next up previous
Next: A unitary discretisation Up: A remark on quantum Previous: A remark on quantum

Introduction

For numerical simulations in quantum mechanics it is important that the orbits tex2html_wrap_inline526 of the quantum evolution can be determined efficiently. Even for a bounded operator L, the computation of tex2html_wrap_inline530 can become tremendous; in higher dimensions it is a task for super computers (e.g. [23]).

Many properties of the orbit tex2html_wrap_inline532 are determined by the spectral measures tex2html_wrap_inline534 which are probability measures on the real line. Dynamical properties of the system depend on the nature of the spectral measure because the Fourier transform of tex2html_wrap_inline534 is tex2html_wrap_inline538 . For example, a Hamiltonian L with absolutely continuous spectrum leads by the Rieman-Lebesgue lemma to the transient behavior tex2html_wrap_inline542 . On the other hand, if an operator L has purely discrete spectrum, then tex2html_wrap_inline546 is almost periodic a fact which is responsible for recurrent behavior of the dynamics.

In this note, we use the fact that it is often irrelevant, whether we evolve with the Hamiltonian L or if we use a deformed Hamiltonian f(L), where f is an invertible smooth real function. The reason is that the spectral measures of f(L) are only distorted versions of the spectral measures of L. It is therefore natural, to look for a function f such that the discrete time step tex2html_wrap_inline560 can be easily computed. Properties of the spectrum, which are unchanged by a replacement tex2html_wrap_inline562 can now be determined faster through numerical experiments because iterating a map is more simple than integrating a differential equation.

If tex2html_wrap_inline564 where L has been rescaled such that aL has norm smaller or equal to 1, the time evolution can be computed by iterating the map tex2html_wrap_inline572 on tex2html_wrap_inline574 . This is the time one map for the unitary evolution of tex2html_wrap_inline576 . This allows an efficient determination of the Fourier coefficients tex2html_wrap_inline578 with tex2html_wrap_inline580 of measures tex2html_wrap_inline534 on the circle, which determine the spectral measures tex2html_wrap_inline584 of L. Some properties of the quantum evolution are not affected by the change tex2html_wrap_inline588 because the spectral measures tex2html_wrap_inline590 of tex2html_wrap_inline576 and the spectral measure tex2html_wrap_inline594 of L are related by tex2html_wrap_inline598 for every interval I.

One possibility to test for discrete spectrum of an operator is to determine numerically the Wiener averages tex2html_wrap_inline602 . We illustrate this method for a tight binding model of an electron in a constant or random magnetic field in the plane. For random magnetic fields, where the existence of point spectrum is not known, we made numerical experiments on a grid of size up to tex2html_wrap_inline604 .

We also illustrate the theoretical usefulness of the discrete time evolution by providing a relation between quantum mechanical return probabilities of the generator for a random walk on a graph and the return probability of the classical random walk on a graph: continuity properties of spectral measures with respect to the tex2html_wrap_inline606 -dimensional Hausdorff measures are related to power-law decays of averaged return probabilities of the random walk.


next up previous
Next: A unitary discretisation Up: A remark on quantum Previous: A remark on quantum

Oliver Knill
Tue Aug 18, 1998