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arXiv:quant-ph/0508111 · Journal of Mathematical Physics

Thin layer quantization in higher dimensions and codimensions

Alexey V. Golovnev

Published Aug 1, 2006• 7 citations
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Abstract

We consider the thin layer quantization with use of only the most elementary notions of differential geometry. We consider this method in higher dimensions and get an explicit formula for quantum potential. For codimension 1 surfaces the quantum potential is presented in terms of principal curvatures, and equivalence with Prokhorov quantization method is proved. It is shown that, in contrast with original da Costa method, Prokhorov quantization can be generalized directly to higher codimensions.

DOI
10.1063/1.2213787

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We consider the thin layer quantization with use of only the most elementary notions of differential geometry. We consider this method in higher dimensions and get an explicit formula for quantum potential. For codimension 1 surfaces the quantum potential is presented in terms of principal curvatures, and equivalence with Prokhorov quantization method is proved. It is shown that, in contrast with original da Costa method, Prokhorov quantization can be generalized directly to higher codimensions.
Abstract A fundamental problem regarding the Dirac quantization of a free particle on an ( ) curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.
In this work, Schrödinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac’s quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared and some notes on the differences between the results will be included.
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.