Studies Of Two-Dimensional Harmonic Oscillator In Non-Commutative (Phase) Space

In this thesis, we propose two different methods, namely, the operator and path integral method, to analyze the spectra and wave functions of the non-commutative two-dimensional (2D) harmonic oscillator. The contents of this thesis are the following:1) The spectra and the wave functions of 2D harmonic oscillator in non-commutative space. Firstly, the non-commutative space is mapped to a commutative one and then the operator form is used to obtain the eigenvalues and the wave functions. Then, the path integral formulation is employed in coordinate space and momentum space, respectively. The propagator is computed both in coordinate space and momentum space. At last, the spectra and wave functions are read off from it.2) The spectra and the wave functions of 2D harmonic oscillator in non-commutative phase space. We propose a mapping-independent method to solve the spectra and wave functions of the 2D harmonic oscillator in non-commutative phase space. The path integral formulation is applied and the propagator is constructed straightly in non-commutative phase space. The spectra are read off directly from the propagator and the question of uniqueness of the operator form is answered.