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

Current density operator in systems with non-parabolic, position-dependent energy bands