| Abstract: | 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.
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