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Journal of Physics A Mathematical and General

Time-dependent Aharonov-Bohm scattering

M. Jursa, Peter Kasperkovitz

Published Apr 7, 1993• 0 citations
Publisher

Abstract

Scattering of Gaussian wavepackets by impenetrable solenoids is studied using both analytical and numerical methods. A formula for the asymptotic (t to infinity ) angular distribution of the scattered packet is derived and used to discuss the physical meaning of cross sections, the optical theorem, and the classical limit. Angular distributions obtained from this formula are found to be in good agreement with angular distributions calculated from numerical solutions of the Schrodinger equation.

DOI
10.1088/0305-4470/26/7/028

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Scattering of Gaussian wavepackets by impenetrable solenoids is studied using both analytical and numerical methods. A formula for the asymptotic (t to infinity ) angular distribution of the scattered packet is derived and used to discuss the physical meaning of cross sections, the optical theorem, and the classical limit. Angular distributions obtained from this formula are found to be in good agreement with angular distributions calculated from numerical solutions of the Schrodinger equation.
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.
The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.
We briefly review the scattering theory for the Schrödinger operator related to the Aharonov-Bohm effect, by exhibiting some explicitly solvable models; an infinitesimally thin solenoid in mathbb{R}^{2}, two infinitesimally thin solenoids with the quantized magnetic fluxes in mathbb{R}^{2}, and an infinitesimally thin ring solenoid with the quantized magnetic flux in mathbb{R}^{3}. We give explicit formulas for the incoming wave solution and the scattering amplitude for these models in terms of some special functions, with the graphs obtained by a numerical method. We also review known high-energy asymptotic formulas related to these examples.
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.
Abstract A uniform asymptotic formula for the Aharonov–Bohm wavefield (that of a plane quantum wave scattered by a thin straight solenoid) far away from the solenoid is obtained in a direct way. Actually quite good accuracy is achieved even down to one wavelength away. The error is numerically of order radius^(−3/2) for all values of polar angle, including directly forwards. Several previous formulas, uniform and otherwise, for the far field limit exist in the literature. All contain the essential ingredient: the Fresnel integral (complex error function), but ordinarily the error in these formulas is of order radius^(−1/2) in the forwards direction where the Fresnel contribution is most important.