Riccati, Ermakov and the Quantum-Classical Connection
From the wave packet solutions of exactly-solvable time-dependent Schroedinger equations, it is known that the dynamics of such systems is governed by a Newtonian equation for the mean value, corresponding to the classical trajectory, and a complex nonlinear Riccati equation for the corresponding uncertainty, or wave aspect, which can be transformed into a real nonlinear Ermakov equation. The coupled pair of Newtonian and Ermakov equations yields a dynamical invariant which is the centre of my discussion. It not only provides the time-dependent Wigner functions of the corresponding systems, it also can be expressed in terms of complex variables that linearize the Riccati equation. The relation between these complex variables and the creation and annihilation operators, their generalization in the form of supersymmetric quantum mechanics will be mentioned. Further, the relation between the invariant and the Feynman kernel, or time-propagator, will be discussed. From this discussion, the connection between classical and quantum dynamics and, in particular, the role of the initial position uncertainty, will become obvious.