In the approach to effective-mass theory proposed by Wannier [7]
and expounded by Slater [8],
the microscopic wavefunction 
is expanded as a linear combination of
localized Wannier functions, each of which is centered within a different unit cell.
The expansion coefficients 
can be regarded as values of a discrete lattice
function which is interpolated between lattice points by a continuous function
, for which an effective-mass Schroedinger equation is derived. If the
electron dispersion relation can be expanded as
then the effective Hamiltonian is
The presence of higher derivatives in the kinetic energy operator is clearly required
if one is to describe anything but a parabolic dispersion relation within the
context of a single-band, continuum model. These higher derivatives, however,
require that the wavefunction and its derivative (in fact all derivatives up to the
order) be continuous. This contradicts the prevailing wisdom, that
current continuity requires a kink in the wavefunction (discontinuous first derivative)
across a heterojunction which creates an effective-mass discontinuity. This view is
incorrect, and the present work attempts to clarify the situation. In the case of a
higher-order hamiltonian, there are of course additional solutions
(roots of the dispersion relation), which will usually
be evanescent at the energies of interest. These evanescent solutions participate in
the interface matching equations, and have the effect of keeping the wavefunction and
the subject derivatives continuous over a microscopic distance scale, but over mesoscopic
scale, the expected kink would appear.
To derive the appropriate J for this Hamiltonian, we require a generalized Green's identity:
(This identity is readily proven by expanding the derivative on the right-hand side; the summation then becomes a telescoping series.) The value of current density is thus
Some formulations of the matching conditions produce terms in the Hamiltonian of the form
Their contribution to the current density may be readily derived from another identity:
leading to contributions to the current density of
It appears (based upon an examination of some low-order cases) that any apparently hermitian differential operator [9] can be manipulated into an expression containing only terms of the form (4) and (7).
