量子台球体系动力学性质研究
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摘要
随着光刻技术与晶体生长技术的日渐成熟,制作小尺度、任意形状的量子台球体系成为可能。特别是近二十年来,由于半导体材料的制备技术和工艺水平的迅速发展使得半导体纳米结构成为研究微结中电子传导的典型实例。这种材料可以由包含高速运动的电子的薄层半导体(异质结)构成。垂直于层方向的运动是量子化的,所以电子的运动被局限在平面内。作为一个理论模型,这种系统被认为是二维电子气体系(量子台球体系),与薄金属膜相比,它的电子密度较低(在低温条件下),且可以通过外加电场来控制。这种低的电子密度意味着有大的费米波长(可以达到40nm)和大的电子平均自由程(可以达到10μm),这样电子间的相互作用可以忽略,而杂质粒子及声子的散射作用影响甚微。除与壁相碰外,电子可以看做是自由运动的经典粒子,所以在量子台球(或者二维电子气)中研究(电导)输运性质非常方便。二维量子台球体系的性质与边界的形状密切相关,改变体系的几何形状可以很方便的控制体系中粒子的运动由规则到混沌的过渡。例如,圆形,正方形,椭圆量子台球体系是可积体系,粒子在这些体系中的运动为规则运动。当体系的几何形状变为Sinai型,体育场型甚至是心脏线型时,体系变为不可积,粒子的运动呈现混沌特征。另外,在外加磁场中,体系的运动性质会发生很大变化。研究表明:当加上均匀磁场后,如果体系的几何边界形状与粒子的运动轨迹不能贴合时,运动性质也会出现混沌。基于此,量子台球体系成为人们研究BEC(波色爱因斯坦凝聚)、测定磁导率、电导性质以及实现量子控制直至研制量子计算机的理想模型。
对于不可积体系,一般无法得到解析解,而只能通过理论计算的方法求得薛定谔方程的数值解。对于量子台球系统,人们已经发展了许多数值求解的方法如:有限差分法、定态展开方法、B-样条法……等,然而数值方法需要大量的数据点和参数,而且由于算法复杂,通常只能得到近似解。而对于混沌过程,即使初值上的微小偏差也会对结果产生巨大的甚至是灾难性的影响。因此量子混沌的研究至今未能顺利得到广泛开展。科研工作者需要努力基于物理直觉,发展理论方法,以期推动量子混沌研究以更便利、简洁而且有效的方式广为人们所接受。这方面,量子谱函数方法、相空间分布函数方法以及传输矩阵结合拓扑学的方法都是成功的例证。
从1970年Sinai首先利用经典力学的方法研究了Sinai量子台球各态历经混沌性质和横向叶脉性质,然后Bunimovich等人又利用微扰理论从直观上解释了体育场型量子台球的性质,在1977年,Berry等人利用统计的方法研究了量子台球的混沌性.还提出了Berry-Kubo公式方法;此外,定态展开方法以及拓扑学等研究方法也相继产生。Landauer猜想电子通过这种微腔的电导可以通过计算传输矩阵,对传输系数求和得到。1986年,杜孟利和Delos等人研究外Rydberg原子在强电磁场中光吸收谱时在Gutzwiller态密度迹公式的基础上提出了闭合轨道理论为研究经典物理与量子力学的对应关系提供了理论基础,被称为是联系经典世界与量子世界的唯一桥梁。这种理论扩展到开轨道情况,也为研究量子台球性质提供新的有用的工具,它导致量子谱函数方法的提出并获得广泛应用。Delos小组利用S矩阵方法研究了圆形量子微腔的传输问题,认为微结中的电导涨落是由于沿着连接不同导线间的经典轨迹的波的相干导致的并考虑了电子在导线接口处的衍射效应。Christopher Stampfer小组利用赝路径半经典近似和衍射散射的Dyson方程研究量子台球体系的传输问题,解决了在导线出口和入口处的尖角效应。为了符合实际情况,研究外场中量子台球的动力学行为是必要的。对于多分量高简并体系,相空间分布函数方法是一个非常有用的工具,因为相空间公式提供了一个利用经典信息描述量子现象的理论框架,为理论研究者提供有用的物理洞见,是其他方法不容易实现的,具有重要的理论意义。
量子力学诞生以来,其方法和计算技术已经成为原子和分子体系中精确计算的主要手段。理论计算和实验测量结果的精确符合消除了人们对量子力学基本概念的任何质疑,直到今天量子力学仍然是人们解决微观体系的精确理论。但是在应用量子力学的方法解决多维不可积体系时需要进行大量的数值计算(虽然可以通过选取合适的基矢改进本征值计算,对角化哈密顿量需要的计算量通常十分庞大),而这些数值计算的结果对于我们了解体系动力学性质的作用甚微。相反,半经典方法可以很好的解释实验结果或应用理论计算得到的数据,这种方法对于我们了解体系的动力学性质以及探索新的发展方向起到了重要的作用。因而微观体系中的量子力学和经典力学的对应关系一直是人们十分感兴趣的重大课题,对人们更深地理解自然的本质有着重要的意义。量子力学与经典物理的对应关系经历了从量子力学诞生之初,Bohr和Sommerfeld的量子化假设到对可积系统存在作用量量子化的本征轨迹EBK定理的提出,再到后来Gutzwiller对于不可积系统发展的周期轨道理论和杜孟利J. B. Delos提出的半经典闭合轨道理论以及谱函数方法和量子力学相空间理论方法的飞跃过程。但是由于Heisenberg的测不准关系的限制,量子力学相空间图像不是唯一的。这种不唯一性主要表现在所使用的数学函数或算符具有一定的任意性,这就是量子力学相空间图像的一大缺陷。因此,探索新的有效的相空间的描述一直是人们长期追求的目标,这也是本文提出采用和发展Husimi分布函数的初衷。
本文首先采用量子谱函数方法研究了三维正方体量子台球的经典-量子对应。通过对经典轨道长度与量子谱函数傅里叶变换谱峰位置的比较发现,每个量子峰位置与一条或者几条经典轨道长度对应。为理解量子输运问题提供了比较清晰的物理图像。把谱函数方法应用于开放系统的输运过程则需考虑传导和散射,根据Dyson方程的微扰展开计算了正方形量子台球的输运性质,进而分析了不同导线宽度对传输系数的傅里叶变换谱的影响。为了充分考虑台球腔内部多重散射和导线开口处衍射效应的影响,我们在研究开放三角形量子台球微结的输运性质时了考虑扭结效应,并通过几何拓扑的关系和稳定相近似得到了体系长度较短的部分经典轨道。计算得到了三角形量子台球内部的波函数态密度分布和传输系数随能量的变化关系,对传输系数进行傅里叶变换得到量子谱,与经典轨迹长度做比较,发现每个傅里叶变换的量子峰位置对应一条或者几条经典轨迹,从而在一定精度允许范围内同样给出量子物理与经典物理的对应关系。
