Advances in quantum waveguide dynamics, with applications to physical and biological nanostructures

Abstract

Advances in quantum waveguide dynamics, with applications to physical and biological nanostructuresIAIN JOHN CLARK It is known [1,2] that as semiconductor microstructures get progressively smaller and more free of impurities, it becomes increasingly necessary to use quantum mechanics to describe accurately these systems, which may have features sized between a few and several hundred nanometers.Recent advances in fabrication technology [11] make it feasible to build such nanostructures or mesoscopic systems, as they are known.The term "quantum waveguide" [6] is used to denote a nanostructure which acts as a conducting wire, along which a wavefunction is guided to move by the boundary conditions.Unlike classical wires, curved quantum waveguides have curvature-induced bound states; see [6] for a recent review.This has implications for nanoelectronic circuit design, since the curvature-induced bound states will appear as resonances.However, the effect of torsion (where the waveguide twists out of a plane) and twisting (where the waveguide changes its orientation along its path) upon bound states has attracted little investigation, and is the motivation for the analysis in this work.This thesis develops a unified mathematical approach by which the energy spectrum of a quantum waveguide with curvature and torsion can be analysed, for any crosssectional geometry that remains uniform along the waveguide.Following the introduction and literature review of Chapter 1, a non-relativistic Hamiltonian H for a particle on a quantum waveguide is set up in Chapter 2. A system of curvilinear coordinates is used to express Schrodinger's equation using the Laplace-Beltrami operator V 2 .The metric tensor G is expressed as the product of a curvaturedependent matrix F and the metric tensor G for a "straightened-out" waveguide.The Hamiltonian H is then shown to be unitarily equivalent to the Hamiltonian Hy for a straightened-out waveguide plus interaction terms, Hy -H o + Ky + Vu, where H o is the Hamiltonian for a "straightened-out" waveguide, Ky is a kinetic energy perturbation term and Vu is an effective potential.In Chapter 3, the spectral theory needed to investigate the curvature-induced bound states is developed.The spectrum of a Hamiltonian is partitioned into the