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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 theparticle and wave aspects, respectively, of the quantum system. The dynamics ofthe maximum, located at the mean value of position, is governed by theNewtonian equation of the corresponding classical problem. The width, which isdirectly proportional to the position uncertainty, obeys a complex nonlinearRiccati equation which can be transformed into a real nonlinear Ermakovequation. The coupled pair of these equations yields a dynamical invariantwhich plays a key role in our investigation. It can be expressed in terms of acomplex variable that linearizes the Riccati equation. This variable alsoprovides the time-dependent parameters that characterize the Green's function,or Feynman kernel, of the corresponding problem. From there, also the relationbetween the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wignerfunction will be shown. Factorization of the dynamical invariant allows forcomparison with creation/annihilation operators and supersymmetry where thepartner potentials fulfil (real) Riccati equations. This provides the link to anonlinear formulation of time-independent quantum mechanics in terms of anErmakov equation for the amplitude of the stationary state wave functionscombined with a conservation law. Comparison with SUSY and the time-dependentproblems concludes our analysis.

Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems