On the geometric potential derived from Hermitian momenta on a curved surface

Abstract

A geometric potential $V_C$ depending on the mean and Gaussian curvatures of a surface $Σ$ arises when confining a particle initially in a three-dimensional space $Ω$ onto $Σ$ when the particle Hamiltonian $H_Ω$ is taken proportional to the Laplacian $L$ on $Ω$. In this work rather than assume $H_Ω\propto L$, momenta $P_η$ Hermitian over $Ω$ are constructed and used to derive an alternate Hamiltonian $H_η$. The procedure leading to $V_C$, when performed with $H_η$, is shown to yield $V_C = 0$. To obtain a measure of the difference between the two approaches, numerical results are presented for a toroidal model.