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arXiv:0910.0769 · Journal of Physics A Mathematical and Theoretical

Constraint-induced mean curvature dependence of Cartesian momentum operators

Q H Liu, C L Tong, M M Lai

Published Mar 23, 2007• 34 citations
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Abstract

The Hermitian Cartesian quantum momentum operator p for an embedded surface M in R 3 is proved to be a constant factor −iℏ times the mean curvature vector field H n added to the usual differential term. With use of this form of momentum operators, the operator-ordering ambiguity exists in the construction of the correct kinetic energy operator and three different operator-orderings lead to the same result.

DOI
10.1088/1751-8113/40/15/007

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The Hermitian Cartesian quantum momentum operator p for an embedded surface M in R 3 is proved to be a constant factor −iℏ times the mean curvature vector field H n added to the usual differential term. With use of this form of momentum operators, the operator-ordering ambiguity exists in the construction of the correct kinetic energy operator and three different operator-orderings lead to the same result.
When the motion of a particle is constrained, excess terms exist using hermitian form of Cartesian momentum p i ( i = 1,2,3) in usual kinetic energy (1/2μ)∑ p i 2 , and the correct kinetic energy turns out to be (1/2μ)∑1/ f i p i f i p i , where f i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper the explicit form of the dummy functions f i is given for a charged rigid planar rotator in the uniform magnetic field with different gauge chosen. Under different gauges, we have different sets of dummy factors. It means that these factors do not have direct observable effect.
For describing particles moving on the two dimensional curved surfaces, we can use either the intrinsic local coordinates or the Cartesian coordinates. The representation of the momentum operators differs from each other in these two kinds of coordinates, the former ones depend on the intrinsic geometrical quantities, but the latter case depend on a geometrical invariant, namely the mean curvature. Taking the operator-ordering problem into consideration, the kinetic operator for the former case can be expressed in a possibly unique way, while that for latter case can be expressed in two different ways.
Abstract The Hermitian surface momentum operator for a particle confined to a 2D curved surface spanned by orthogonal coordinates and embedded in 3D space is expressed as a symmetric expression in derivatives with respect to the surface coordinates and so is manifestly along the surface. This is an alternative form to the one reported in the literature and usually named geometric momentum, which has a term proportional to the mean curvature along the direction normal to the surface, and so “apparently” not along the surface. The symmetric form of the momentum is the sum of two symmetric Hermitian operators along the two orthogonal directions defined by the surface coordinates. The centripetal force operator for a particle on the surface of a cylinder and a sphere is calculated by taking the time derivative of the momentum and is seen to be a symmetrization of the well-known classical expressions.