Coupling curvature to a uniform magnetic field; an analytic and numerical study

The Schrodinger equation for an electron near an azimuthally symmetric curved surface $\Sigma$ in the presence of an arbitrary uniform magnetic field $\mathbf B$ is developed. A thin layer quantization procedure is implemented to bring the electron onto $\Sigma$, leading to the well known geometric potential $V_C \propto h^2-k$ and a second potential that couples $A_N$, the component of $\mathbf A$ normal to $\Sigma$ to mean surface curvature, as well as a term dependent on the normal derivative of $A_N$ evaluated on $\Sigma$. 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.