Studies On Some Problems Of Physical Systems In Noncommutative Space

The theory of quantum mechanics in noncommutative space has some differences with that of normal quantum mechanics, so there are some new methods and technolo-gies to deal with the quantum mechanics problems in noncommutative space. In this thesis we discuss the quantum mechanics problems in noncommutative space in two ways. First we discuss how to extend the deformation quantization methods of the normal space to that of the noncommutative space. Respectively, we investigate the case of only the spatial coordinates is noncommutative and also that of both spatial and momentum coordinates are noncommutative. We derive the expressions of the corre-sponding *-genvalue equations and Wigner functions of both the Hamiltonian systems and non-Hamiltonian systems, and discuss the equivalence and differences between dif-ferent methods. Then we apply the deformation quantization method to some physical models in noncommutative space. First we investigate the coupled harmonic oscillators on noncommutative plane, and calculate the corresponding energy spectra and Wigner function, and derive a formula for a class of Hamiltonian with special form in defor-mation quantization. We also investigate some typical non-Hamiltonian systems such as the damped systems and Bateman system in noncommutative space, and obtain the energy spectra and generalized Wigner function of these systems. In the second part of the thesis we discuss the construction of some quantum representations in noncom-mutative space and the properties and applications of these representations. Firstly, based on the commutation relations of the coordinates of generalized noncommutative phase space we construct a type of deformed creation and annihilation operators, and these operators satisfy the deformed boson algebraic relations. Based on these operators and the deformed boson algebra, we construct the noncommutative coherent states and squeezed states, and after some calculations we find that these states are normalized but not orthogonal to each other, and besides, they are over-complete, so we can construct the coherent representation and squeezed representation. Furthermore we derive the transformation relations between these representations. We investigate the Heisenberg uncertainty relations of the one-and two-mode quadrature operators, and we find that the Heisenberg uncertainty relation can give a restricted relation between the parame-ters of the noncommutativity of the space and the Planck constant. We also construct the entangled state representations for the noncommutative phase space, and calculate the matrix elements of the coordinate operators of noncommutative space on entangled state representations. We also obtain the expression of the Wigner operator on entan-gled state. Finally we show some applications of the entangled state representation by the coupled harmonic oscillators on 2D noncommutative plane.