Spectra Analysis Of Two-dimensional Sinai Billiards

Since the development of Periodic orbit theory for chaotic systems by Gutzwiller, it has become an important tool to study the connections between the quantized energy eigenvalues of a bound state and the classical motions of the corresponding classical point particle. 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. we studied sinai billiard using a new quantum spectry defined in the closed-orbit theory.The electronic moving is controlled by a gate voltage in the devices, and restricted in one or two dimensional cavities with arbitrary shapes which are made of materials such as semiconductor heterogeneity. At low temperature, the size of high quality semiconductor heterogeneity can be adjusted less than the electronic free path and larger than the electronic Fermi wave length, so that the electron is regarded as a free particle when it moves through the cavity. As a theoretical model of orderly and chaotic behavior, quantum billiards has been an active research for many years. Because of its unique characteristic, Sinai billiard has become a typical model to study quantum chaos. In last 20 years, with the remarkable advances in the growth on the crystal and artificial etching technology, the nanotechnology receives more and more people's favor, and founded the manufacture nanometer component new method, sinai quantum billiard has been made, which is a basis for experiment.We extend the closed-orbit theory method to open trajectories, computed the eigenvalues and eigenfunctions of Sinai billiard using the expansion method for stationary states (EMSS),and got the Fourier transformation of the quantum spectral functionρ( L),the peaks in plots ofρ( L)2 are compared with the length L ,of the classical trajectories in these geometries. It is demonstrated that the locations of quantum spectral peaks are consonant with the lengths of classical orbits, which testifies Closed-orbit theory is well, and allows us to understand the nature of chaos. This is a new method to study the correspondence between quantum and classical dynamics. In the thesis, we also computed statistic distribution of the nearest energy intervals, and revealed that distribution has a change from Poisson to Wigner with the changed geometrical shape of the billiard. this also proved the unique character of the billiard, so we can make use of it enough later.This thesis is divided into four chapters. The first chapter is summarization, which briefly introduces the chaos concept, aftertastes the development history of semiclassical Period orbit theory (Closed-orbit theory) and current situation of the studying on quantum billiard. The second chapter the expansion for stationary states method and the theory about the nearest energy interval statistic distributions are explained in detail. In the third chapter, we analyzed the quantum spectral function of Sinai billiard. Comparing it with classical orbit, we found that there are good correspondences, and we also computed the nearest energy interval statistic distribution. The results reveal that the Sinai billiards is dynamic chaos in original nature. The last chapter is the conclusion of our study and plan for the future.