Current Density and Continuity in Discretized Models
Date of this Version7-21-2010
This work was supported in part by Semiconductor Research Corporation. nanohub.org computational resources provided by the Network for Computational Nanotechnology, funded by the National Science Foundation were used.
This document has been peer-reviewed.
Discrete approaches have long been used in numerical modelling of physical systems in both research and teaching. Discrete versions of the Schr ¨ odinger equation employing either one or several basis functions per mesh point are often used by senior undergraduates and beginning graduate students in computational physics projects. In studying discrete models, students can encounter conceptual difﬁculties with the representation of the current and its divergence because different ﬁnite-difference expressions, all of which reduce to the current density in the continuous limit, measure different physical quantities. Understanding these different discrete currents is essential and requires a careful analysis of the current operator, the divergence of the current and the continuity equation. Here we develop point forms of the current and its divergence valid for an arbitrary mesh and basis. We show that in discrete models currents exist only along lines joining atomic sites (or mesh points). Using these results, we derive a discrete analogue of the divergence theorem and demonstrate probability conservation in a purely localized-basis approach.
Nanoscience and Nanotechnology