Spectra Analysis Of Circular Billiards Systems

Since the development of periodic orbit theory for chaotic systems by Gutzwiller, ithas become an important tool of the study of the connections between the quantizedenergy eigenvalues of a bound state and the classical motions for the correspondingpoint particle The study of integrabel and chaotic interface is continue between classicalmechanics and quantum mechanics.The semi-classical method has become a necessaryinstrument to study the classical movement of the particle.For the correspondingconnection of the systems between quantum-classical description,the theory has given adeeper explanation.In recently years the study of "artificial atoms"(quantum well) and nanodevices hasbeen of great interest in the relatively new field. The study of microjunctions and theirtransport behaviors have become an important field.The models have becomeincreasingly interesting and important and will development very well in theory andexperiment during to study the dynamic character and especially quantum chaos.In the early quantum mechanics, the quantized method ofWKB( Wentzel-Kramers-Brillouin) and EBK(Einstein-Brillouin-Keller) presented bysemi-classical technology are apply respective in one-dimension and n-dimensions. Butsemi-classical quantized methods are applicable in integrabel system.Gutzeiller startedfrom the exact expression of trace of Green function insteads of its quantized form, andgained the stated density of semi-classical system. Which made the semi-classical formof Gutzwiller were perfectly applied in complete chaos systems. In quantum mechanicsthere are kinds of semi-classical calculational means. Which may help us to understandthe interface between classical and quantum.We use a recently quantum spectral function popularized it from closed-orbittheory to the opened-orbit theory. The quantum spectra function contain richinformation of all classical orbits connecting two arbitrary points in well. We study thecorrespondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevantfor both types of analyses. As a simple example of the application to a billiard or infinitewell system of Periodic orbit theory we compute the Fourier transform( ρ ( L)) of thequantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ( )ρ L2 versus L are compared to the lengths of theclassical trajectoried in these geometries. The locations of peaks in ρ ( L) agree withthe lengths of classical orbits perfectly,which testifies the correspondence of quantummechanics and classical mechanics.Furthermore, the connections between the energyeigenvalues spectrum of two-dimensional billiard systems and the classical dynamics ofparticle an be explored through the time-dependence of wave packet solutions of theSchrodinger equation. In the thesis, we study the correspondence between quantumspectra and classical orbits in the circular billiard, 1/2 circular and 1/4 circular billiardsby applying the analytic and numerical method. We find that the peak positions in theFourier-transformed quantum spectra match accurately with the lengths of the classical.This thesis id divided into four chapters. The first chapter is summarization,whichbriefly introduces the development of semiclassical Period orbit theory(Closed orbittheory) and quantum packet wave revival theory. In the second chapter we introducedthe periods orbits theory which applied in the infinite-circular well. In the third chapter,we study the correspondence between quantum spectra and classical orbits in thecircular billiard, 1/2 circular and 1/4 circular billiards by applying the analytic andnumerical method. The locations of peaks in ρ ( L) agree with the lengths of classicalorbits perfectly. In which the orbits theory air open fashion. In the last chapter, Weextend the Closed orbits theory to teo-dimensional ciucular billiard. The energyeigenvalues and wavefunctions of the system have been derived in a variety of differentcontext by different groups.We calculate the Fourier transform of the quantum spectraand examine the connections between the length of the classical path and the locationsof the quantum spectral peaks.In addition, in a numerical evaluations one must use afinite number of wave numbers to limit the energy region, witch brings into a resolutionproblem. These examples show evidently that semi-classical methods provides a bridgebetween quantum and classical mechanics.