  Since the initial development of one-dimensional electron gases (1DEG) two
decades ago, there has been intense interest in both the fundamental physics
and the potential applications, including quantum computation, of these quantum
transport systems. While experimental measurements of 1DEGs reveal the
conductance through a system, they do not probe critical other aspects of the
underlying physics, including energy eigenstate distribution, magnetic field
effects, and band structure. These are better accessed by theoretical modeling,
especially modeling of the energy and wavefunction distribution across a
system: the local density of states (DOS). In this thesis, a numerical Greens
function model of the local DOS in a 1DEG has been developed and implemented.
The model uses an iterative method in a discrete lattice to calculate Greens
functions by vertical slice across a 1DEG. The numerical model is adaptable to
arbitrary surface gate geometry and arbitrary finite magnetic field conditions.
When compared with exact analytical results for the local DOS, waveband
structure, and real band structure, the model returned very accurate results. A
second numerical model was also developed that measured the transmission and
reflection coefficients through the quantum system based on the
Landauer-Buttiker formalism. The combination of the local DOS model with the
transmission coefficients model was applied to two current research topics:
antidot behavior and zero-dimensional to one-dimensional tunneling. These
models can be further applied to investigate a wide range of quantum transport
phenomena.
