Gaussian functions and their integrals
Dynamics on a quadratic potential and relation between the mean values of the Gaussian distribution
It is well-known that a normal distribution can be fully characterised by its first two moments. Its cumulants of the order higher than 2 are all zero (see for example, Abramowitz, Stegun).
However, in terms of the dynamics of the density which can be closely approximated by a Gaussian distribution it becomes an interesting feature. One can derive explicitly the relation between the mean values of the higher powers of coordinate and momentum using analytical form of the Gaussian integrals and their mean values and variance.
Relation between the Gaussian integrals needed to compute the mean values of different powers of R:
