Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9809319 · Europhysics Letters (EPL)

Magnetic force exerted by the Aharonov-Bohm line

Andrei L. Shelankov

Published Sep 15, 1998• 82 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/epl/i1998-00408-4

Related papers

Some aspects of the Aharonov-Bohm effect

Walter C. Henneberger

It is shown that the Aharonov-Bohm scattering cross section diverges. This presents a problem since the range of the scattering force (Lorentz force) is finite. The cross section found by Aharonov and Bohm (AB) results whether one takes an incident wave function corresponding to particles moving in the incident direction, as was done by AB, or just takes the incident state to be a plane wave. The mechanism by which a localized force results in an infinite cross section is not clear. This is an apparent contradiction to recent proofs that electrodynamics is a local theory.

Physical Review A•Oct 1, 1980•38 citations

Force and impulse from an Aharonov-Bohm flux line

Jonathan P. Keating, Jonathan M. Robbins

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.

arXiv:cond-mat/0212146•Journal of Physics A Mathematical and General•Jan 18, 2001•14 citations

Aharonov-Bohm beam deflection: Shelankov's formula, exact solution, asymptotics and an optical analogue

Michael Berry

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.

Journal of Physics A Mathematical and General•Jul 20, 1999•32 citations

Scattering from a magnetic flux line due to the Lorentz force of the return flux

Donald H. Kobe, J.-Q. Liang

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.

Physical Review A•Feb 1, 1988•6 citations

Aharonov-Bohm scattering on two antiparallel flux lines of the same magnitude-without return flux

Gu Zhi-Yu, Qian Shang-Wu

The problem of Aharonov-Bohm (AB) scattering on two parallel flux lines of arbitrary magnitude is solved exactly; the expression for the scattering cross-section in the region of geometric shadow is derived. It is shown that in the particular case of two antiparallel lines of the same magnetic flux, though the return flux does not exist, the AB scattering still exists.

Journal of Physics A Mathematical and General•Sep 7, 1993•2 citations

A possible interplay between electron beams and magnetic fluxes in the Aharonov-Bohm effect

Ruifeng Wang

Most studies on the magnetic Aharonov-Bohm (A-B) effect focus on the action exerted by the magnetic flux on the electron beam, but neglect the back-action exerted by the electron beam on the magnetic flux. This paper focuses on the latter, which is the electromotive force Δ U across the solenoid induced by the time-dependent magnetic field of the electron beam. Based on the back-action analysis, we observe that the magnetic A-B effect arises owing to the interaction energy between the magnetic field of the electron beam and the magnetic field of the solenoid. We also demonstrate that the interpretation attributing the magnetic A-B effect to the vector potential violates the uncertainty principle.

Frontiers of Physics•Apr 17, 2015•10 citations
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.
It is shown that the Aharonov-Bohm scattering cross section diverges. This presents a problem since the range of the scattering force (Lorentz force) is finite. The cross section found by Aharonov and Bohm (AB) results whether one takes an incident wave function corresponding to particles moving in the incident direction, as was done by AB, or just takes the incident state to be a plane wave. The mechanism by which a localized force results in an infinite cross section is not clear. This is an apparent contradiction to recent proofs that electrodynamics is a local theory.
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 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.
Most studies on the magnetic Aharonov-Bohm (A-B) effect focus on the action exerted by the magnetic flux on the electron beam, but neglect the back-action exerted by the electron beam on the magnetic flux. This paper focuses on the latter, which is the electromotive force Δ U across the solenoid induced by the time-dependent magnetic field of the electron beam. Based on the back-action analysis, we observe that the magnetic A-B effect arises owing to the interaction energy between the magnetic field of the electron beam and the magnetic field of the solenoid. We also demonstrate that the interpretation attributing the magnetic A-B effect to the vector potential violates the uncertainty principle.