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

Gravesen J., Willatzen M.

A set of ordinary differential equations is derived employing the method of differentiableforms so as to describe the quantum mechanics of a particle constrained tomove on a general two-dimensional surface of revolution. Eigenvalues and eigenstatesare calculated quasianalytically in the case of a finite cylinder (finite alongthe axis) and compared with the eigenvalues and eigenstates of a full threedimensionalSchrödinger problem corresponding to a hollow cylinder in the limitwhere the inner and outer radii approach each other. Good agreement between thetwo models is obtained for a relative difference less than 20% in inner and outerradii”