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A remark on quantum dynamics

Oliver Knill
Department of Mathematics,
University of Arizona, Tucson, AZ, 85721, USA,
This paper has apperared in Helvetica Physica Acta, 71, 233-241, 1998.


Some computations in classical quantum dynamics can be simplified by substituting the Schrödinger Hamiltonian with a different operator. The time evolution can then be obtained by iterating a map. This allows efficiently to determine the Fourier coefficients of the spectral measures of the new Hamiltonian. Many properties of the quantum evolution are not affected by the deformation of the Hamiltonian because the spectral measures are only distorted. For example, a numerical computations of the Wiener averages allows to test numerically for the existence of bound states. We illustrate the time discretisation for a tight binding model of an electron in a constant or random magnetic field in the plane. As a theoretical illustration, we relate the return probability for the quantum evolution on a graph to the return probability of the corresponding random walk.

Oliver Knill, Aug 18, 1998