Energy Spectra And Wavefunctions Of Nonrelativistic And Relativistic Oscillators In Noncommutative Space

Quantum mechanics and relativity build theoretical foundations of modern physics. The mathematical framework of quantum mechanics is the theory of lin-ear Hermitian operators on Hilbert spaces. Commutativity of operators is in a very important position in quantum mechanics. Recently, with the development of noncommutative geometry, string theory, and quantum gravity, noncommuta-tive space have attracted considerable attention. Besides the noncommutativity of coordinate and momentum brought by quantization of physical systems, in the frame of noncoommutative quntum mechanics, there have been the noncommuta-tive relations between coordinates or momenta induced by the phase space. The theory of quantum mechanics in noncommutative space has some differences with that of normal quantum mechanics. There are some new methods to deal with the quantum mechanics problems in noncommutative space. The study of noncom-mutative quantum mechanics may help us find some low energy relics of effect of noncommutativity. In this thesis we discuss nonrelativistic and relativistic oscil-lators at the level of noncommutative quantum mechanics. This thesis is divided into the following four parts.The first part includes chapter1and chapter2. In the first chapter, the back-ground and the basic knowledge of noncommutative spaces are described. In the second chapter, we introduce some basic theories of noncommutative space, includ-ing Moyal or star product. Wigner function and the algebra of noncommutative space.The second part consists of chapter3,4, and5. In chapter3. In this part, we study a noncommutative-space harmonic oscillator in a uniform magnetic field. A key observation that we make is that the effect of noncommutativity may vanish for a specific choice of the magnetic field. We also study thermodynamic properties of a harmonic oscillator in noncommutative phase space. In chapter4, we obtain energy spectra and wavefunctions of Klein-Gordon and Duffin-Kemmer-Petiau rel-ativistic oscillators in a noncommutative space. It is shown that the effect of the magnetic field may compete with that of the noncommutative space. In chapter5, we obtain energy spectrum and wavefunetion of a noncommutative-space Dirac particle in a magnetic field. We show that the velocity relates to configutation space noncommutativity, and the momentum noncommutativity corresponding to a small magnetic field. We investigate the dynamics of the Dirac oscillator ex-actly in2+1dimensional noncommutative space. We find an exact mapping of this quantum-relativistic system onto a Jaynes-Cummings model which indicates that Zitterbewegung effect relates to noncommutativity. We present a physical implementation of the Dirac oscillator in noncommutative space.The third part is chapter6, we introduce a general noncommutativitv of coordinates, and discuss properties of uncertainty relations in a noncommutativc phase space. It is shown that there are two phases in a noncommutative phase space. We construct the noncommutative two-mode coherent states and squeezed states based on deform boson algebra, which change to single-mode coherent states and squeezed states at the critical point, respectively.Chapter7is the fourth part. In this chapter, we conclude this thesis and give some outlook in the future.