Max Tegmark's quantum mechanics library: gaussians

Please click here to download the paper described below. Click here if you are interested in other research of mine.



Max Tegmark & Harold Shapiro


We study the behavior of infinite systems of coupled harmonic oscillators as the time t approaches infinity, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a chain of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT is generally not applicable if the densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class.

Reference info:

Published in Phys. Rev. E, 50, 2538 (1994)

Online references:

This site also contains the latest versions of a paper that is referenced in the text; Tegmark & Yeh 1994, ("TY").

Return to my home page
This page was last modified July 1, 1998.