  In this paper, we find the quantum propagator for a general time-dependent
quadratic Hamiltonian. The method is based on the properties of the propagator
and the fact that the quantum propagator fulfills two independent partial
differential equations originating from Heisenberg equations for positions and
momenta. As an application of the method, we find the quantum propagator for a
linear chain of interacting oscillators for both periodic and Dirichlet
boundary conditions. The state and excitation propagation along the harmonic
chain in the absence and presence of an external classical source is studied
and discussed. The location of the first maxima of the probability amplitude
$P(n,\tau)$ is a straight line in the $(n,\tau)$-plane, indicating a constant
speed of excitation propagation along the chain.
