Coupling curvature to a uniform magnetic field: An analytic and numerical study

The Schr\"odinger equation for a spinless electron near an azimuthally symmetric curved surface Σ in the presence of an arbitrary uniform magnetic field B is developed. A thin-layer quantization procedure is implemented to bring the particle onto Σ, leading to the well-known geometric potential VC∝h²-k and a second potential that couples AN, the component of A normal to Σ to mean surface curvature, as well as a term dependent on the normal derivative of AN evaluated on Σ. Numerical results in the form of ground-state energies as a function of the applied field in several orientations are presented for a toroidal model.