## Geometric momentum for a particle constrained on a curved hypersurface    [PDF]

Q. H. Liu
A strengthened canonical quantization scheme for the constrained motion on curved surface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas those between positions and moments are categorized into the first. As an $N-1$ ($N\geq2$) dimensional surface is embedded in an $N$ dimensional Euclidean space, we obtain the geometric momentum $\mathbf{p}=-i\hbar (\mathbf{\nabla}_{S}+M\mathbf{n}/2)$ where $\mathbf{\nabla}_{S}$ denotes the gradient operator on the surface and the $M\mathbf{n}$ is the mean curvature vector. For the surface is the spherical one of radius $r$, we resolve in a lucid and unambiguous manner a long-standing problem of the geometric potential that proves to be $V_{g}=(N-1)(N-3)\hbar^{2}/(8mr^{2})$.
View original: http://arxiv.org/abs/1305.0970