Constraint-induced mean curvature dependence of Cartesian momentum operators

The Hermitian Cartesian quantum momentum operator $\mathbf{p}$ for an embedded surface $M$ in $R^3$ is proved to be a constant factor $-i\mathrm{\hslash }$ 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