Husimi分布函数是投影到相空间量子态的分布函数,作为经典系统中Poincaré截面的一种表述方式被广泛的应用于经典-量子研究对应中。在本文中我们研究了外磁场作用下的圆形量子台球。通过求解磁场下的Schr?dinger方程得到体系的能量本征值与本征波函数。对不同磁场下的能量本征值进行能级统计分析发现最近邻能级间隔分布符合Poisson分布,说明对称性比较高的圆形量子台球即使在加外磁场以后仍然是可积体系。利用体系的能量本征态计算得到相空间的Husimi分布,发现随着外加磁场的增加,Husimi分布函数的峰先在r方向分裂,然后在p方向分裂,同时峰位向台球的边缘处聚集。Husimi分布函数的每个峰代表经典周期轨迹在相平面上穿过的点。这里得到的分布性质很容易过渡到混沌运动情况。当粒子在磁场量子台球腔内运动的圆轨迹与腔边界不相吻合,如磁场作用下的矩形腔中电子的运动将出现混沌。
论文共分五章。第一章为研究背景综述,简要介绍介观物理,量子台球体系的特点以及选题的意义以及主要已经进行的工作。第二章介绍了开轨道的量子谱函数,计算了三维量子台球的量子谱,寻找量子与经典之间的对应关系。第三章利用Dyson方程和考虑扭结效应分别讨论了开放矩形量子台球和正三角形量子台球的输运性质。第四章通过求解外加磁场下Schr?dinger方程,讨论了外加磁场下圆量子台球体系的能级统计,利用相空间Husimi分布函数讨论了相空间量子-经典对应关系。第五章是本文的结束语,对已完成研究进行简要的总结,并提出今后可以进一步探索的问题。
With the development of the technology in lithography and the growth in crystal become perfect day by day, it is possible to produce the little scale and random shape quantum billiard system. Especially in the recent twenty years, the structure of the nano-semiconductor has become the ideal model to research the conductor that the electrons transport through the micro-cavity, with the perfect development of the semiconductor material and the crystal’s purity, the heterojunction structure can be formed by folium of the electrons in higher speed. In the perpendicular direction of the layer, the motivation of the electrons is quantization, then the motivation of the electrons can be located in the plane. As a theoretical model, this system can be treated as two dimensional electronic gas (2DEG) or quantum billiards system and it is different from mental thin film in their low electric number. The low electronic density means that the electrons have large Fermi wavelength (which can achieve 40nm) and large mean free path (which can achieve 10μm). Then the coulomb interaction between the electrons can be ignored, the scattering impact between electrons and the impurity particles is very little, then the electrons can be seen as the classical particles, the model of quantum billiards or two dimensional electrons gas is used very convenient to research the quantum transmission. The property of the two dimensional quantum billiard system is related to the shape of the boundary closely. It is easy to control the motivation of the particles in the cavity transmitting to chaos from regular conveniently by changing the shape of the billiards system. For example, circular billiards, square billiards and ellipse billiard systems are integrable systems, which motivation of the particles is regular in these systems. When the shape of the systems changed into Sinai or stadium billiard, the systems becomes nonintegrable and the motivation of the particles becomes chaotic. In addition, when the system is applied a magnetic field, the character of the motion will also happen a large change. The research results show that if the trajectories of the particles after applied the uniform magnetic field do not match with the shape of the boundary for the system, the property of the system will become chaos. In other hand, based on these reason, the quantum billiard system becomes an ideal theoretical model to study the dynamic property of the quantum transport system.
