Exact quantum-statistical dynamics of time-dependent generalized oscillators

Using a pair of linear invariant operators satisfying the Liouville-von Neumann equation, we find the most general thermal density operator and Wigner function for time-dependent generalized oscillators. The general Wigner function has five free parameters and describes the thermal Wigner function about a classical trajectory in phase space. The contour of the Wigner function depicts an elliptical orbit with a constant area moving about the classical trajectory, whose eccentricity determines the squeezing of the initial vacuum.