Quantum Spectral Analysis Of Two-dimensional Annular Billiards Systems

There exist a wide variety of semi-classical calculatingly tools in quantum mechanics which help to make clear the substantial overlap between the classical and quantum mechanical descriptions of a system. Since the periodic orbit (PO) theory for chaotic systems is developed by Gutzwiller, it is rapidly becoming one of the most useful semiclassical tools of the study of the connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle. The study of integrabel and chaotic interface continue between classical mechanics and quantum mechanics. The semi-classical method has become a necessary instrument to study the classical movement of the particle. For the corresponding connection of the systems between quantum and classical description, the theory has given a deeper explanation.In recently years the study of"artificial atoms"(quantum well) and nanodevices has been of great interest in the relatively new field. The study of microjunctions and their transport behaviors have become an important field. The models have become increasingly interesting and important and will development very well in theory and experiment during to study the dynamic character and especially quantum chaos.In the early quantum mechanics, the quantized method of WKB (Wentzel-Kramers-Brillouin) and EBK(Einstein-Brillouin-Keller) presented by semi-classical technology are apply respective in one-dimension and n-dimensions. But semi-classical quantized methods are applicable in integrabel system. Gutzeiller started from the exact expression of trace of Green function instead of its quantized form, and gained the stated density of semi-classical system. Which made the semi-classical form of Gutzwiller were perfectly applied in complete chaos systems. In quantum mechanics there are kinds of semi-classical calculatingly means. Which may help us to understand the interface between classical and quantum.We use a quantum spectral function which contain rich information of classical orbits in well. We study the correspondence between quantum spectra and classical orbits in the annular. Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses. As a simple example of the application to a billiard or infinite well system of Periodic orbit theory we compute the Fourier transform (ρ( L)) of the quantum mechanical energy level density of two-dimensional annular billiard systems. The resulting peaks in plots of ( )ρL2 versus L are compared to the lengths of the classical trajectories in these geometries. The locations of peaks inρ( L) agree with the lengths of classical orbits perfectly, which testifies the correspondence of quantum mechanics and classical mechanics. We find that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical.As the parameter (in this case f = RRoiunt)is continuously varied, the system of annular is largely different from other systems .Providing the outer circular is same and inner circular changes ,we compare the quantum spectra of different systems in this paper. We also compare the quantum spectra of circular system and annular system in this paper. It is surprising for us that valuable something appears. This thesis is divided into four chapters. The first chapter is summarization, which briefly introduces the development of semiclassical period orbit theory. In the second chapter we introduced the period orbits theory. In the third chapter, we study the correspondence between quantum spectra and classical orbits in the annular billiard by applying the analytic and numerical method. The locations of peaks inρ( L) agree with the lengths of classical orbits perfectly. In the last chapter are the conclusions of our study and plan in the future. This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.With the development of modern experimental techniques, the study of the classical-quantum interface has become an important subject. Advances in crystal growth and lithographic techniques have made it possible to produce very small and clean devices, known as nanodevices. The electrons in such devices through gate voltage are confined to one or two spatial dimensions then should be regarded as an experimental realization of a quantum billiard. Our results possess an important insight for experiment research.