Reduction method for the linear quantum or classical oscillator with time-dependent frequency, damping, and driving

We elucidate the relationship between the linear oscillator with time-dependent frequency, damping, and driving, and the autonomous oscillator with unit frequency and no damping or driving. Such a relationship can be derived from a canonical transformation and a redefinition of the time. A study of the scalars of the transformation makes it possible to write down the evolving quantum states for the nonautonomous Hamiltonian given the evolving states for the autonomous harmonic oscillator.