Semiclassical Asymptotics of the Aharonov‐Bohm Interference Process

Abstract

We systematically derive the semiclassical limit of a charged particle's motion in the presence of an infinitely long and infinitesimally thin solenoid carrying magnetic flux. Our limit establishes the connection of the particle's quantum mechanical canonical angular momentum to the latter's classical counterpart. A picture of Aharonov-Bohm interference of two half-waves acquiring Dirac's magnetic phase when passing on either side of the solenoid naturally emerges from the quantum propagator. The resulting interference pattern is fully determined by the ratio of the angular part of Hamilton's principal function to Planck's constant, and the wave interference smoothes out discontinuities in the semiclassical propagator which is recovered in the limit when the above ratio diverges. We discuss the relation of our results to the whirling-wave representation of the exact propagator, and to previous approaches on the system's asymptotics.

Forward-smooth high-order uniform Aharonov–Bohm asymptotics

“…represents the universal scaling law for the AB wave [15], which, as is well understood, it shares with edge diffraction [3]. Note that as a function of θ this is smooth (all derivatives continuous) in the forward direction θ=0: the discontinuity in the pole contribution (2.4) has been cancelled by the discontinuity in (2.13).…”

“…Moreover, and importantly for the present paper, the AB wave has been approximated in terms of Fresnel integrals (complex error functions) [3,[12][13][14][15], originating in an exact representation of the AB wave for half-integer flux [1]. Such approximations are physically intuitive because there is a gauge in which the magnetic flux line is the edge of a phasechanging half-plane of vector potential, inspiring analogies with edge diffraction and the Cornu spiral.…”