Quantum scattering theory and applications.

摘要

Scattering theory provides a convenient framework for the solution of a variety of problems. In this thesis we focus on the combination of boundary conditions and scattering potentials and the combination of non-overlapping scattering potentials within the context of scattering theory. Using a scattering t-matrix approach, we derive a useful relationship between the scattering t-matrix of the scattering potential and the Green function of the boundary, and the t-matrix of the combined system, effectively renormaliaing the scattering t-matrix to account for the boundaries. In the case of the combination of scattering potentials, the combination of t-matrix operators is achieved via multiple scattering theory. We also derive methods, primarily for numerical use, for finding the Green function of arbitrarily shaped boundaries of various sorts.; These methods can be applied to both open and closed systems. In this thesis, we consider single and multiple scatterers in two dimensional strips (regions which are infinite in one direction and bounded in the other) as well as two dimensional rectangles. In 2D strips, both the renormalization of the single scatterer strength and the conductance of disordered many-scatterer systems are studied. For the case of the single scatterer we see non-trivial renormalization effects in the narrow wire limit. In the many scatterer case, we numerically observe suppression of the conductance beyond that which is explained by weak localization.; In closed systems, we focus primarily on the eigenstates of disordered many-scatterer systems. There has been substantial investigation and calculation of properties of the eigenstate intensities of these systems. We have, for the first time, been able to investigate these questions numerically. Since there is little experimental work in this regime, these numerics provide the first test of various theoretical models. Our observations indicate that the probability of large fluctuations of the intensity of the wavefunction are explained qualitatively by various field-theoretic models. However, quantitatively, no existing theory accurately predicts the probability of these fluctuations.