Generally, the analysis results can not be obtained for the quantum chaos systems, one can only obtain the numerical results by the numerical method to solve the Schr?dinger equation. Then some people have developed some numerical methods to solve the problem, for example, Finite Difference method, Vector-based expansion, and Bsplines method, et al, but the numerical calculation needs a great deal of database, the arithmetic is very complex and in the end we only obtain the approximate results. Therefore, the research of the quantum chaos can not be widely studied until nowadays. This faultiness in the mathematic will be overcome by the research partner in the nearly future, and expect that it can drive the quantum chaos obtained by people in more convenient and simple fashion.
In 1970, Mr. Sinai studied the chaotic property in the Sinai quantum billiard system by semiclassical dynamics firstly, and then, Bunimovich et al gave out the properties of the stadium billiard system with the perturbation theory. In 1977, Berry et al researched on the chaotic property of quantum billiard systems with statistic method, meanwhile there were other method used to study the quantum billiards systems, for example Berry-Kubo method, the expansion method for stationary states and topology method et al. In 1986, Mengli Du and J. B. Delos et al took out the closed orbit theory based on the Gutzwiller trace formula, the closed orbit theory made the research developed quickly in the quantum billiard field. It provides the principle to study the correspondence between quantum physics and classical physics in quantum billiard system and the closed orbit theory is called the only bridge between the classical world and quantum world. The Delos group have study the transport problem in the circular micro-cavity with the diffraction S matrix method, they considered that the fluctuation of the conductance caused by the coherence between the wave of the classical orbits connecting the different lead. The Christopher Stampfer group have studied the transport problem in the quantum billiard system with the pseudo path semiclassical approximate and the Dyson equation and solved the sharp edge effect in the input and output mouth. Considering the real situation, the research of the dynamics on the quantum billiard system with applied magnetic field is need. The distribution function in the phase-space is an useful tool, since the phase-space formula give out a frame to describe the quantum phenomenon with classical massage, it provides a physics insight for the theoretical researcher which can not be achieved by other method easily, so it has an important theoretical signification.
