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.
Gravesen, J., Willatzen, M., & Voon, L. C. L. Y. (2005). Quantum-Mechanical Particle Confined to Surfaces of Revolution - Truncated Cone and Elliptic Torus Case Studies. Physica Scripta, 72(2-3), 105-111. https://doi.org/10.1238/Physica.Regular.072a00105