We construct the effective Hamiltonian describing the motion of electr
ons in compositionally graded crystals which is valid throughout a giv
en energy band and part way into the gaps. Near the edges of a simple
or degenerate band this effective Hamiltonian reduces to an effective
mass Hamiltonian with position dependent effective mass. Next, we exam
ine more general states-not restricted to the vicinity of a band edge-
in crystals with composition and applied potential variation in one di
rection. We obtain a WKB-type solution for the envelope functions, as
well as the appropriate turning point connection rules.