Smooth quantum potential for the hydrodynamic model

An effective stress tensor and energy density for the quantum hydrodynamic (QHD) equations are derived in the Born approximation to the Bloch equation. The quantum potential appearing in the stress tensor and energy density is valid to all orders of [Formula Presented] and to first order in βV, and involves both a smoothing integration of the classical potential over space and an averaging integration over temperature. In the presence of discontinuities in the classical potential (which occur, for example, at potential barriers in semiconductors), the effective stress tensor and energy density are more tractable analytically and numerically than in the original O([Formula Presented]) QHD theory. By cancelling the leading singularity in the classical potential at a barrier and leaving a residual smooth effective potential (with a lower potential height) in the barrier region, the effective stress tensor makes the barrier partially transparent to the particle flow and provides the mechanism for particle tunneling in the QHD model.