Quantum Motion on 2D Surface of Nonspherical Topology

An excess term exists when using hermitian form of Cartesian momentum p _i (i = 1, 2, 3) in usual kinetic energy 1/(2) p _i ² for a particle moving on the 2D surface, and the correct kinetic energy turns to be 1/(2) 1/f_ip_i f_i p_i where the f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper, the explicit form of the dummy functions f _i is given for some surfaces of nonspherical topology, such as toroidal surface, paraboloid of revolution, the hyperboloid of revolution of two sheets, and the hyperboloid of revolution of one sheets.