Classical and quantum chaos: Strongly interacting particles in a confined geometry.

摘要

This dissertation details the classical and quantum dynamics of two mechanical systems. The first one represents a charged particle confined inside a square elastic boundary acted on by a uniform magnetic field—the Square Magnetic Billiard. The second system, called the Circular Coulomb Billiard, consists of two particles, interacting by virtue of the Coulomb potential, and enclosed inside a circular boundary. One of the particles is considered to be massive and remains stationary.; The first two chapters give a brief history of classical and quantum chaos, and review the major theoretical concepts. The third chapter analyzes the classical dynamics of the Square Magnetic Billiard. A number of approaches were used for numerical experiments: which shows that the system's classical behavior ranges from completely integrable to fully chaotic, but then the system restores it's integrability as the magnetic field continues to grow.; The fourth chapter examines the Square Magnetic Billiard quantum mechanically. The eigenvalues for intermediate strengths of the magnetic field exhibit a great deal of an inter-level repulsion and the eigenfunctions demonstrate quantum scars. As the classical analogue restores its integrability, the quantum spectrum tends to Landau levels. The time evolution of the system also displays chaotic features for intermediate strength of the magnetic field. A model of a quantum dot based on the Square Magnetic Billiard show resonant character in the dependence of the transition.; The last two chapters focus on the Circular Coulomb billiard. The classical dynamics display a transition from integrability to a mixed phase space as a measure of asymmetry grows. A second parameter, the strength of interaction, suppresses chaos for small degrees of asymmetry and intensifies it for higher values. The quantum energy eigenvalues show strong correlation and eigenfunctions display quantum scars for a range of parameters corresponding to chaotic classical analogue. However, some uncorrelated levels persist in the spectrum, which can be attributed to the mixed phase space of the classical dynamics. A model of a quantum dot based on the Circular Coulomb Billiard displays an extremely sharp decay of a transport in a symmetric case.