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Force and Impulse from an Aharonov-Bohm Flux Line
Keating, J. P.; Robbins, J. M.
Keyword(s): Aharonov-Bohm effect
Abstract: We calculate the force operator for a charged particle in the field of an Aharonov-Bohm flux line. Formally this is the Lorentz force, with the magnetic field operator modified to include quantum corrections due to anomalous commutation relations. For stationary states, the magnitude of the force is proportional to the product of the wavenumber k with the amplitudes of the 'pinioned' components, the two angular momentum components whose azimuthal quantum numbers are closest to the flux parameter. The direction of the force depends on the relative phase of the pinioned components. For paraxial beams, the transverse component of our expression gives an exact version of Shelankov's formula [Shelankov A 1998 Europhys. Lett. 43, 623-8], while the longitudinal component gives the force along the beam. Nonstationary states are treated by integrating the force operator in time to obtain the impulse operator. Expectation values of the impulse are calculated for two kinds of wavepackets. For slow wavepackets, which spread faster than they move, the impulse is inversely proportional to the distance from the flux line. For fast wavepackets, which spread only negligibly before their closest approach to the flux line, the impulse is proportional to the probability density transverse to the incident direction evaluated at the flux line. In this case, the transverse component of the impulse gives a wavepacket analogue of Shelankov's formula. The direction of the impulse for both kinds of wavepackets is flux dependent. We give two derivations of the force and impulse operators, the first a simple derivation based on formal arguments, and the second a rigorous calculation of wavepacket expectation values. We also show that the same expressions for the force and impulse are obtained if the flux line is enclosed in an impenetrable cylinder, or distributed uniformly over a flux cylinder, in the limit that the radius of the cylinder goes to zero.
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