  A novel derivation of quantum propagator of a system described by a general
quadratic Lagrangian is presented in the framework of Heisenberg equations of
motion. The general corresponding 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.
