meaning to the sum over the partial phase shifts, which turns out to be meaningless as it stands.…”

Note on wavefront dislocation in surface water waves

“…Experimental observations in a neon discharge in two-dimensional space-time are reported by Krása [16]. A study of it in the Aharonov-Bohm effect is done by Berry [17]. An analysis for constructing a theory of wavefront dislocation using catastrophe theory is developed by Wright [18].…”