Spectra Analysis Of Two-dimensional Billiards Systems

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. The elctronic moving is controlled by a gate voltage of devices,and restricted in the one or two dimentional cavities with arbitrary shapes which are made of varieties of materials such as semiconductor heterogeneity. At low temperature,the size of high quality semiconductor heterogeneity can be ajusted less than the electronic free path and langer than the electronic Fermi wave length, so that the eletron is regared as a free particle when it moves through the cavity,the systerm is dubbed billards. 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 .Two-dimensional billiard systems have provided easily visual examples due to both its types and analyses method. As a simple example of the application to a billiardor infinite well system of Periodic orbit theory we compute the Fourier transformation of the quantum transformation coefficient of two-dimensional elliptic billiards and quarter stadium billiard. For the variables cannot be separated in these systems (elliptic and quarter stadium and cardioid billiard systems), we adopt the expansion method for stationary states to obtain the eigenvalues and eigenfunctions of these systems. The resulting peaks in plots ofρ~2 (L) are compared to the lengths L of the classical trajectories in these geometries .The locations of peaks inρ(L) agree with the lengths of classical orbits perfectly, which testifies the correspondence of quantum mechanics and classical mechanics. In this thesis the quantum billiards in the magnetic field is also mentioned and the eigenvalues and eigenfunctions are calculated, a briefly analyses allows us to obtain the rough quantum spectra of the system.This thesis is consisted of five chapters. The first chapter is summarization, in which we introduces briefly the semiclassical Period orbit theory (Closed orbit theory),the development of history and present situation of the quantum billiard study also are concerned. The second chapter presents the basic methods to compute the eigenvalues and eigenfunctions of the quantum billiards with different geometries, the expansion method for stationary states can also be applied to other general billiards. In third chapter, as examples, the dynamic properties of the two-dimensional cardioid billiard is discussed by using the closed orbit theory. Comparing the quantum behavior with the classical behavior we discovered that the classical behavior (classical track information) and the quantum behavior have very good correspondences in the cardioid quantum billiard system, the quantum spectra in the cardioid billiards is quite complex because the system itself is chaotic, but the classical behavior and quantum behavior correspond quite good in short time scale and short orbits. The last chapter is the conclusions of our study and plan for the future.