Quantum Diffusion-Limited Aggregation

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, only now has research begun in this area in regards to quantum random motion. 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 important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schr\"odinger equation and a classical random walk are both diffusion 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 particles moving according to Schr\"odinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.