The Study Of The Dynamics In The Two-dimensional Billiard Systems

In the last 20 years, with the remarkable advances in the growth on the crystal andartificial etching technology, the nanotechnology receives more and more people's favor,and founded the manufacture nanometer component new method. The electronicmoving is controlled by a gate voltage in the devices, and restricted in the one or twodimensional cavities with arbitrary shapes which are made of varieties of materials suchas semiconductor heterogeneity. At low temperature, the size of the high qualitysemiconductor heterogeneity can be adjusted less than the electronic free path and largerthan the electronic Fermi wave length, so that the electron is regarded as a free particlewhen it moves through the cavity, the system is dubbed billiards. The study of "artificialatoms"(quantum well) and nanodevices have been of great interest in the relatively newfield, this study of microjunctions and their behaviors of transportation would havesignificant influence to the micro-dispositions and to bring about of micro-devices andnew generations of computers. As a theoretical model of orderly and chaotic behaviorwhich we are working quantum billiards has been an active research for many years. Inthis thesis, we will utilize Closed orbits theory to compute the correspondence betweenquantum spectra and classical trajectories, and to analyze the dynamic properties ofelliptic and quarter stadium systems.Since the development of Periodic orbit theory for chaotic systems by Gutzwiller,it has become an important tool of the study on the connections between the quantumspectra of a bound state and the classical motions. Periodic orbit theory and Closed orbittheory which is developed by Du and Delos open a way to a deep understanding of thesystem's dynamics, furthermore they provide a bridge link the classical mechanics ofmacroscopic world to the quantum counterparts of microscopic systems.Two-dimensional billiard systems have provided easily visual examples due toboth 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 transformationof the quantum transformation coefficient of two-dimensional elliptic billiards andquarter stadium billiard. For the variables cannot be separated in these systems (ellipticand quarter stadium billiard systems), we adopt the expansion method for stationarystates to obtain the eigenvalues and eigenfunctions of these systems. The resulting peaksin plots of ( )ρ L2 are compared to the lengths, L , of the classical trajectories in thesegeometries .The locations of peaks in ρ ( L) agree with the lengths of classical orbitsperfectly, which testifies the correspondence of quantum mechanics and classicalmechanics. In this thesis the quantum billiards in the magnetic field is also mentionedand the eigenvalues and eigenfunctions are calculated, a briefly analyses allows us toobtain the rough quantum spectra of the system.This thesis is consisted of five chapters. The first chapter is summarization, inwhich we introduces briefly the semiclassical Period orbit theory (Closed orbit theory),the development of history and present situation of the quantum billiard study also areconcerned. The second chapter presents the basic methods to compute the eigenvaluesand eigenfunctions of the quantum billiards with different geometries, the expansionmethod for stationary states can also be applied to other general billiards;such asquantum billiards in additional magnetic field is mentioned, and the approach of thestatistical nearest energy level spacing is utilized in this section. In third chapter, asexamples, the dynamic properties of the two-dimensional elliptic billiard and quarterstadium billiards are discussed by using the closed orbit theory. Comparing the quantumbehavior with the classical behavior we discovered that the classical behavior (classicaltrack information) and the quantum behavior have very good correspondences in theelliptic quantum billiard system, the quantum spectra in the quarter stadium billiards isquite complex because the system itself is chaotic, but the classical behavior andquantum behavior correspond quite good in short time scale and short orbits. The lastchapter is the conclusions of our study and plan for the future.