Quantum Chaos For A Periodically Kicked Harmonic Oscillator System

Since 1960s, nonlinear science has achieved encouraging results, especially the periodically kicked Hamilton system exhibits very complicated dynamics phenomenon. Asκincreases, the system emergence chaos. In classical mechanics, chaos means deterministic unpredictability. The fundamental reason of such chaotic unpredictability is the near orbits diverge exponently in the chaotic motion, the character of chaos is called "sensitivity to initial conditions". People begin to pay attention to the relevance of chaos and quantum after they realized the importance of chaos in classical mechanics. In fact, according to Bohr correspondence principle, 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.In this thesis, we construct a periodically kicked harmonic oscillator system as our research model, we have done following work in the paper:In classical dynamics, through our research on the space phase diagram of the system, we find that the periodic web will exist in the phase space when the ratio of against frequency and natural frequency is a rational number. As we increase the kicking strengthκ, there will be chaos near the grid, then it will become stochastic web and fractal will take place in classical phase space.In quantum dynamics, we study the distribution of eigenvalue and quasienergy. Through our research, we find that as the kicking strengthκincreases, the eigenvalue distribution of the one step time evolution operator will diffuse toward the center of the unit circle, from on the unit circle. When the eigenvalues are normalized, the normalized eigenvalue distribution will back to on the unit circle. On the basis of our research, we also study the energy level statistics, we specifically calculate the nearest-neighbor-spacing distribution (NNSD) and the average spectral rigidity distribution asκincreases. Through our study, we find that the nearest-neighbor-spacing distribution is similar to the Poisson distribution no matter how largeκis, but the average spectral rigidity distribution shows a complex phenomenon. When L is small, the average spectral rigidity distribution is similar to the Poisson distribution, with the increase ofκ, the interval of L becomes smaller in which the Poisson distribution appears. The GOE distribution and harmonic oscillator distribution will appear whenκtakes specific values. The average spectral rigidity distribution of the system has greater impact byκthan the nearest-neighbor-spacing distribution (NNSD).