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arXiv:1310.6651 · Journal of Mathematical Physics

Quantum dynamics of a particle constrained to lie on a surface

Gustavo de Oliveira

Published Sep 1, 2014• 12 citations
arXivHTMLPDF

Abstract

We consider the quantum dynamics of a charged particle in Euclidean space subjected to electric and magnetic fields under the presence of a potential that forces the particle to stay close to a compact surface. We prove that, as the strength of this constraining potential tends to infinity, the motion of this particle converges to a motion generated by a Hamiltonian over the surface superimposed by an oscillatory motion in the normal directions. Our result extends previous results by allowing magnetic potentials and more general constraining potentials.

DOI
10.1063/1.4895761

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We consider the quantum dynamics of a charged particle in Euclidean space subjected to electric and magnetic fields under the presence of a potential that forces the particle to stay close to a compact surface. We prove that, as the strength of this constraining potential tends to infinity, the motion of this particle converges to a motion generated by a Hamiltonian over the surface superimposed by an oscillatory motion in the normal directions. Our result extends previous results by allowing magnetic potentials and more general constraining potentials.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.
We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on a nonsingular static surface in ordinary three-dimensional space in the presence of arbitrary configurations of time independent electric currents. Starting from the canonical action in the embedding space we show that a charged particle with charge qqq couples to a term linear in qA3Mq{A}^{3}MqA3M, where A3{A}^{3}A3 is the transverse component of the electromagnetic vector potential and MMM is the mean curvature in the surface. This term cancels exactly a curvature contribution to the orbital magnetic moment of the particle. It is shown that particles, independently of the value of the charge, in addition to the known couplings to the geometry also couple to the mean curvature in the surface when a Neumann type of constraint is applied on the transverse fluctuations of the wave function. In contrast to a Dirichlet constraint on the transverse fluctuations a Neumann type of constraint on these degrees of freedom will in general make the equations of motion nonseparable. The exceptions are the equations of motion for electrically neutral particles on surfaces with constant mean curvature. In the presence of electric currents the equation of motion of a charged particle is generally nonseparable independently of the coupling to the geometry and the boundary constraints.
We derive the Schrödinger equation for a spinless charged particle constrained to move on a curved surface in the presence of an electric and magnetic field. The particle is confined on the surface using a thin-layer procedure, which gives rise to the well-known geometric potential. The electric and magnetic fields are included via the four potential. We find that there is no coupling between the fields and the surface curvature and that, with a proper choice of the gauge, the surface and transverse dynamics are exactly separable. Finally, we derive an analytic form of the Hamiltonian for spherical, cylindrical, and toroidal surfaces.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.
Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.
We discuss the motion of a quantum-mechanical system constrained to move on an arbitrary submanifold M of its configuration space R n by a real confining potential. A complete perturbative expansion for the Hamiltonian describing the dynamics of the system is obtained in terms of the quantities characterizing the intrinsic and extrinsic geometric properties of the constraint's surface M. In accordance with Heisenberg's principle the zeroth-order term of the expansion represents strong fluctuations of the system in directions normal to M, whereas the first-order term is naturally interpreted as the Hamiltonian describing the effective dynamics along the constraint's surface. The effective motion on M results in coupling with Abelian/non-Abelian gauge fields and quantum potentials. The rest of the perturbative expansion allows one to take into account further interactions between normal and effective degrees of freedom. As a concrete example we consider the dynamics of an electron constrained on a circle by a hypothetical semiconductor device. Dunham's expansion for the rotovibrational energy of a rigid diatom is also obtained.