The Study Of The Thermodynamic Properties Of The Quantum Oscillators In Noncommutative Space

The emergence of noncommutative geometry in string theory lead to profound changes in people's understanding and awareness of whole theory of fundamental physics, which proposes a new physical space time---noncommutative(NC) space time. In the past decades, noncommutative effects have received much attention and have been extensively studied in the fields of quantum mechanics, field theory, condensed matter physics and astrophysics.In order to increase our knowledge of the noncommutative formalism and to explore the influence of noncommutative effect on traditional physical systems, we study the eigenvalues problem of several quantum systems and thermodynamic properties of the spinless Duffin-Kemmer-Petiau oscillator as well as the graphene in NC(phase) space. The work focuses on the following aspects:We have studied the motion of a non-relativistic as well as relativistic charged particle in the presence of an exponentially decaying magnetic field in NC space. The energy eigenvalue and corresponding wave function are obtained by employing Bopp shift and the functional analysis method. It shows the noncommutative parameter shifts the energy levels of the system and eliminates degenerate. We also discuss the Landau problem via the vanishing NC parameter and obtain the energy spectrum for a Dirac electron in graphene under the exponentially decaying magnetic field via the parametric mappings. Moreover, due to the Bopp maps from the NC phase space to a commutative one are not unique, and different maps will lead to different representations in usual phase space. We adopt an algebraic approach without any type of Bopp shifts to deal with the noncommutative eigenvalues problem. The basic idea of the algebraic procedure is to substitute coordinate and momentum operators with the usual creation and annihilation operators, and obtain the eigenvalues with a diagonalization. By solving the model of noncommutative Klein-Gordon Oscillator via the two methods, we show that they can give the consistent result for a same eigenvalues problem in NC quantum mechanics.Thermodynamic properties of the DKP oscillator for spin-0 particle in NC space are also investigated. We first obtained energy spectrum as well as corresponding spinor? of the system by employing the functional analysis method. Subsequently, we study the thermodynamic properties of the system by employing Euler-MacLaurin method and depict our numerical results on the evaluation of the thermodynamic functions such as free energy F, mean energy U, specific heat C and entropy S. We show that the thermodynamic functions depend explicitly on the NC parameter? which characterizes the noncommutativity of the space. For the high temperature limit, we also analyze the characters of these thermodynamic functions from the asymptotic behavior. The effect of the NC parameter on free energy and entropy is observable, while is ignorable on mean energy and specific heat. In addition, the work also discusses thermal properties of graphene in noncommutative phase space. By employing the same method, we obtain the expression of thermodynamic functions. It is observed that the energy spectrum and corresponding thermodynamic functions depend on the NC parameter. We expect that once obtain experimental measurements of high accuracy involving the thermodynamical properties of grapheme in the future, our results may be used as a good tool to test the noncommutativity of the space.