Near-integrable Systems In Classical And Quantum Behavior Of The Control

In classical mechanics, chaos means deterministic unpredictability .It leads to random in the chaotic motion that near orbits diverge exponently. The character of chaos is called "sensitivity to initial conditions". The more exact scientific term is exponently local unstability of the motion. Lyapunov characteristic exponent is defined to describe the local unstability.With the extensive development of the chaotic theory in classical mechanics, it is natural that the notion of deterministic chaos is carried in quantum mechanics. In fact, according to Bohr corresponding theorem, the result achieved from carrying quantum mechanics to macroscopic motion corresponds with the one of classical mechanics, so chaotic characters of a dynamic system can be manifested in quantum mechanics. Now the studies of chaos in a quantum system include mainly the following several aspects:(1)the study of the dynamic behavior in the Hamitonian which corresponds with a certain quantum system. The numerical method which is used to analyze the integrable and chaotic systemis the Poincare section of the orbits which abides by the classical Hamitionian.(2) The study of quantum energy spectra and wavefunction in the classical chaotic system .It has been known that the energy spectra statistic of a chaotic system agrees with Wigner distribution which is achieved from Random Matrix Theory and the one of a integrable system is Possion distribution achieved originally from the Non-regular spectra. When the energy spectras of a chaotic systemchange with a parameter, the phenomena of a lot of "energy avoiding crossing" are found and can be a signature of quantum chaos.In this thesis, we construct a new quasi-integrable system which is based on the Henon-Heils model . we find the newly constructed system is quite suitable for the quantum-classical direct comparison.Our important results include the following points:(l)The classical integrable system is sensible to the external perturbation at the resonant points. To the quantum system , the evident correspondence has been found. We have also observed a transitional process of energy level structure from a type of intersection to a type of avoided crossing.(2) The best method to manifest the classical nonlinear resonance is the classical Poincare section for the system of high dimensions. But to the quantum system, we suggest a scheme for the construction of quantum Poincare section plot for the system of high dementions which is based on so-called coherent states. We realized a direct comparison between the phase space structures of the classical and quantum version of a two-dimensional system.By using coherent states, we have constructed quantum Poincare section by which we researched the area of high energy levels. When the system's energy is very high , the system should be close to quasi-classical or standard-classical system naturally. I have found that the quantum corresponding states of several closer energies are corresponding to one classical state which has the average energy.With the method suggested in (2) by which we construct quantum sections, when we study the quasi-integrable system, an obvious correspondance between quantum section plots and classical section plots has been found. Under the quantum and classical conditions, the two corresponding cases have the same property-sensitivity to the external perturbation, but they have different indications. Their implications are unified in our scheme of section plots.(3) When the classical dynamic system is changed from the integrablesystem to the chaotic one, according to Random Matrix Theory , the distribution of its spectra statistic has achieved the great transition from "similar Possion distribution"to "'Wigner distribution".