After the quantum mechanics appeared, the method and the compute technique have been an effective tool to calculate the atomic and molecular systems explicitly. The results of the quantum calculation and the experiment can eliminate people’s any query on the quantum mechanics, so the quantum mechanics is still the exact theory to solve the micro systems. When we deal with the multiple dimensional and nonintegrable systems, the quantum calculation needs large numerical calculation, although we can select right basic vector to optimize the Hailtonian, the diagonal of the Hamiltonian needs still large computational works. So the numerical results almost can not help us to understand the dynamic property of the system. The other way round, the semiclassical method can explain the experiment results or the database that obtained by the quantum mechanics method. This method plays an important role to understand the dynamic property of the system for us. The correspondence between the quantum mechanics in the micro systems and the classical mechanics in the macro systems is still the hot topic for people nowadays, knowing about this correspondence is very important to understand the natural essence deeply for people. The classical-quantum correspondence has gone through a long history. When the quantum mechanics was born, Plank and Einstein was puzzled by the disagreement between black-body radiation and classical physics. Then Hersenberg established a quantum method to understand classical mechanics, and the quantum system was finished when the Schr?dinger equation was provided by Schr?dinger. Later Gutzwiller developed the semiclasscial method and the closed orbit theory was established by Du and J. B. Delos. When the quantum phase-space theory was given out, the classical-quantum correspondence was developed quickly. But for the limitation of the Heisenberg uncertainty relation, the picture of quantum phase-space is not only one. This uncertain property is the largest defect that mainly reflected in the mathematic function and operator with arbitrary property. Then to find an effective method to describe the phase-space is a goal that people pursue in long time.
In this work, we calculate the open orbit quantum spectra in the 3-dimensional cubic billiard system, and obtain the Fourier transformation of the quantum spectra in three dimension. According the topology property of the system, we get the classical trajectory which connect the arbitrary two points in the 3-dimensional quantum billiards. Comparing the length of the classical orbits and the peaks position of the quantum spectra, we find that each of the quantum peaks position corresponding one or several length of the classical trajectories. Then it gives out the correspondence between the quantum physics and classical physics, and provide a clear physic picture to understand the quantum transport problems. According to perturbation expansion of the Dyson equation, we calculate the transport property of the square quantum billiard system, and then we analyze the effects of the different width of the lead on the Fourier transform of the transition coefficient. In order to consider the multiple scattering and diffraction on the lead mouth in the cavity, we deal with the triangular billiard junction under the kinks effects. We obtain change relation of the square of the transition coefficient with energy of the system and the quantum spectra of transmit coefficient after the Fourier transform. After compared with the length of the classical orbit, we find that each of the quantum peak corresponds one or several classical trajectories, then we obtain the correspondence of the quantum physics and classical physics in some precision. Some shorter classical trajectories are obtained by the topologic character and the stationary phase approximation. We also give out the distribution of the states density inter the equilateral triangle billiard system by solving the Schrodinger equation of the triangular quantum billiard system.
The Husimi distribution is a distribution function which includes many quantum states in the phase space, it is applied to the quantum mechanics widely as a counter part of Poincarésurface section in the classical systems. In this work we study on the circular quantum billiard applied a external magnetic field. The eigen-value and the eigen-wave function are obtained by solving the Schrodinger equation of the system. In the different magnetic field, we calculated the nearest energy space level statistic, we find that the nearest energy level spacing follow the Poisson distribution, so it is still an integrable system when applied external magnetic field for the higher symmetry circular quantum billiard system. We calculate the Husimi distribution with the eigenstates of the system and find that the peak of the Husimi function splits into two peaks in the r direction firstly, and then it splits in the p direction with the magnetic field enhance, meanwhile the peaks locates at the edge of the quantum billiard. Each of the peaks of the Husimi function stands for a point on the phase plane which the classical trajectory cross through. For the electrons can be treated as classical particles that move in the system, according to the classical motivation equation, we can obtain the classical orbit of the electrons traveling in the cavity. From the classical trajectories that obtained by different initial conditions, we find that the classical trajectories of the electrons depends on the initial condition sensitively. So whether the property of the system is chaotic or regular depend on the circular trajectories of particles that travels in the magnetic cavity does not match with the boundary shape of the quantum billiard.
