  We approach quantum dynamics in one spatial dimension from a systematic study
of moments starting from the dynamics of the mean position. This is
complementary to the approach of Brizuela whose starting point was generalized
recursion relations between moments. The infinite set of coupled equations is
truncated which allows us to use the techniques used in the study of dynamical
systems. In particular we predict for what initial variance the purely quartic
oscillator will time develop with minimal change in the shape of the initial
packet and what the frequency of oscillation of the mean position will be. We
show how quantum fluctuations will cause a particle to escape from the well of
a volcano potential and how they will cause an oscillation between the two
wells of a double well potential. Further, we consider an oscillatory external
field in addition to the double well potential and work near the separatrix
where the classical system is known to be chaotic. We show how the quantum
fluctuations suppresses the chaotic behaviour after a time interval inversely
proportional to the strength of the quantum fluctuations.
