Evolution and invariants of oscillator moments

Moments are expectation values over wave functions (or averages over a set of classical particles) of products of powers of position and momentum. For the harmonic oscillator, the evolution in the quantum case is very closely related to that of the classical case. Here we consider the non-relativistic evolution of moments of all orders for the oscillator in one dimension and investigate invariant combinations of the moments. In particular, we find an infinite set of invariants that enable us to express the evolution of any moment in terms of sinusoids. We also find explicit expressions for the inverse of these relations, thus enabling the expression of the evolution of any moment in terms of the initial set of moments. More detailed attention is given to moments of the third and fourth order in terms of the invariant combinations.