Quantum particle on a surface: Catenary surface and paraboloid of revolution

Abstract

We revisit the Schrodinger equation of a quantum particle that is confined on a curved surface. Inspired by the seminal work of da Costa (1980 Phys. Rev. A 23 1982) we find the field equation in a more convenient notation. The contribution of the principal curvatures in the effective binding potential on the surface is emphasized. Furthermore, using the so-called Monge-Gauge we construct the approximate Schrodinger equation for a flat surface with small fluctuations. Finally, the resulting Schrodinger equation is solved for some specific surfaces. In particular, we give exact solutions for a particle confined on a Catenary surface and a paraboloid of revolution.