  Moments are expectation values of products of powers of position and
momentum, taken over quantum states (or averages over a set of classical
particles). For free particles, the evolution in the quantum case is closely
related to that of a set of classical particles. Here we consider the evolution
of symmetrized moments for free particles in one dimension, first examining the
geometric properties of the evolution for moments up to the fourth order, as
determined by their extrema and inflections. These properties are specified by
combinations of the moments that are {\it invariant} in that they remain
constant under free evolution. An inequality constrains the fourth-order
moments and shows that some geometric types of evolution are possible for a
quantum particle but not possible classically, and some examples are examined.
Explicit expressions are found for the moments of any order in terms of their
initial values, for the invariant combinations, and for the moments in terms of
these invariants.
