Numerical study of models of quantum chaos

Abstract

This work is dedicated to the study of models of quantum chaos. It is clear that one cannot define chaos in quantum mechanics the same way it is defined in classical dynamics. The classical definition of chaos relies upon the ideas of sensitive dependence on initial conditions and exponentially diverging trajectories. This type of definition is simply not possible in quantum mechanics. One approach is to ignore the question of defining chaos and to concentrate on identifying features of quantum systems that correspond to chaos in classical systems. A great deal of progress has been made along these lines in the second part of the last century and Random Matrix Theory (RMT), created by Wigner, was found to statistically described the spectrum of quantum systems whose classical counterpart is chaotic. In contrast, regular systems obey a Poissonian nearest neighbor level distribution. But most studies have focused on simple systems in the semi-classical limit ℏ → 0. No random matrix theory model is presently known to reproduce the measured statistics of mixed systems (with phase space containing both regular and chaotic components). These mixed systems are not the exception but rather correspond to the usual situation in physical examples. One of the motivation of this work is to test the validity of a class of RMT models to describe the statistics of mixed systems: the Porter-Rosenzweig model (PR model). To this sake we look at two cases: quantum graphs which are quite useful because of their simplicity and the hydrogen atom in uniform magnetic field because of its obvious interest. We observe a crossover from the Poisson statistics to the Wigner's one in the chaotic region. Close to the ionization threshold, a crystalline structure of energy levels is revealed. We think that these crossover effects are important in more general situations and in a sense are characteristic of mixed systems. The first part of this dissertation is concerned with the physics of the problems. We compute the statistical behavior of energy levels at short and long range and compare with the PR model. There exist very few results on the statistical functions of the PR model in the case of interest. Its underlying complexity makes it very difficult to achieve simple analysis without approximations. Although some results exist for time-irreversal Hamiltonians, the case of time-reversal invariance, which one encounters more often, is more challenging. It is virtually solved but in its present form not usefully exploitable. For the case of the hydrogen atom in uniform magnetic field, we develop a new technique to compute the integrated density of states up to third order. To the best of our knowledge, this has never been done before. We further demonstrate that the first order term, namely the Thomas-Fermi approximation, is not sufficient to faithfully describe numerical results. At last, the question of the wave packet dynamics is addressed by looking at the survival probability of a given initial state. Prediction of RMT are compared with the numerical results in the chaotic region. In agreement with RMT, it is shown that the histogram of the survival probability at fixed time is independent of the initial state and is exponential. This is fully done for quantum graphs, whereas for the hydrogen atom in uniform magnetic field only limited results have been obtained. The second part is dedicated to the numerical aspects and theories linked to the solution of the hydrogen atom in uniform magnetic field using an engineering method, namely the Finite Elements Method (FEM). Emphasis will be given to the unusual requirement of high precision within a dense part of the spectrum. It is indeed quite unusual to apply FEM in the field of quantum physics where spectral decomposition techniques are dominating. Although computing the highly excited energies of the hydrogen atom in uniform magnetic field is especially difficult, we shall demonstrate that one can achieve a high degree of accuracy. Furthermore, the technique is not restricted to the extremely high or low magnetic field strength but contrary to spectral decomposition techniques, can be applied to the entire range.