The Luttinger-Kohn approach to effective-mass theory [10,11,12]
more conveniently includes
the effects of several bands. In this scheme the microscopic
wavefunction is decomposed into the 
Bloch functions and a set of
slowly-varying envelope functions 
, m being a band index. Differing
numbers of bands may be included, with perhaps the most general scheme being that
derived by Bastard [11].
By regrouping the various terms of Bastard's Hamiltonian with respect to the
derivatives,
it can be written in the form [13]
(summations implied over repeated indices)
where A and C are hermitian matrices and B a general matrix. A, B, and C are indexed by the band label m, and are z-dependent in a heterostructure. Using the above identities, the current density is readily shown to be
Similar expressions have been obtained by Altarelli [14] and by Burt [12].
