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Physical Review A

Aharonov-Bohm scattering of wave packets by Maxwell coils

M. Jursa, Peter Kasperkovitz

Published May 1, 1993• 1 citation
Publisher

Abstract

Maxwell coils are solenoids of spheroidal shape which produce a constant magnetic field inside the coil; at large distances the field of a Maxwell coil becomes that of a magnetic dipole. The current in systems consisting of several confocal Maxwell coils can be adjusted in such a way that the magnetic field inside the smallest coil is constant and the back flux is confined to the interior of the largest coil. Quantum-mechanical scattering of charged particles by Maxwell coils, or systems thereof, is first formulated as a three-dimensional problem and then treated within a two-dimensional approximation. The evolution in time of wave packets, which are Gaussian wave packets in the beginning, is calculated numerically. In all cases an Aharonov-Bohm effect is observed in the angular distribution of the scattered wave packets. The back flux results only in an additional deflection of the packets, which corresponds to the classical deflection caused by the Lorentz force.

DOI
10.1103/physreva.47.3602

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Maxwell coils are solenoids of spheroidal shape which produce a constant magnetic field inside the coil; at large distances the field of a Maxwell coil becomes that of a magnetic dipole. The current in systems consisting of several confocal Maxwell coils can be adjusted in such a way that the magnetic field inside the smallest coil is constant and the back flux is confined to the interior of the largest coil. Quantum-mechanical scattering of charged particles by Maxwell coils, or systems thereof, is first formulated as a three-dimensional problem and then treated within a two-dimensional approximation. The evolution in time of wave packets, which are Gaussian wave packets in the beginning, is calculated numerically. In all cases an Aharonov-Bohm effect is observed in the angular distribution of the scattered wave packets. The back flux results only in an additional deflection of the packets, which corresponds to the classical deflection caused by the Lorentz force.
The quantum-mechanical Aharonov-Bohm effect in the diffraction of charged particles by a toroidal solenoid containing a magnetic field is investigated. The total and differential elastic scattering cross sections depend on the magnetic flux inside the solenoid, even in the presence of a “black” ring-shaped screen which prevents charged particles from entering the region where the magnetic field is localized. Relations describing the momentum-transfer cross section for the elastic scattering of charged particles by a toroidal solenoid are obtained in the eikonal approximation and in a unitary model of scattering with a sharp jump in the partial amplitudes. The momentum-transfer scattering cross section is proportional to the average transfer of the longitudinal momentum of the scattered particle and can be expressed in terms of a force operator. It is shown that in the absence of a screen the momentum-transfer scattering cross section of toroidal solenoid is indeed determined only by the part of the incident beam that intersects the inner region of the toroidal solenoid, where the magnetic field intensity and, therefore, the Lorentz force are nonzero. At the same time, the momentum-transfer cross section for the scattering of charged particles by a toroidal solenoid covered by a “black” ring-shaped screen does not depend on the magnetic flux inside the solenoid and is identical to the momentum-transfer cross section for diffraction by the same screen. The contribution from scattering by an opening in the screen, which depends on the magnetic flux, is completely compensated by the contribution of the interference of the scattering amplitudes of the opening and the “black” screen.
The Aharonov-Bohm (AB) scattering of charged particles by the magnetic field of an infinitely long and infinitely thin solenoid (magnetic string) in an absorbing medium is studied. We discuss the partial-wave approach to this problem and show that the standard partial-wave method can be adjusted to this case. The effect of absorption leads to oscillations of the AB cross section. Based on this we investigate the scattering of neutral atoms with induced electric dipole moments by a charge wire of finite radius, which is placed in a uniform magnetic field. The physical realistic and practically important case that all atoms that collide with the wire are totally absorbed at its surface is studied in detail. The dominating terms of the scattering amplitude are evaluated analytically for different physical constellations. The remaining terms are written in a form suitable for a numerical computation. We show that if the magnetic field is absent, the absorbing charged wire causes oscillations of the cross section. In the presence of the magnetic field the cross section increases and the dominating Aharonov-Bohm peak appears in the forward direction, suppressing the oscillations.
The Aharonov--Bohm effect in a model system described by the generalized Schrödinger equation is considered. The scattering cross section is calculated in the standard formulation: an electron beam impinges on a long impenetrable solenoid encompassing a magnetic field. It is shown that the incident wave is scattered regardless of whether the magnetic flux through the solenoid is an integer of flux quanta whereby the scattering cross section becomes a continuously nonzero function of the total magnetic flux. The problem may find its application in investigations of the soliton-magnon interaction in the low-dimensional magnetism.
In this paper we consider the scattering of charged particles by magnetic strings for several configurations of the incident state and of the enclosed flux. We analyze the scattering of a plane wave by an infinite magnetic string and show that the scattering contribution arises from flux-dependent phase shifts followed by quantum diffusion. Then we report that the amplitude for the scattering of a wave packet of finite extension by a magnetic string becomes exponentially small as the width of the packet goes to zero. At the same time the phase at the center of the incident wave packet is progressively shifted by an amount proportional to the enclosed flux, as predicted on the basis of the quasiclassical approximation. We consider further the scattering by a pair of parallel strings carrying opposite fluxes and show that the probability pattern is affected by the magnetic fluxes mainly in the vicinity of a finite surface spanning the two strings. We finally discuss the relevance for the Aharonov-Bohm effect of the multivalued gauge transformations which eliminate the potentials from the field-free region. The results reported in this paper confirm the reality of the quantum effects of remote fluxes, and prove that the state of the electromagnetic field is not completely determined by the local knowledge of the field strengths.
We study the Aharonov–Bohm (AB) effect through resonances for magnetic scattering in two dimensions. The scattering system consists of three scatterers, one bounded obstacle, and two scalar potentials with compact supports at large separation, where the obstacle is placed between two supports and the support of the magnetic field is completely shielded by the obstacle. The field does not influence particles from a classical mechanical point of view, but quantum particles are influenced by the corresponding vector potential which does not necessarily vanish outside the obstacle. This quantum phenomenon is called the AB effect. The resonances are shown to be generated near the real axis by the trajectories oscillating between two supports of the scalar potentials as the distances between the three scatterers go to infinity. The location is described in terms of the backward amplitudes for scattering by each of the scalar potentials and by the obstacle, and it depends heavily on the magnetic flux of the field.