Connection between quantum-mechanical and classical time evolution via a dynamical invariant

The time evolution of a quantum system with at most quadratic Hamiltonian is described with the help of different methods, namely the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function. It is shown that all three methods are connected via a dynamical invariant, the so-called Ermakov invariant. This invariant introduces explicitly the quantum aspect via the position uncertainty and its possible time dependence. The importance of this aspect, also for the difference between classical and quantum dynamics, and in particular the role of the initial position uncertainty is investigated.