Schr?dinger Equation of a Particle on a Rotating Curved Surface

We derive the Schr?dinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potential V_\rm g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates. This novel effective potential, which is included in the surface Schr?dinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.