Schr?dinger Equation of a Particle on a Rotating Curved Surface
Long Du1, Yong-Long Wang1,2, Guo-Hua Liang1, Guang-Zhen Kang1, Hong-Shi Zong2,3,4
1Department of Physics, Nanjing University, Nanjing 210093 2Department of Physics, School of Science, Linyi University, Linyi 276005 3Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093 4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190
Abstract：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.
2016, Vol. 33 
