  In this work we present the simplest generic form of the propagator for the
time-dependent quadratic Hamiltonian. We manifest the simplicity of our method
by giving explicitly the propagators for a free particle in time-dependent
electric field, forced harmonic oscillator and the Paul trap. Exact transition
amplitudes and uncertainties are calculated analytically for the Paul trap and
harmonic oscillator. The results show that near the instability regions very
large quantum mechanical uncertainties are obtained as demonstrated in a
special figure. The method is also applied to calculating the trajectory of a
classical forced time-dependent harmonic oscillator.
