Studies On Deformation Quantization For The Physical Systems On Noncommutative Space

Besides the noncommutativity of coordinate and moment brought by quantization of physical systems, in the frame of noncommutative quantum mechanics, there has the noncommutative relation between coordinates or momentum included by the phase space. So one can not find the common eigen states of some coordinates or momentum operators. It is difficult for us to study the quantum physical systems on the noncommutative phase space. Usually one adapts the following process to overcome the difficulty: one denotes the operators in the noncommutative space in the term of the ones in the ordinary space, and discusses the corresponding physical model in the ordinary space. But with the procedure we changes the property of the space, so we can not realize the intrinsic content of the systems in the noncommutative space. In this paper, we research the noncommutative quantum mechanics by deformation quantization.In the frame of deformation quantization, one deals with functions on the phase space, and the star product deforms the commutative classical algebra of observables into the noncommutative quantum algebra of observables. In order to get the transformation from the ordinary space to the noncommutative space, one needs to expand the star product, and the extension is simple and natural. By deformation quantization, we do not make the transformation of variables and can study the physical model directly on the noncommutative phase space, therefore our results are credible and scientific.Usually, Wigner functions are obtained by the integration of the wave functions. However, in the noncommunicative quantum theory, Wigner function can not be got similarly due to the change of physical observables. In this thesis, starting from fundamental principle of the Weyl correspondence, we derive explicit form of the Wigner function in noncommunicative phase space. Our results not only imply the noncommunicative relation of variables of phase space, but also satisfy a generalized *gen-equation. Furthermore, we discuss a harmonic oscillator in noncommunicative space to support our results. At last, we study two physical models including coupled oscillators and damped oscillators. By using the algebraic skills, we get the time evolution function of the Hamiltonian and expand it to the Wigner functions in the noncommutative space. Compared to the the related papers, our presentation gave out both the specific expression of Wigner functions and the investigation of excited states.