We study approximate solutions to the time-dependent Schrodinger equation
$i\epsi\partial_t\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x)\,\psi_t(x)$
with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$
taking values in the space of bounded operators on
the Hilbert space $\Hi _{\rm f}$ of fast ''internal'' degrees of freedom.
By assumption $H(q,p)$ has an isolated energy band.
Using a method of Nenciu and Sordoni \cite{NS}
we prove that interband transitions are suppressed to any order in $\epsi$.
As a consequence, associated to that energy band there exists a subspace of
$L^2(\mathbb{R}^d,\Hi _{\rm f})$ almost invariant under the unitary
time evolution. We develop a systematic perturbation scheme
for the computation of effective Hamiltonians which govern approximately
the intraband time evolution. As examples for the general perturbation scheme
we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we
reconsider also the time-adiabatic theory.