  The time-evolution of the maximum and the width of exact analytic wave packet
(WP) solutions of the time-dependent Schr\"odinger equation (SE) represents the
particle and wave aspects, respectively, of the quantum system. The dynamics of
the maximum, located at the mean value of position, is governed by the
Newtonian equation of the corresponding classical problem. The width, which is
directly proportional to the position uncertainty, obeys a complex nonlinear
Riccati equation which can be transformed into a real nonlinear Ermakov
equation. The coupled pair of these equations yields a dynamical invariant
which plays a key role in our investigation. It can be expressed in terms of a
complex variable that linearizes the Riccati equation. This variable also
provides the time-dependent parameters that characterize the Green's function,
or Feynman kernel, of the corresponding problem. From there, also the relation
between the classical and quantum dynamics of the systems can be obtained.
Furthermore, the close connection between the Ermakov invariant and the Wigner
function will be shown. Factorization of the dynamical invariant allows for
comparison with creation/annihilation operators and supersymmetry where the
partner potentials fulfil (real) Riccati equations. This provides the link to a
nonlinear formulation of time-independent quantum mechanics in terms of an
Ermakov equation for the amplitude of the stationary state wave functions
combined with a conservation law. Comparison with SUSY and the time-dependent
problems concludes our analysis.