The thesis works include five chapters. The first chapter is the summarization, which briefly introduces the mesoscopic physics and the background of the quantum billiard system. The signification of the subject we choose and the main work we have done. The second chapter introduce the open orbit quantum spectral function, we give out the calculation results of the quantum spectra of the 3-dimensional quantum billiards, and search the quantum-classical correspondence. In the third chapter, we discuss the transport property of the open square billiard and the open triangular billiard by Dyson equation and considering the Kink effect separately. In the next chapter, we solve the Schrodinger equation with addition magnetic field, and discuss the nearest energy level spacing distribution in the circular quantum billiards with magnetic field. We also discuss the quantum-classical correspondence in the phase-space with Husimi distribution function. As the conclusion, in the last chapter, we briefly summarize the total subject and give an outlook for the future work.
对于不可积体系,一般无法得到解析解,而只能通过理论计算的方法求得薛定谔方程的数值解。对于量子台球系统,人们已经发展了许多数值求解的方法如:有限差分法、定态展开方法、B-样条法……等,然而数值方法需要大量的数据点和参数,而且由于算法复杂,通常只能得到近似解。而对于混沌过程,即使初值上的微小偏差也会对结果产生巨大的甚至是灾难性的影响。因此量子混沌的研究至今未能顺利得到广泛开展。科研工作者需要努力基于物理直觉,发展理论方法,以期推动量子混沌研究以更便利、简洁而且有效的方式广为人们所接受。这方面,量子谱函数方法、相空间分布函数方法以及传输矩阵结合拓扑学的方法都是成功的例证。
从1970年Sinai首先利用经典力学的方法研究了Sinai量子台球各态历经混沌性质和横向叶脉性质,然后Bunimovich等人又利用微扰理论从直观上解释了体育场型量子台球的性质,在1977年,Berry等人利用统计的方法研究了量子台球的混沌性.还提出了Berry-Kubo公式方法;此外,定态展开方法以及拓扑学等研究方法也相继产生。Landauer猜想电子通过这种微腔的电导可以通过计算传输矩阵,对传输系数求和得到。1986年,杜孟利和Delos等人研究外Rydberg原子在强电磁场中光吸收谱时在Gutzwiller态密度迹公式的基础上提出了闭合轨道理论为研究经典物理与量子力学的对应关系提供了理论基础,被称为是联系经典世界与量子世界的唯一桥梁。这种理论扩展到开轨道情况,也为研究量子台球性质提供新的有用的工具,它导致量子谱函数方法的提出并获得广泛应用。Delos小组利用S矩阵方法研究了圆形量子微腔的传输问题,认为微结中的电导涨落是由于沿着连接不同导线间的经典轨迹的波的相干导致的并考虑了电子在导线接口处的衍射效应。Christopher Stampfer小组利用赝路径半经典近似和衍射散射的Dyson方程研究量子台球体系的传输问题,解决了在导线出口和入口处的尖角效应。为了符合实际情况,研究外场中量子台球的动力学行为是必要的。对于多分量高简并体系,相空间分布函数方法是一个非常有用的工具,因为相空间公式提供了一个利用经典信息描述量子现象的理论框架,为理论研究者提供有用的物理洞见,是其他方法不容易实现的,具有重要的理论意义。
量子力学诞生以来,其方法和计算技术已经成为原子和分子体系中精确计算的主要手段。理论计算和实验测量结果的精确符合消除了人们对量子力学基本概念的任何质疑,直到今天量子力学仍然是人们解决微观体系的精确理论。但是在应用量子力学的方法解决多维不可积体系时需要进行大量的数值计算(虽然可以通过选取合适的基矢改进本征值计算,对角化哈密顿量需要的计算量通常十分庞大),而这些数值计算的结果对于我们了解体系动力学性质的作用甚微。相反,半经典方法可以很好的解释实验结果或应用理论计算得到的数据,这种方法对于我们了解体系的动力学性质以及探索新的发展方向起到了重要的作用。因而微观体系中的量子力学和经典力学的对应关系一直是人们十分感兴趣的重大课题,对人们更深地理解自然的本质有着重要的意义。量子力学与经典物理的对应关系经历了从量子力学诞生之初,Bohr和Sommerfeld的量子化假设到对可积系统存在作用量量子化的本征轨迹EBK定理的提出,再到后来Gutzwiller对于不可积系统发展的周期轨道理论和杜孟利J. B. Delos提出的半经典闭合轨道理论以及谱函数方法和量子力学相空间理论方法的飞跃过程。但是由于Heisenberg的测不准关系的限制,量子力学相空间图像不是唯一的。这种不唯一性主要表现在所使用的数学函数或算符具有一定的任意性,这就是量子力学相空间图像的一大缺陷。因此,探索新的有效的相空间的描述一直是人们长期追求的目标,这也是本文提出采用和发展Husimi分布函数的初衷。
本文首先采用量子谱函数方法研究了三维正方体量子台球的经典-量子对应。通过对经典轨道长度与量子谱函数傅里叶变换谱峰位置的比较发现,每个量子峰位置与一条或者几条经典轨道长度对应。为理解量子输运问题提供了比较清晰的物理图像。把谱函数方法应用于开放系统的输运过程则需考虑传导和散射,根据Dyson方程的微扰展开计算了正方形量子台球的输运性质,进而分析了不同导线宽度对传输系数的傅里叶变换谱的影响。为了充分考虑台球腔内部多重散射和导线开口处衍射效应的影响,我们在研究开放三角形量子台球微结的输运性质时了考虑扭结效应,并通过几何拓扑的关系和稳定相近似得到了体系长度较短的部分经典轨道。计算得到了三角形量子台球内部的波函数态密度分布和传输系数随能量的变化关系,对传输系数进行傅里叶变换得到量子谱,与经典轨迹长度做比较,发现每个傅里叶变换的量子峰位置对应一条或者几条经典轨迹,从而在一定精度允许范围内同样给出量子物理与经典物理的对应关系。
