Notes on the quantum mechanics of particles constrained to curved surfaces

摘要

In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schr?dinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schr?dinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.