Dynamics of wave packets in the quantum Lorentz gas

Abstract

This dissertation addresses the dynamics of a quantum particle moving in an array of fixed scatterers. The system is known as the Lorentz gas. The scatterers are taken to be two- or three-dimensional hard-spheres. The quantum Lorentz gas is analyzed in two dynamical regimes: (i) semiclassical regime, and (ii) high-energy diffraction regime. In both regimes the dynamics of the quantum particle is found to be determined by properties characterizing chaotic dynamics of the counterpart classical Lorentz gas. Thus, this dissertation provides an attempt to more deeply understand the role that classical chaos plays in quantum mechanics of nonintegrable systems.

In the semiclassical regime, the quantum particle is represented by a small Gaussian wave packet immersed in the array of scatterers. The de Broglie wavelength of the particle is considered to be much smaller than both the scatterer size and the typical separation between scatterers. It is found that for times, during which the wave packet size remains smaller than the scatterer size, the spreading of the quantum wave packet is exponential in time, and the spreading rate is determined by the sum of positive Lyapunov exponents of the corresponding classical system.

The high-energy diffraction approximation allows one to analytically describe the dynamics of large wave packets in dilute scattering systems for times far beyond the Ehrenfest time. The latter is defined as the time during which the evolution of the wave packet is predominantly classical-like. The following two conditions are satisfied by the system in the high-energy diffraction regime: (i) the ratio of the particle s de Broglie wavelength to the scatterer size is much smaller than unity, and (ii) this ratio is much larger than the ratio of the scatterer size to the typical separation between scatterers. The time-dependent autocorrelation function is calculated for wave packets in hard-disk and hard-sphere geometrically open billiard systems. The envelope of the autocorrelation function is shown to decay exponentially with time, with the decay rate determined by the mean Lyapunov exponents and the Kolmogorov-Sinai entropy of the counterpart classical system.