  We develop a technique for finding the dynamical evolution in time of an
averaged density matrix. The result is an equation of evolution that includes
an Effective Hamiltonian, as well as decoherence terms in Lindblad form.
Applying the general equation to harmonic Hamiltonians, we confirm a previous
formula for the Effective Hamiltonian together with a new decoherence term
which should in general be included, and whose vanishing provides the criteria
for validity of the Effective Hamiltonian approach. Finally, we apply the
theory to examples of the AC Stark Shift and Three- Level Raman Transitions,
recovering a new decoherence effect in the latter.
