Schrodinger eigenstates for surface-confinement problems

The theory of a quantum-mechanical particle confined to a surface is described using differential geometry arguments including the simplification of the three-dimensional Schr\"{o}dinger problem into three ordinary differential equations in curved coordinates for the case of an arbitrary surface of revolution. These equations are solved - in terms of eigenvalues and eigenstates - either completely analytically or by use of a simple one-dimensional finite-difference scheme for the cases of a cylinder, a cone, an elliptic torus, a sinusoidal-shaped surface of revolution, and a catenoid. A comparison with an exact three-dimensional treatment of the hollow cylinder problem shows that the surface-confinement approximation (corresponding to assuming zero thickness of the particle domain perpendicular to the surface) is excellent in cases where the (hollow) cylinder thickness is less than approximately 10{\%} of the cylinder radius, hence justifying the rationale in employing a similar analysis for the other (above-mentioned) more complicated surface-confinement problems. Symmetry properties of the various eigenstates are finally discussed and compared.