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_i² 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.