  We derive a new expansion of the Heisenberg equation of motion based on the
projection operator method proposed by Shibata, Hashitsume and Shing\=u. In
their projection operator method, a certain restriction is imposed on the
initial state. As a result, one cannot prepare arbitrary initial states, for
example a coherent state, to calculate the time development of quantum systems.
In this paper, we generalize the projection operator method by relaxing this
restriction. We explain our method in the case of a Hamiltonian both with and
without explicit time dependence. Furthermore, we apply it to an exactly
solvable model called the damped harmonic oscillator model and confirm the
validity of our method.
