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. 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...