Fractional Angular Momentum In Noncommutative Generalized Chern-Simons Quantum Mechanics

In this thesis, we study the fractional angular momentum of reduced models in both commutative and noncommutative generalized Chern-Simons quantum mechanics. The generalized Chern-Simons quantum mechanics describes a charged particle confined by a harmonic potential interacting with both a uniform magnetic field and magnetic vector potentials produced by a long thin solenoid. We find that the angular momentum operator in commutative space is similar with the Hamiltonian of a one-dimensional harmonic oscillator when mass takes the zero limit. We solve the angular momentum exactly and find that there are two parts in the eigenvalue of angular momentum in reduced model of the generalized Chern-Simons quantum mechanics, one is the ordinary part the other is proportional to the flux inside the solenoid. Therefore, the angular momentum of the reduced model in the generalized Chern-Simons quantum mechanics can take arbitrary values when the magnetic flux changes.Then, we generalize the model to the noncommutative plane. We show that there are two different reduced models in the noncommutative generalized Chern-Simons mechanics. They correspond to setting the mass and a dimensionless parameter to zero respectively. The angular momentum of each reduced model takes fractional values. Thus, there are two different approaches to get the fractional angular momentum in the noncommutative generalized Chern-Simons quantum mechanics. One of the approaches is analogous to the commutative case, i.e., we set the mass to zero and find that the angular momentum is similar to the Hamiltonian of a one-dimensional harmonic oscillator. The other approach is to set a dimensionless parameter to zero. We find that the eigenvalues of the angular momentum of this reduced model are also fractional. The uniform magnetic plays subtle roles in getting the fractional angular momentum in each reduced model:Although the uniform magnetic do not contribute to the fractional angular momentum, the fractional angular momentum would not appear if only the magnetic potential produced by the solenoid are present.