An Algebra Analysis For An Operator With N×L On Spherical Surface

In quantum mechanics, angular momentum has its own definition that includes spin as a special case and the spin is a pure quantum quantity without classical correspondence. Though the square of angular momentum represents the Hamiltonian of the rigid rotor, how to define a coherent state on spherical surface remains a long standing problem. This dissertation studies a relevant problem as how to define a novel algebra on operators defined on the surface of a sphere, because coherent state is formed on the base of the closeness of a set of generators.Researches reveal that for free particles moving on the spherical surface, there are operators that have algebraic structure related to spherical harmonics. Simple observation leads to a fact that the core part of these operators is a cross product of the directional and the angular momentum operator. An operator containing this core part will take the following general form: where N is directional operator, L is angular momentum operator, and a, b are parameters.Each operator in quantum mechanics must be hermitian, if it represents physical quantity. We first require the hermiticity of the operator, and find out its condition between J1+ and J2- which deform form J, when parameters of J2- are c and d respectively, then carry out the commutation relations with N, Lz and operators related to get the closure of the algebra. Finally, a, b, c and d are given. After analysis, two closed algebras have been determined. In detail, when a, b, c and d are set to be special values, the redefined operator K+, K-, Kz constructed from J have su(1,1) and u(1,1) Lie algebra structure. Their Casmir operators also have been given, and they do not commute with the Hamiltonian of particles on spherical surface, so yet do not represent any new invariant.