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International Journal of Modern Physics C

MONTE CARLO EVALUATION OF THE AHARONOV-BOHM EFFECT

George C. John, Vijay A. Singh

Published Feb 1, 1995• 1 citation
Publisher

Abstract

The electron propagator in the Aharonov-Bohm effect is investigated using the Feynman path integral formalism. The calculation of the propagator is effected using a variation of the Metropolis Monte Carlo algorithm. Unlike “exact” calculations, our approach permits us to include a nonvanishing solenoid radius. We investigate the dependence of the resulting interference pattern on the magnetic field as well as the solenoid radius. Our results agree with the exact case in the limit of an infinitesimally small solenoid radius.

DOI
10.1142/s012918319500006x

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Journal of the Optical Society of America B•Dec 1, 2000•1 citation
The electron propagator in the Aharonov-Bohm effect is investigated using the Feynman path integral formalism. The calculation of the propagator is effected using a variation of the Metropolis Monte Carlo algorithm. Unlike “exact” calculations, our approach permits us to include a nonvanishing solenoid radius. We investigate the dependence of the resulting interference pattern on the magnetic field as well as the solenoid radius. Our results agree with the exact case in the limit of an infinitesimally small solenoid radius.
The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.
In the usual Aharonov-Bohm effect, representing an especial case of the Berry phase phenomenon, classical magnetic field within long and thin solenoid (or classical vector potential of the electromagnetic field without this solenoid) causes phase difference and interference shape translation of the quantum propagating (external) electron. In this work we consider a variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. In this way we obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept. As it is well-known in the usual experimental arrangement corresponding to the AharonovBohm effect [1]-[4] (representing an especial case of the Berry phase phenomenon [4], [5]) behind a diaphragm with two slits there is a long and very thin solenoid (placed perpendicularly, i.e. in the z-axis direction, to the x-0-y plane of the propagation of the (external, without solenoid
The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric (m=0)(m=0)(m=0) partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.
The Aharonov–Bohm effect is formulated in terms of a constrained path integral. The path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths. The interference terms are also calculated for an infinitely thin solenoid, which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment.
A method of partial-wave analysis is used to extend the treatment of Aharonov and Bohm (1959) for the scattering from an unscreened solenoid of infinitesimal radius to include screening and non-zero radius. The method yields a scattering amplitude which bears a formal resemblance to that appropriate to modified Coulomb scattering. The scattering amplitude and the momentum transfer cross section retain a dependence on the enclosed flux as the screening barrier becomes infinite, resulting in a component of force, periodic in the enclosed flux, being exerted on the barrier. The resulting implications concerning the locality of electromagnetic interactions are discussed. The force is shown to persist when the conditions of strict impenetrability are relaxed and the possibility of directly observing this quantum force is examined.
The Aharonov–Bohm solenoid effect is related to the inequality of the forces of action and reaction of two subsystems in a three-system configuration. During the motion of the electron around the solenoid there are momentum exchanges between the solenoid and the electromagnetic field that are produced by the electron and the solenoid. The integral of the momentum of interference over the trajectory of the electron, which is equal to the integral of the energy of interference over the corresponding time, gives the Aharonov–Bohm phase shift.