Husimi分布函数是投影到相空间量子态的分布函数,作为经典系统中Poincaré截面的一种表述方式被广泛的应用于经典-量子研究对应中。在本文中我们研究了外磁场作用下的圆形量子台球。通过求解磁场下的Schr?dinger方程得到体系的能量本征值与本征波函数。对不同磁场下的能量本征值进行能级统计分析发现最近邻能级间隔分布符合Poisson分布,说明对称性比较高的圆形量子台球即使在加外磁场以后仍然是可积体系。利用体系的能量本征态计算得到相空间的Husimi分布,发现随着外加磁场的增加,Husimi分布函数的峰先在r方向分裂,然后在p方向分裂,同时峰位向台球的边缘处聚集。Husimi分布函数的每个峰代表经典周期轨迹在相平面上穿过的点。这里得到的分布性质很容易过渡到混沌运动情况。当粒子在磁场量子台球腔内运动的圆轨迹与腔边界不相吻合,如磁场作用下的矩形腔中电子的运动将出现混沌。
论文共分五章。第一章为研究背景综述,简要介绍介观物理,量子台球体系的特点以及选题的意义以及主要已经进行的工作。第二章介绍了开轨道的量子谱函数,计算了三维量子台球的量子谱,寻找量子与经典之间的对应关系。第三章利用Dyson方程和考虑扭结效应分别讨论了开放矩形量子台球和正三角形量子台球的输运性质。第四章通过求解外加磁场下Schr?dinger方程,讨论了外加磁场下圆量子台球体系的能级统计,利用相空间Husimi分布函数讨论了相空间量子-经典对应关系。第五章是本文的结束语,对已完成研究进行简要的总结,并提出今后可以进一步探索的问题。
With the development of the technology in lithography and the growth in crystal become perfect day by day, it is possible to produce the little scale and random shape quantum billiard system. Especially in the recent twenty years, the structure of the nano-semiconductor has become the ideal model to research the conductor that the electrons transport through the micro-cavity, with the perfect development of the semiconductor material and the crystal’s purity, the heterojunction structure can be formed by folium of the electrons in higher speed. In the perpendicular direction of the layer, the motivation of the electrons is quantization, then the motivation of the electrons can be located in the plane. As a theoretical model, this system can be treated as two dimensional electronic gas (2DEG) or quantum billiards system and it is different from mental thin film in their low electric number. The low electronic density means that the electrons have large Fermi wavelength (which can achieve 40nm) and large mean free path (which can achieve 10μm). Then the coulomb interaction between the electrons can be ignored, the scattering impact between electrons and the impurity particles is very little, then the electrons can be seen as the classical particles, the model of quantum billiards or two dimensional electrons gas is used very convenient to research the quantum transmission. The property of the two dimensional quantum billiard system is related to the shape of the boundary closely. It is easy to control the motivation of the particles in the cavity transmitting to chaos from regular conveniently by changing the shape of the billiards system. For example, circular billiards, square billiards and ellipse billiard systems are integrable systems, which motivation of the particles is regular in these systems. When the shape of the systems changed into Sinai or stadium billiard, the systems becomes nonintegrable and the motivation of the particles becomes chaotic. In addition, when the system is applied a magnetic field, the character of the motion will also happen a large change. The research results show that if the trajectories of the particles after applied the uniform magnetic field do not match with the shape of the boundary for the system, the property of the system will become chaos. In other hand, based on these reason, the quantum billiard system becomes an ideal theoretical model to study the dynamic property of the quantum transport system.
