A novel derivation of the quantum propagator of a system described by a general quadratic Lagrangian is presented in the framework of the Heisenberg equations of motion. The general correspond- ing density matrix is obtained for a derived quantum harmonic oscillator and a particle confined in a one-dimensional Paul trap. Total mean energy, work and absorbed heat, Wigner function and excitation probabilities are found explicitly. The method presented here is based on the Heisenberg representation of position and momentum operators and can be generalized to a system consisting of a set of linearly interacting harmonic oscillators straightforwardly.
