非对易含时与中心场问题的相关研究

【作者】 龙超云；

【导师】 秦水介；

【作者基本信息】 贵州大学 ， 微电子学与固体电子学， 2009， 博士

【摘要】 众所周知,非对易并非是一种新的思想,它的起源可以追述到1930年Landau的工作。近年来,由于受到D-膜非零背景场低能效应的研究的推动,非对易理论的研究又引起了人们的广泛兴趣。它的研究将对深入认识小尺度系统所出现的物理效应具有重要的意义。尽管非对易效应仅仅出现在高能标区域,然而在探索其是否存在低能效应也是十分重要的。人们希望通过对非对易量子力学的研究来进一步揭示非对易的本质和非对易效应。在有关非对易的研究中,除了非对易场论的研究兴趣外,目前对非对易量子力学已有相当多的研究工作,在非相对论情况下,如:非对易量子Hall效应、非对易Landau问题和非对易任意中心场问题等。在相对论情况下,如:Dirac谐振势的非对易Dirac方程、Klein-Gordon方程和Duffin-Kemmer-Petiau方程等。虽然在人们在非对易量子力学层面已做了大量的研究,但值得指出的是,目前有关非对易量子力学的研究工作和兴趣主要集中在非含时量子系统的研究,而非含时量子系统只是含时量子系统的特殊情况,它也仅仅是真实物理过程的一种近似。因此,非对易含时量子系统的研究不仅具有重要的理论价值,同时在物理学的众多领域也可能存在广泛的实际应用,如:量子输运、量子Hall效应、自旋电子学和量子光学等。中心场问题在物理学的理论和实验研究中也具有重要的地位,但有关中心场问题的研究被局限在非对易空间。然而,非对易空间仅仅是非对易相空间的特殊情况,在非对易相空间中不仅坐标是非对易的,同时动量也是非对易的。因此,非对易相空间中的中心场问题也是值得研究的。基于波函数在量子理论研究中的重要性和以上的考虑,在本论文中,我们已就非对易量子力学框架下的以下问题进行了研究:1、在非对易空间和非对易相空间中,我们应用不变量理论和李代方法分别研究研究了含时线性势Schrodinger方程、含时电磁场Schrodinger方程和一般含时线性谐振子Schrodinger方程。构造相应含时系统的不变量和时间演化算符,通过不变量和时间演化算符获得了系统波函数的解析表达式。2、在非对易相空间中,我们应用幺正变换和代数方法研究了任意中心场和电磁场存在时的定态Schrodinge方程,获得了相应的能谱。3、在非对易相空间中,我们应用不变量理论研究了含时线性势Dirac方程,构造时系统的不变量,通过不变量的解获得了Dirac方程的解析解;同时,还研究了具有Dirac谐振势的定态Duffin-Kemmer-Petiau方程,获得了该方程的解析波函数和能谱;由于非对易效应的存在,量子态是不简并。本文的研究表明:在非对易量子力学中,描述量子系统重要性质的能谱和波函数都依赖于非对易参数,具有非对易效应。此外,非对易效应主要出现在高能区,检验空间非对易效应是一个富有挑战性的研究课题,在现有实验条件下一般难以探测。而非对易量子力学出现的非对易效应,在某程度上使得人们在未来的低能实验中对非对易量子理论进行验证成为可能,从而对推动该领域的进一步深入研究具有一定的意义。我们希望本文研究结果所得到的非对易效应在未来的实验中能得到验正,并在凝聚态物理和量子光学等领域获得应用。更多还原

【Abstract】 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.更多还原