Generally, the analysis results can not be obtained for the quantum chaos systems, one can only obtain the numerical results by the numerical method to solve the Schr?dinger equation. Then some people have developed some numerical methods to solve the problem, for example, Finite Difference method, Vector-based expansion, and Bsplines method, et al, but the numerical calculation needs a great deal of database, the arithmetic is very complex and in the end we only obtain the approximate results. Therefore, the research of the quantum chaos can not be widely studied until nowadays. This faultiness in the mathematic will be overcome by the research partner in the nearly future, and expect that it can drive the quantum chaos obtained by people in more convenient and simple fashion.
In 1970, Mr. Sinai studied the chaotic property in the Sinai quantum billiard system by semiclassical dynamics firstly, and then, Bunimovich et al gave out the properties of the stadium billiard system with the perturbation theory. In 1977, Berry et al researched on the chaotic property of quantum billiard systems with statistic method, meanwhile there were other method used to study the quantum billiards systems, for example Berry-Kubo method, the expansion method for stationary states and topology method et al. In 1986, Mengli Du and J. B. Delos et al took out the closed orbit theory based on the Gutzwiller trace formula, the closed orbit theory made the research developed quickly in the quantum billiard field. It provides the principle to study the correspondence between quantum physics and classical physics in quantum billiard system and the closed orbit theory is called the only bridge between the classical world and quantum world. The Delos group have study the transport problem in the circular micro-cavity with the diffraction S matrix method, they considered that the fluctuation of the conductance caused by the coherence between the wave of the classical orbits connecting the different lead. The Christopher Stampfer group have studied the transport problem in the quantum billiard system with the pseudo path semiclassical approximate and the Dyson equation and solved the sharp edge effect in the input and output mouth. Considering the real situation, the research of the dynamics on the quantum billiard system with applied magnetic field is need. The distribution function in the phase-space is an useful tool, since the phase-space formula give out a frame to describe the quantum phenomenon with classical massage, it provides a physics insight for the theoretical researcher which can not be achieved by other method easily, so it has an important theoretical signification.
After the quantum mechanics appeared, the method and the compute technique have been an effective tool to calculate the atomic and molecular systems explicitly. The results of the quantum calculation and the experiment can eliminate people’s any query on the quantum mechanics, so the quantum mechanics is still the exact theory to solve the micro systems. When we deal with the multiple dimensional and nonintegrable systems, the quantum calculation needs large numerical calculation, although we can select right basic vector to optimize the Hailtonian, the diagonal of the Hamiltonian needs still large computational works. So the numerical results almost can not help us to understand the dynamic property of the system. The other way round, the semiclassical method can explain the experiment results or the database that obtained by the quantum mechanics method. This method plays an important role to understand the dynamic property of the system for us. The correspondence between the quantum mechanics in the micro systems and the classical mechanics in the macro systems is still the hot topic for people nowadays, knowing about this correspondence is very important to understand the natural essence deeply for people. The classical-quantum correspondence has gone through a long history. When the quantum mechanics was born, Plank and Einstein was puzzled by the disagreement between black-body radiation and classical physics. Then Hersenberg established a quantum method to understand classical mechanics, and the quantum system was finished when the Schr?dinger equation was provided by Schr?dinger. Later Gutzwiller developed the semiclasscial method and the closed orbit theory was established by Du and J. B. Delos. When the quantum phase-space theory was given out, the classical-quantum correspondence was developed quickly. But for the limitation of the Heisenberg uncertainty relation, the picture of quantum phase-space is not only one. This uncertain property is the largest defect that mainly reflected in the mathematic function and operator with arbitrary property. Then to find an effective method to describe the phase-space is a goal that people pursue in long time.
