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Physics of Atomic Nuclei

Anisotropic Aharonov-Bohm Effect

Riccardo Adami

Published Nov 1, 1998• 0 citations
OpenAlex

Abstract

Results obtained for the nonrelativistic Aharonov-Bohm effect, together with contact interactions, are presented and discussed. These results are compared with those concerning the spectrum and the scattering amplitude for two interactions separately. We use the formalism of first quantization and refer, for technical tools, to the von Neumann-Krein theory for the construction of self-adjoint extensions of symmetric operators and to scattering theory. Despite the rotational invariance of both magnetic and contact interactions, their superposition can give rise to an anisotropy effect

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Results obtained for the nonrelativistic Aharonov-Bohm effect, together with contact interactions, are presented and discussed. These results are compared with those concerning the spectrum and the scattering amplitude for two interactions separately. We use the formalism of first quantization and refer, for technical tools, to the von Neumann-Krein theory for the construction of self-adjoint extensions of symmetric operators and to scattering theory. Despite the rotational invariance of both magnetic and contact interactions, their superposition can give rise to an anisotropy effect
The Aharonov-Bohm effect including spin-noncommutative effects is considered. At linear order in $\ensuremath{\theta}$, the magnetic field is gauge invariant although spatially strongly anisotropic. Despite this anisotropy, the Schr\"odinger-Pauli equation is separable through successive unitary transformations and the exact solution is found. The scattering amplitude is calculated and compared with the usual case. In the noncommutative Aharonov-Bohm case the differential cross section is independent of $\ensuremath{\theta}$.
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.
It is shown that there is an exact equivalence between the Aharonov-Bohm effect for spin-1/2 particles and the Aharonov-Casher effect. The demonstration of this precise relationship between the two is seen furthermore to be independent of whether relativistic or nonrelativistic kinematics are used. The only remaining substantive distinction between the two effects may well be the fact that the scattering cross section for polarized beams has a considerably greater sturcture in the Aharonov-Casher case despite the mathematical equivalence of the scattering amplitudes for the two effects.