Quantum Transport And Spectral Analysis Of Two-dimensional Annular Billiard System

The development of laser theory and technique gives rise to extensive influence on the traditional atomic and molecular physics, the attention of the people convert from emphasizing particularly on the energy level structure of system to its correlative dynamic property in the mesoscopic physical field. Gutzwiller's theory of the density of states provides a basal theory for analyzing the dynamic behaviors of many model systems. Closed-orbit theory which is developed by M.L.Du and J.B.Delos open up a way to a deep understanding of the system's dynamics, furthermore they give a bridge linking the classical mechanics of macroscopic world to the quantum mechanics of microscopic systems. Semi-classical approximations are among the most useful tools in describing and analyzing ballistic transport in mesoscopic systems, and have become a necessary instrument to process certain quantum question in studying the movement of the microscopic particle. We studied quantum transport and spectral analysis of two-dimensional annular billiard by using closed-orbit theory.In recent years it has been recognized that nano-technology is more and more widely used in field of micro-junction. The study of microjunctions and their transport behaviors have become an important field. The prototypes have become increasingly interesting and important and will develop very well in theory and experiment in investigating the dynamic character and especially quantum chaos. It is becoming possible to structure arbitrary shapes of two-dimensional billiard system to limit the ultra cold atom by using the laser. The exploitation of nanometer apparatus impels the development of electronic technology enormously, and has become one kind of high precision technology in the semiconductor manufacturing industry.In the early quantum mechanics, the quantized method of WKB (Wentzel-Kramers-Brillouin) and EBK(Einstein-Brillouin-Keller) presented by semi-classical technology are applied respectively in one-dimension and n-dimensions, but those semi-classical quantized methods are applicable in integrabel system alone. Gutzeiller started from the exact expression of trace of Green function instead of its quantized form, and gained the state density of semi-classical system. It makes the semi-classical form of Gutzwiller perfectly applicable to complete chaotic systems. In quantum mechanics there are kinds of semi-classical calculating means which may help to understand the classical and quantum characters of a system and their correlations.Quantum billiard has been a theory model for researching mesoscopic system widely, in this content we analyze the quantum spectra of the two-dimensional annular billiard system based on the opened-orbit quantum spectrum function, and calculate the Fourier-transformed quantum spectra by utilizing combined numerical and analytical method. The quantum spectra function contains rich information of all classical orbits connecting two arbitrary points in well. The resulting peaks in plots of ( )ρL2 versus L are compared to the lengths of the classical trajectories in these geometries. We extend closed-orbited theory to open and compare the locations of the spectral peaks with the lengths of classical trajectories; the results show that there is a good correspondence between quantum spectra and classical orbits,which testifies the correspondence between quantum and classical mechanics.As the parameter (in this case f = RRoiunt) is continuously varied, the system of annular is largely different from other systems. While the energy eigenvalues and the path lengths in the annular well both change dramatically as a function of the parameter f. The classical information of the Fourier-transformed spectrum becomes weak with the dimension of inner circular becoming small, especially for the inner radius is small to the order of the de Broglie wavelength of the particle, the feature of the quantum spectra is purely wavelike phenomena, which is due to diffraction around the inner annulus,and Fresnel-Kirchhoff Diffraction theorem supply a good explanation. For systems with different scales, we choose different processing method and separately use the superiority of the classical theory and the quantum theory to pick out our useful information.Simultaneously we apply the expanding of the closed orbit theory, promote the semi-classical analysis of two-dimensional billiard system to a more common situation, and have made the discussion slightly to transmission nature of the two-dimensional annular billiard with open mouths. We calculated transmission factor of the two-dimensional annular billiard in simplest modes, which shows strong fluctuations as a function of the Fermi wave number. In order to further research the transmission coefficient, we have it a best shown by the examination of the Fourier transform, in which contain many classical orbital characteristics. In the short orbital scope, the locations of energy spectral are consonant with the lengths of classical orbits; these short classical tracks were playing the crucial role in the micro cavity transportation problems.This thesis is divided into five chapters: The first chapter is summarization, which briefly introduces the development of semi-classical theory and the character of two-dimensional billiard systems. In the second chapter we introduced the quantum spectra function, Green function, density of states and so on. In the next chapter, based on the extended closed-orbit theory together with spectral analysis we studied the correspondence between quantum mechanics and the classical counterpart in a two-dimensional annular billiard. We change the scale of the system and find the corresponding relations among the systems. In the fourth chapter, we presented a semi-classical theory for transport through open annular billiards that includes diffractively scattered paths at the lead openings. The conductance factor 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 ~t1 1 (L)2are 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 are the conclusions of our study and plan for the future.