Moment equations for probability distributions in classical and quantum mechanics

The equations of motion for phase-space moments and correlations are derived systematically for quantum and classical dynamics, and are solved numerically for chaotic and regular motions of the Henon-Heiles model. For very narrow probability distributions, Ehrenfest's theorem implies that the centroid of the quantum state will approximately follow a classical trajectory. But the error in Ehrenfest's theorem does not scale with (h) over bar, and is found to be governed essentially by classical quantities. The difference between the centroids of the quantum and classical probability distributions, and the difference between the variances of those distributions, scale as li?, and so are the true measures of quantum effects. For chaotic motions, these differences between quantum and classical motions grow exponentially, with a larger exponent than does the Variance of the distributions. For regular motions, the variance of the distributions grows as t(2), whereas the differences between the quantum and classical motions grow as t(3).