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arXiv:math/0701224

The Aharonov-Bohm effect: Mathematical Aspects of the Quantum Flow

Luis Fernando Mello, Yuri Candido Ribeiro

Published Jan 8, 2007• 1 citation
arXivHTMLPDF

Abstract

This paper addresses the scattering of a beam of charged particles by an infinitely long magnetic string in the context of the hydrodynamical approach to quantum mechanics. The scattering is qualitatively analyzed by two approaches. In the first approach, the quantum flow is studied via a one-parameter family of complex potentials. In the second approach, the qualitative theory of planar differential equations is used to obtain a one-parameter family of Hamiltonian functions which determine the phase portraits of the systems.

DOI
10.48550/arxiv.math/0701224

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This paper addresses the scattering of a beam of charged particles by an infinitely long magnetic string in the context of the hydrodynamical approach to quantum mechanics. The scattering is qualitatively analyzed by two approaches. In the first approach, the quantum flow is studied via a one-parameter family of complex potentials. In the second approach, the qualitative theory of planar differential equations is used to obtain a one-parameter family of Hamiltonian functions which determine the phase portraits of the systems.
Hydrodynamical formalism of quantum mechanics is revisited for a particle under external electromagnetic field, with detailed explanations as to its subtle aspects such as the role of singular vortex lines with quantized circulation. The theory is applied to several problems where a charged particle moves under certain new configurations of external magnetic field, which are suitable to provide examples of Aharonov·Bohm effect in different situations. A remarkable consequence is given about the flow in the case of string magnetic field and is employed to obtain solutions explicitly for the movement under the field of a circular magnetic string.
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.
In this article, we start our discussion on the Aharonov-Bohm effect then raise a plausible interpretation within the quantum electrodynamics (QED) framework. We provide a quantum treatment of the source of the electromagnetic potential and argue that the underlying mechanism in AB effect can be viewed as interactions between electrons described by QED theory where the interactions are mediated by virtue photons. On further analysis, we claim that the framework of one particle quantum mechanics (OPQM) can be shown, in general, as a mathematically approximated model which can be reformulated from QED theory. In addition, the classical Maxwell equations are derived from QED scattering process while both classical electromagnetic fields and potentials serve as mathematical tools that are constructed to approximate the interactions among elementary particles described by QED physics, i.e. neither classical fields nor potentials represent any real entities of nature. At the conclusion of this article, we make a few remarks on the hypnosis of the existence of magnetic monopoles.
Motion of a nonrelativistic particle on a cone with a magnetic flux running through the cone axis (a ``flux cone'') is studied. It is expressed as the motion of a particle moving on the Euclidean plane under the action of a velocity-dependent force. The probability fluid (``quantum flow'') associated with a particular stationary state is studied close to the singularity, demonstrating nontrivial Aharonov-Bohm effects. For example, it is shown that, near the singularity, quantum flow departs from classical flow. In the context of the hydrodynamical approach to quantum mechanics, quantum potential due to the conical singularity is determined, and the way it affects quantum flow is analyzed. It is shown that the winding number of classical orbits plays a role in the description of the quantum flow. The connectivity of the configuration space is also discussed.
We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.
Abstract In this paper we review some aspects of the scattering Aharonov–Bohm effect and Berry’s phase. Specifically, the problem of scattering of free 2d electrons on the system of an arbitrary number of parallel, infinitely thin and infinitely long coils (non-interacting, point magnetic fluxes) is modeled as free semi-classical particle propagation on a topologically non-trivial background. We show that in this case it is possible to obtain Berry’s phase by integration. First the case of a single coil is analyzed, upon which a particular solution for the case of scattering on an arbitrary number of coils is found. Considering the solution in the momentum representation, an analytic continuation of the angle coordinate in the momentum space is introduced, in accord with the non-trivial geometry of the configuration space. Finally, some simple experiments with possible applications for quantum computing are proposed.