Dynamic Property Of Annular Billiard

With the technology of lithography and the growth of crystal become perfect day by day, it is possible to produce the little scale and random shape quantum billiard system. Especially in the recent twenty years, the structure of the nano-semiconductor has become the ideal model to research the conductor that the electrons transport through the micro-cavity, with the perfect development of the purity of the semiconductor material and the crystal, the heterojunction structure can be formed by folium of the electrons in higher speed. In the perpendicular direction of the layer, the motivation of the electrons is quantization, then the motivation of the electrons can be located in the plane. As a theoretical model, this system can be treated as two dimensional electronic gas (2DEG) or quantum billiards system and it is different from mental thin film in their low electric number. The low electronic density means that the electrons have large Fermi wavelength (which can achieve 40nm) and large mean free path (which can achieve 10μm). Then the coulomb interaction between the electrons can be ignored, the scattering impact between electrons and the impurity particles is very little, then the electrons can be seen as the classical particles, the model of quantum billiards or two dimensional electrons gas is used very convenient to research the quantum transmission. The property of the two dimensional quantum billiard system is related to the shape of the boundary closely. It is easy to control the motivation of the particles in the cavity transmitting to chaos from regular conveniently by changing the shape of the billiards system. For example, circular billiards, square billiards and ellipse billiard systems are integrable systems, which motivation of the particles is regular in these systems. When the shape of the systems changed into Sinai billiard and stadium billiard, the systems becomes nonintegrable and the motivation of the particles becomes chaos. In addition, when the system applied a magnetic field, the character of the motion will also happen a large change. The research results show that if the trajectories of the particles after applied the uniform magnetic field do not match with the boundary shape of the system, the property of the system will become chaos. In other hand, based on these reason, the quantum billiard system become an ideal theoretical model to study the dynamic property of the quantum transport system.Generally, the analysis results can not be obtained for the quantum chaos systems, one can only obtain the numerical results by the numerical method to solve the Schr?dinger equation. Then some people have developed some numerical methods to solve the problem, for example, Finite Difference method, Vector-based expansion, and Bsplines method, et al, but the numerical calculation needs a great deal of database, the arithmetic is very complex and in the end we only obtain the approximate results.In 1970, Mr. Sinai studied the chaotic property in the Sinai quantum billiard system by semiclassical dynamics firstly, and then, Bunimovich et al gave out the properties of the stadium billiard system with the perturbation theory. In 1977, Berry et al researched on the chaotic property of quantum billiard systems with statistic method, meanwhile there were other method used to study the quantum billiards systems, for example Berry-Kubo method, the expansion method for stationary states and topology method et al. In 1986, Mengli Du and J. B. Delos et al took out the closed orbit theory based on the Gutzwiller trace formula, the closed orbit theory made the research developed quickly in the quantum billiard field. It provides the principle to study the correspondence between quantum physics and classical physics in quantum billiard system and the closed orbit theory is called the only bridge between the classical world and quantum world. The Delos group have study the transport problem in the circular micro-cavity with the diffraction S matrix method, they considered that the fluctuation of the conductance caused by the coherence between the wave of the classical orbits connect the different lead. The Christopher Stampfer group have studied the transport problem in the quantum billiard system with the pseudo path semiclassical approximate and the Dyson equation and solved the sharp edge effect in the input and output mouth.After the quantum mechanics appeared, the method and the compute technique have been an effective tool to calculate the atomic and molecular systems explicitly. The results of the quantum calculation and the experiment can eliminate people's any query on the quantum mechanics, so the quantum mechanics is still the exact theory to solve the micro systems. When we deal with the multiple dimensional and nonintegrable systems, the quantum calculation needs large numerical calculation. But the numerical results almost can not help us to understand the dynamic property of the system. The other way round, the semiclassical method can explain the experiment results or the database that obtained by the quantum mechanics method. This method plays an important role to understand the dynamic property of the system for us. The correspondence between the quantum mechanics in the micro systems and the classical mechanics in the macro systems is still the hot topic for people nowadays, knowing about this correspondence is very important to understand the natural essence deeply for people. The classical-quantum correspondence has gone through a long history. When the quantum mechanics was born, Plank and Einstein was puzzled by the disagreement between black-body radiation and classical physics. Then Hersenberg established a quantum method to understand classical mechanics, and the quantum system was finished when the Schr?dinger equation was provided by Schr?dinger. Later Gutzwiller developed the semiclasscial method and the closed orbit theory was established by Du and J.B. Delos. When the quantum phase-space theory was given out, the classical-quantum correspondence was developed quickly. But for the limitation of the Heisenberg uncertainty relation, the picture of quantum phase-space is not only one. This uncertain property is the largest defect that mainly reflected in the mathematic function and operator with arbitrary property. Then to find an effective method to describe the phase-space is a goal that people pursue in long time.The thesis works include four chapters. The first chapter is the summarization, which briefly introduces the mesoscopic physics and the background of the quantum billiard system. The signification of the subject we choose and the main work we have done. The second chapter we choose an annular billiard as example, study its dynamical behavior by describing phase space Poincare surface of section.. In the next chapter , we present a semi-classical theory for transport through open annular billiards that includes diffractively scattered paths at the lead openings. The conductance of a ballistic microstructure shows strong fluctuations as a function of the Fermi wave number. We perform a more detailed analysis of the fluctuation pattern by using Fourier transform. The results demonstrate that the peak positions of power spectra t% 1 1 ( L)are accordance with the lengths of the classical ballistic trajectories very well. This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics, in the last chapter, we briefly summarize the total subject and give an outlook for the future work.