  The present manuscript was written in 1994 and was not published. It
addresses the form that the quantum-mechanical current density must take in
mesoscopic treatments of semiconductor heterostructures, in which the electron
dispersion relations are non-parabolic and position dependent, rendering the
textbook expressions inapplicable. The approach is to derive the continuity
equation for the specific model under consideration, using generalizations of
Green's identities to higher-order derivatives and to discrete models of
different topological structure. A new addendum addresses two issues of more
current interest: the use of irregular meshes in discrete formulations, and the
identification of the Heisenberg velocity operator to evaluate current density.
It is demonstrated that on discrete domains the velocity operator fails to
satisfy a sensible continuity equation, and therefore cannot be identified with
the current density.
