| The theoretical study of micron-scale quantum-mechanical systems generally begins with two assumptions about the potential: that there is no background potential, and that any confining potential is hard-walled. In this thesis, we will look at a phenomenon that is seen when these assumptions are not made, in the context of electron conductance through two-dimensional electron gasses (2DEGs).; We begin by setting out two different mathematical frameworks for studying systems with smooth potentials. The discrete variable representation method treats closed systems, where one is solving for eigenstates and eigenvalues. The inverse Green's function method treats open systems, where one is solving for the scattering matrix and steady-state electron flux. It is the latter method that we will apply to the case of 2DEG conductance.; Our study is motivated by recent experiments which probed the spatial pattern of electron flux. In agreement with these experiments, we find that electrons follow narrow branches rather than a diffusive spreading pattern. We conclude that the branches are the result of small-angle scattering off of a weak, smooth disordered background potential, generated by the layer of donor atoms in the 2DEG crystal. We then consider the experimentally observed interference fringes, which persist to ranges of several microns. We present a novel model that explains the persistence and character of these fringes, relying only on first-order scattering off of the sharp potentials generated by impurities in the crystal.; We then turn to the methods of classical mechanics to study the branching pattern. Using classical trajectory stability analysis, we show that the locations of branches can be predicted by a projection of the stability matrix onto the initial manifold. We also consider the scaling laws for various statistical aspects of the classical flow (for example, the momentum-relaxation time). We find that these properties of the branched flow adhere to our theoretical predictions. Finally, we consider what one-dimensional maps can tell us about the dynamics in these systems. The map gives us an understanding of the observed correlation between branch stability properties and turning points in the evolving manifold. |
