Constraint-induced mean curvature dependence of Cartesian momentum operators

TL;DR

This paper derives a form of the Cartesian quantum momentum operator on a surface, showing it depends on mean curvature, and discusses operator-ordering ambiguities in kinetic energy operators.

Contribution

It introduces a curvature-dependent form of the Cartesian momentum operator on embedded surfaces, clarifying operator-ordering issues in surface quantum mechanics.

Findings

Momentum operator includes mean curvature term

Operator-ordering ambiguity is resolved with multiple orderings

Different orderings yield the same kinetic energy result

Abstract

The Hermitian Cartesian quantum momentum operator $\mathbf{p}$ for an embedded surface $M$ in $R^{3}$ is proved to be a constant factor $-i\hbar $ times the mean curvature vector field $H\mathbf{n}$ 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. PACS: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization