Ouantum-mechanical particle confined to surfaces of revolution-truncated cone and elliptic torus case studies

The theory of a quantum-mechanical particle confined to a surface of revolution is described using differential geometry methods including the derivation of a general set of three ordinary differential equations in curved coordinates. The problem is shown to be completely separable with the present hard-wall boundary conditions. Two case studies of recent experimental interest. the nanocone and torus-shaped nanoring structures. are analyzed in terms of eigenstates, energies. and symmetry characteristics based on the theory presented.