The time-dependent Harmonic oscillator - A general and simple solution with application to an ion-trap

The harmonic oscillator (HO) is a fundamental model both in classical and in quantum mechanics. It is one of the few models that can be solved analytically and leads to many important insights. The time-dependents HO is encountered most frequently in ion traps.
We present the simplest possible formula for the propagator (and complete set of wave-functions) of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second order ordinary differential equation corresponding to the classical quadratic Hamiltonian.