Schrödinger problems for surfaces of revolution—the finite cylinder as a test example

Abstract

A set of ordinary differential equations is derived employing the method of differentiable forms so as to describe the quantum mechanics of a particle constrained to move on a general two-dimensional surface of revolution. Eigenvalues and eigenstates are calculated quasianalytically in the case of a finite cylinder (finite along the axis) and compared with the eigenvalues and eigenstates of a full three-dimensional Schrödinger problem corresponding to a hollow cylinder in the limit where the inner and outer radii approach each other. Good agreement between the two models is obtained for a relative difference less than 20% in inner and outer radii.