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Journal of Physics A Mathematical and General

Scattering on a hyperbolic torus in a constant magnetic field

MATHIAS ANTOINE, Alain Comtet, Stéphane Ouvry

Published Aug 21, 1990• 34 citations
Publisher

Abstract

The authors study the quantum mechanical scattering of a particle on a two-dimensional non-compact Riemann manifold of constant negative curvature embedded in a constant external magnetic field. The interplay between fundamental group operations and gauge transformations allows one to compute the scattering states. A closed expression for the phase shift of a plane wave entering the torus through its leak is shown to involve the quantized magnetic flux.

DOI
10.1088/0305-4470/23/16/018

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The authors study the quantum mechanical scattering of a particle on a two-dimensional non-compact Riemann manifold of constant negative curvature embedded in a constant external magnetic field. The interplay between fundamental group operations and gauge transformations allows one to compute the scattering states. A closed expression for the phase shift of a plane wave entering the torus through its leak is shown to involve the quantized magnetic flux.
The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.
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.
The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.
Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.
The motion of a charged particle moving on a flat surface is studied through the constants of motion associated with the system, given the magnetic gauge. The usual Landau’ solution and the non-separable solution for the Landau’s gauge are found, and a new non-separable solution for the symmetric gauge is given. As a consequence of this, the quantization of the magnetic flux results from the invariance of the solutions under the unitary transformations which arise from the operator’s constants of motion.
Abstract The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum m , their energies ( E m ) on a sphere are related to those on a 2D plane ( − ∣ E m , o ∣ ) as E m = − ∣ E m , o ∣ + E R m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 2 ) , and R is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies E ∼ E R differ significantly from those on a 2D plane. As energy E approaches zero, the phase shift in the planar case approaches 0, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale R .