Abstract
Quantum time-evolution propagators are derived for a particle on a fractal lattice. It is shown that the path of such a particle represents a self-affine fractal. A relationship is obtained between the fractal dimension of the path (as determined by three different operational definitions) and the fractal dimension of the lattice. From these relationships it is seen that a quantum-mechanical particle in a one-dimensional Euclidean space also has a path which is a self-affine fractal. A strong analogy exists between these observations and the difference between Brownian motion and fractional Brownian motion. The uncertainty principle for a particle on a fractal lattice is derived and the phenomena of persistence and antipersistence is related to the uncertainty in momentum. The fractal quantum propagators are then used to develop a quantum theory of fractal rate constants. The theory of Miller and co-workers [J. Chem. Phys. 79, 4889 (1983); 90, 904 (1989)] is used to derive expressions for a bimolecular rate constant. Using the position-flux cross correlation function, scaling laws for the time dependence of the ‘‘rate constant’’ is obtained. These results are discussed in relation to experimental and theoretical results for time-dependent rate constants in percolation clusters.
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