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

Abstract

The authors 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 Phi . Defining the quantum flux parameter by alpha =q Phi /h, they show that for the case R=0 the wave psi AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to alpha . When alpha passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but they devise an analogue where surface waves on water encounter an irrotational 'bathtub' vortex; in this case alpha depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. psi AB is an unusual function in which incident and scattered waves cannot be clearly separated in all asymptotic directions; they discuss its properties using a new asymptotic method.