Non-center Field In The Easy-containing Study

It is well-known that the idea of noncommutativity is not a new concept and in physics , first example of noncommutativity was probably studied by Landau in 1930. Recently there has been a renewed interest in noncommutative space. This is motivated by studies of the low energy effective theory of D-brane with a non-zero NS-NS B field back-ground. The study on noncommutative space is much important for understanding phenomena at short distances beyond the present test of QCD. Although the effects of noncommutativity only appear at very high energy scales, it is meaningful to speculate whether there might be some low- energy effects of the fundamental quantum field. One expects that quantum mechanics in noncommutative space may clarify some low energy phenomenological consequences and may lead to deeper understanding of effects of noncommutativity. Besides the string theory interests, in recent years there are many papers devoted to the study of various aspects of non-relativistic quantum mechanics on noncommutative space, such as Quantum Hall effect, noncommutative Landau problem on plane, the two-dimensional quantum system with arbitrary central potential, etc. In the relativistic aspect, the Dirac,Klein-Gordon and Duffin-Kemmer-Petiau oscillator has been discussed by in noncommutative space separately. However, to our knowledge these studies of noncommutative quantum mechanics were restricted to the time-independent problems which are only an approximation to the true physics. Recently the study of system depending explicitly on time raised considerable interest because of their varied application in various domain of physics such as quantum optics, quantum transport, spintronics ect. So it is worth while studing noncommutative time-dependent quantum mechanics. In addition to, center field in a magnetic field has proved to an extremely rich subject for theoretical and experimental investigation, but study on it was restricted to noncommutative space and the noncommutative space can be considered as a special case of noncommutative phase space. In noncommutative phase space, not only the confinguration but also momentum is noncommutative. Since study of center field on noncommutative phase space is also interesting. Based on the importance of wave function and above consideration, in this work we study following some problems in at the level of noncommutative quantum mechanics:1,On noncommutative phase space and noncommutative space the Schrodinger equations with time-dependent linear, time-dependent magnetic field and general time-dependent oscillator potential have been studied by invariant theory and Li algebra separately.The invariant and evolution Operator have been constructed. The corresponding wave functions have been obtained.2,On noncommutative phase space, the time-independent Schrodinger equations with arbitrary central potential have been studied by the unitary transformation. The corresponding energy spectrums have been gotten.3,On noncommutative phase space, the Dirac equation with with time-dependent linear and the time-independent DKP equation with oscillator potential have been studied. For the Dirac equation, the analytical wave function has obtained by invariant theory and for DKP equation, the analytical wave function and energy spectrum have been gotten separately. As a result of the noncommutative effect the energy spectrum is not degenerate.The results show that the wave functions and energies obtained here depend on the noncommutativity parameters. On the other hand, it is very difficult to test noncommutative effect under current experiment condition because the effects of noncommutativity only appear at very high energy scales. We hope that the noncommutative effecst obtained here can be tested in the future experiments at the level of Quantum mechanics and can be applied to many ares such as condensed state physics and quantum optics etc.