In this work, we calculate the open orbit quantum spectra in the 3-dimensional cubic billiard system, and obtain the Fourier transformation of the quantum spectra in three dimension. According the topology property of the system, we get the classical trajectory which connect the arbitrary two points in the 3-dimensional quantum billiards. Comparing the length of the classical orbits and the peaks position of the quantum spectra, we find that each of the quantum peaks position corresponding one or several length of the classical trajectories. Then it gives out the correspondence between the quantum physics and classical physics, and provide a clear physic picture to understand the quantum transport problems. According to perturbation expansion of the Dyson equation, we calculate the transport property of the square quantum billiard system, and then we analyze the effects of the different width of the lead on the Fourier transform of the transition coefficient. In order to consider the multiple scattering and diffraction on the lead mouth in the cavity, we deal with the triangular billiard junction under the kinks effects. We obtain change relation of the square of the transition coefficient with energy of the system and the quantum spectra of transmit coefficient after the Fourier transform. After compared with the length of the classical orbit, we find that each of the quantum peak corresponds one or several classical trajectories, then we obtain the correspondence of the quantum physics and classical physics in some precision. Some shorter classical trajectories are obtained by the topologic character and the stationary phase approximation. We also give out the distribution of the states density inter the equilateral triangle billiard system by solving the Schrodinger equation of the triangular quantum billiard system.
The Husimi distribution is a distribution function which includes many quantum states in the phase space, it is applied to the quantum mechanics widely as a counter part of Poincarésurface section in the classical systems. In this work we study on the circular quantum billiard applied a external magnetic field. The eigen-value and the eigen-wave function are obtained by solving the Schrodinger equation of the system. In the different magnetic field, we calculated the nearest energy space level statistic, we find that the nearest energy level spacing follow the Poisson distribution, so it is still an integrable system when applied external magnetic field for the higher symmetry circular quantum billiard system. We calculate the Husimi distribution with the eigenstates of the system and find that the peak of the Husimi function splits into two peaks in the r direction firstly, and then it splits in the p direction with the magnetic field enhance, meanwhile the peaks locates at the edge of the quantum billiard. Each of the peaks of the Husimi function stands for a point on the phase plane which the classical trajectory cross through. For the electrons can be treated as classical particles that move in the system, according to the classical motivation equation, we can obtain the classical orbit of the electrons traveling in the cavity. From the classical trajectories that obtained by different initial conditions, we find that the classical trajectories of the electrons depends on the initial condition sensitively. So whether the property of the system is chaotic or regular depend on the circular trajectories of particles that travels in the magnetic cavity does not match with the boundary shape of the quantum billiard.
The thesis works include five chapters. The first chapter is the summarization, which briefly introduces the mesoscopic physics and the background of the quantum billiard system. The signification of the subject we choose and the main work we have done. The second chapter introduce the open orbit quantum spectral function, we give out the calculation results of the quantum spectra of the 3-dimensional quantum billiards, and search the quantum-classical correspondence. In the third chapter, we discuss the transport property of the open square billiard and the open triangular billiard by Dyson equation and considering the Kink effect separately. In the next chapter, we solve the Schrodinger equation with addition magnetic field, and discuss the nearest energy level spacing distribution in the circular quantum billiards with magnetic field. We also discuss the quantum-classical correspondence in the phase-space with Husimi distribution function. As the conclusion, in the last chapter, we briefly summarize the total subject and give an outlook for the future work.
引文
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