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arXiv:cond-mat/0212146 · Journal of Physics A Mathematical and General

Force and impulse from an Aharonov-Bohm flux line

Jonathan P. Keating, Jonathan M. Robbins

Published Jan 18, 2001• 14 citations

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 α. 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 wavepacket. 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 wavepacket 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.

DOI: 10.1088/0305-4470/34/4/308

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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 α. 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 wavepacket. 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 wavepacket 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.

Using a paraxial analysis, Shelankov (1998 Europhys. Lett. 43 623) has shown that charged particles in a beam of small angular width 1/ w , aimed at a magnetic flux line with quantum flux , are deflected through an angle D proportional to sin(2 )/ w , vanishing in the classical limit and also vanishing if the incident beam has zero intensity at the flux line. These properties are confirmed by numerical calculations based on an exact solution of the Schrödinger equation, and the paraxial wavefunction is obtained as an asymptotic approximation for large w . Paraxial theory suggests that the same deflection will occur for a light beam reflected by a mirror containing a step of height . A theory of this optical phenomenon, based on an exact solution (for which D is not periodic in ), shows that convergence to the paraxial D is fast for <<1, and slow for near ½.

The controversial question of the transverse force exerted by the Aharonov-Bohm (AB) magnetic-flux line is reconsidered with the help of a new paraxial solution to the AB-scattering problem. It is shown that despite the left-right symmetry in the AB scattering cross-section, a beam of finite width is deflected by the AB-line as if by the “Lorentz” force. The asymmetry and the magnetic force originate from the quantum interference in the forward direction within the angular size of the incident wave.

We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.

The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux,

Δφ_{\rm AB} = \frac{1}{T} \int_0^T e Φ(t) \, dt

, with the total phase shift, including kinetic contributions, equaling

e Φ(0)

. For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius

R

, the AB phase shift is also proportional to the time average of the enclosed flux

Φ_{\rm enc}(R,t)

, with the total phase shift depending only on the initial enclosed flux

e Φ_{\rm enc}(R,0)

; for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.

The differential cross section for the scattering of an electron by an impenetrable, infinitely long magnetic flux line with return magnetic flux uniformly distributed on the surface of a cylinder of radius a is calculated. The scattering is due to the Lorentz force from the magnetic field of the return flux. In the limit as the radius of the return flux goes to infinity (a\ensuremath{\rightarrow}\ensuremath{\infty}), the differential scattering cross section is the same as that obtained by Aharonov and Bohm (AB) for the scattering of an electron by an impenetrable, infinitely long magnetic flux line without return flux. The AB scattering cross section is, however, attributed to the effect of the vector potential in a field-free region so no force acts on the electron.