The Dynamics Research Of The Quantum Billiards Systems

With the development of the technology in lithography and the growth in 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 semiconductor material and the crystal's purity, 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 or stadium billiard, the systems becomes nonintegrable and the motivation of the particles becomes chaotic. In addition, when the system is 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 shape of the boundary for the system, the property of the system will become chaos. In other hand, based on these reason, the quantum billiard system becomes 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. Therefore, the research of the quantum chaos can not be widely studied until nowadays. This faultiness in the mathematic will be overcome by the research partner in the nearly future, and expect that it can drive the quantum chaos obtained by people in more convenient and simple fashion.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 connecting 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. Considering the real situation, the research of the dynamics on the quantum billiard system with applied magnetic field is need. The distribution function in the phase-space is an useful tool, since the phase-space formula give out a frame to describe the quantum phenomenon with classical massage, it provides a physics insight for the theoretical researcher which can not be achieved by other method easily, so it has an important theoretical signification.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, although we can select right basic vector to optimize the Hailtonian, the diagonal of the Hamiltonian needs still large computational works. So 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.In this work, we calculate the open orbit quantum spectra in the 3-dimensional cubic billiard system, and obtain the Fourier transformation of the quantum spectra in three dimension. According the topology property of the system, we get the classical trajectory which connect the arbitrary two points in the 3-dimensional quantum billiards. Comparing the length of the classical orbits and the peaks position of the quantum spectra, we find that each of the quantum peaks position corresponding one or several length of the classical trajectories. Then it gives out the correspondence between the quantum physics and classical physics, and provide a clear physic picture to understand the quantum transport problems. According to perturbation expansion of the Dyson equation, we calculate the transport property of the square quantum billiard system, and then we analyze the effects of the different width of the lead on the Fourier transform of the transition coefficient. In order to consider the multiple scattering and diffraction on the lead mouth in the cavity, we deal with the triangular billiard junction under the kinks effects. We obtain change relation of the square of the transition coefficient with energy of the system and the quantum spectra of transmit coefficient after the Fourier transform. After compared with the length of the classical orbit, we find that each of the quantum peak corresponds one or several classical trajectories, then we obtain the correspondence of the quantum physics and classical physics in some precision. Some shorter classical trajectories are obtained by the topologic character and the stationary phase approximation. We also give out the distribution of the states density inter the equilateral triangle billiard system by solving the Schrodinger equation of the triangular quantum billiard system.The Husimi distribution is a distribution function which includes many quantum states in the phase space, it is applied to the quantum mechanics widely as a counter part of Poincarésurface section in the classical systems. In this work we study on the circular quantum billiard applied a external magnetic field. The eigen-value and the eigen-wave function are obtained by solving the Schrodinger equation of the system. In the different magnetic field, we calculated the nearest energy space level statistic, we find that the nearest energy level spacing follow the Poisson distribution, so it is still an integrable system when applied external magnetic field for the higher symmetry circular quantum billiard system. We calculate the Husimi distribution with the eigenstates of the system and find that the peak of the Husimi function splits into two peaks in the r direction firstly, and then it splits in the p direction with the magnetic field enhance, meanwhile the peaks locates at the edge of the quantum billiard. Each of the peaks of the Husimi function stands for a point on the phase plane which the classical trajectory cross through. For the electrons can be treated as classical particles that move in the system, according to the classical motivation equation, we can obtain the classical orbit of the electrons traveling in the cavity. From the classical trajectories that obtained by different initial conditions, we find that the classical trajectories of the electrons depends on the initial condition sensitively. So whether the property of the system is chaotic or regular depend on the circular trajectories of particles that travels in the magnetic cavity does not match with the boundary shape of the quantum billiard.The thesis works include five 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 introduce the open orbit quantum spectral function, we give out the calculation results of the quantum spectra of the 3-dimensional quantum billiards, and search the quantum-classical correspondence. In the third chapter, we discuss the transport property of the open square billiard and the open triangular billiard by Dyson equation and considering the Kink effect separately. In the next chapter, we solve the Schrodinger equation with addition magnetic field, and discuss the nearest energy level spacing distribution in the circular quantum billiards with magnetic field. We also discuss the quantum-classical correspondence in the phase-space with Husimi distribution function. As the conclusion, in the last chapter, we briefly summarize the total subject and give an outlook for the future work.