【作者】 高峰；

【导师】 林圣路；

【作者基本信息】 山东师范大学 ， 原子与分子物理， 2004， 硕士

【摘要】 近二十年来，人造原子（量子阱）和纳米器件逐渐成为一个新的热门课题，研究这些微腔结构及其输运问题对于新一代计算机的研制将产生重大的影响。量子弹子球（特别是二维弹子球）作为这些研究的理论模型和应用半经典方法研究规则和混沌行为的典型例子，一直是人们感兴趣的一个体系。本文将通过周期轨道理论（闭合轨道理论）和量子波包回归理论两种半经典方法对该体系进行谱分析和动力学性质研究。 自从Gutzwillet提出量子体系态密度迹公式以来，周期轨道理论已经成为人们研究定态体系的量子谱和所对应粒子经典运动的关系的主要工具。应用该理论，特别是在此基础上发展的闭合轨道理论，能深刻了解所研究体系的动力学性质。对于体系的量子描述和经典描述的对应关系，该理论也给出了深层次的解释。 应用波包分析量子体系的动力学性质也是近年来研究量子——经典对应关系的一个重要方面。根据定态体系的能量本征值和波函数构造含时高斯波包，把波包中的含时部分以体系的某一个本征值E（n₀）为中心进行泰勒展开，定义经典周期，量子回归时间等。根据计算得到体系的自动关联函数分析其动力学性质。 对于这两种谱分析方法，二维弹子球体系（例如：正方形弹子球体系和正三角形弹子球体系）提供了最直观的例子。本文应用二维无限深方势阱中的能量本征值和本征函数，计算该体系的量子能态密度的傅立叶变换ρ（L）。在|ρ（L）|²随L变化的函数图像中出现了一系列的峰，量子峰的位置与用经典方法得到的轨道长度符合得很好，这不但说明了周期轨道理论（闭合轨道理论）的正确性，还给出了量子描述和经典描述精确符合的典型例子。除了周期轨道理论以外，我们还应用高斯波包理论研究了正方形弹子球体系中能量本征值谱和经典轨道的关系。根据正方形势阱中的能量本征值和本征函数构造含时高斯波包，计算了经典回归周期，量子波包回归和超级回归等重要的时间标度量。应用一维无限深势阱中高斯波包的展开系数α_n的简单求和形式，在高能量（初始动量P₀=400π）下讨论波包的经典周期行为（计算关联函数），其结论与经典结果符合得很摘要好。我们还讨论了量子波包的回归和部分回归，取p。=o，即不存在经典周期，只有量子波包回归，选取特殊的初始坐标（x。，y0）二（a/2，a/2）研究了量子波包的部分回归的情况。 本文的结构如下:第一章介绍了半经典周期轨道理论（闭合轨道理论）和量子波包回归理论的发展。第二章给出了量子波包的构造和分析的一般过程，主要讨论了自动关联函数，量子波包的经典回归周期和量子波包回归的基本概念，并把它推广到二维体系。在第三章中，以二维方形弹子球体系为例，应用周期轨道理论和量子波包理论对其动力学性质进行了详细的分析。把体系的量子行为和经典行为对照后，我们发现方形弹子球体系的经典行为（经典轨道信息）和量子行为具有很好的对应。在本文的最后一章，应用闭合轨道理论的思想，选取正三角形弹子球体系作为研究对象，把对弹子球体系的半经典分析进一步推广到更一般的开轨道情况。我们用几何的方法详细的给出了经典轨道的信息（形状，轨道长度），并把这些轨道和体系的傅立叶变换的量子谱的峰一一对照。这种半经典方法更具有现实性。最后我们还对傅立叶变换的精度进行了讨论。更多还原

【Abstract】 In last 20 years, the study of "artificial atoms"(quantum well) and nanodevices has been of great interest in the relatively new field, this study of microjunctions and their transport behaviors would become useful in future generations of computers. As a theoretical model of this study and a model of orderly and chaotic behavior, quantum billiards has been an active research for many years. In this thesis, we will analyze the quantum spectra and dynamics of this system using Periodic orbits theory (Closed orbits theory) and wave packet dynamics method.Since the development of Periodic orbit theory for chaotic systems by Gutzwiller, it has become an important tool of the study of the connections between the quantized energy eigenvalues of a bound state and the classical motions of the corresponding classical point particle. Periodic orbit theory and Closed orbit theory which is developed by Du and Delos open a way to a deep understanding of the system’s dynamics, furthermore they give a bridge link the classical mechanics of macroscopic world to the quantum mechanics of microscopic systems andThe use of wave packet dynamics to analyze the quantum mechanical systems is also an increasingly important aspect of the study of the classical-quantum interface. We construct the time-dependent Gaussian wave packet solutions of Schrodinger equation with the energy eigenvalues and eigenfunctions of the bound state systems, and define the classical period, quantum mechanical revival and superrevival times by expanding the energy eigenvaluesabout the central value of the quantum number n0. Finally, we analyze the dynamics ofsystems by computing the autocorrelation function of the systems.Two-dimensional billiard systems have provided easily visualizible examples relevant for both types of analyses. As a simple example of the application to a billiard or infinite wellsystem of Periodic orbit theory we compute the Fourier transform (p(L)) of the quantummechanical energy level density of two-dimensional square billiard systems and equilateral triangle billiard systems. The resulting peaks in plots of p(L)\ versus L are compared tothe lengths of the classical trajectories in these geometries .The locations of peaks in p(L)agree with the lengths of classical orbits perfectly, which testifies the correspondence of quantum mechanics and classical mechanics. Furthermore, the connections between the energy eigenvalues spectrum of two-dimensional billiard systems and the classical dynamics of particles can be explored through the time-dependence of wave packet solutions of theSchrodinger equation. First we define the expansion coefficients an for a general Gaussianwave packet in one-dimensional infinite well and give the approximation for the expansion coefficients. The classical period, quantum mechanical revival and superrevival times are determined with the energy eigenvalues and eigenstates of the two-dimensional billiardsystems. We compute the autocorrelation function inp0 = 400;r and compare to the locationof the classical closed (corresponding to the periods for the classical closed orbits deduced from simple geometric arguments). We also discuss the revival time by considering zero momentum (p0 = 0) and the fractional revivals that are related with the special case(x0,y0) = (a/2,a/2).This thesis is divided into four chapters. The first chapter is summarization, which briefly introduces the development of semiclassical Period orbit theory (Closed orbit theory) and quantum packet wave revival theory. The second chapter introduces the construction quantum wave packet and some basic concepts. In the third chapter, as a test of Periodic orbit theory and quantum wave packet revival theory, we analyze a two-dimensional square billiard system. In the last chapter, we extend the Closed orbits theory to two-dimensional equilateral triangle billiard, in which the orbits are open fashion. Although the system is integrable, the method of separation of variables can’t be employed. The energy eigenvalues and wavef更多还原