Date of Award

8-15-2011

Degree Type

Thesis

Degree Name

Honors Thesis

Department

Physics

Abstract

Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, nobody has yet pursued this avenue of research in the realm of quantum random walks. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is very important to consider the quantum variant of diffusion-limited aggregation. Quantum diffusion-limited aggregation is an important step forward in understanding particle aggregation in areas where quantum effects are dominant, such as low temperature chemistry and the development of techniques for forming thin films. Recognizing that the Schrödinger equation and a classical random walk are both di usion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that the quantum di usion process can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum di usion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.