Smooth quantum potential for the hydrodynamic model

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Abstract

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 ℏ2 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(ℏ2) 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.

Original languageEnglish (US)
Pages (from-to)157-167
Number of pages11
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume53
Issue number1 SUPPL. A
StatePublished - 1996

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Hydrodynamic Model
Stress Tensor
Quantum Hydrodynamics
hydrodynamics
Energy Density
stress tensors
flux density
Born Approximation
Hydrodynamic Equations
Effective Potential
Averaging
Smoothing
Semiconductors
Discontinuity
Singularity
Valid
First-order
hydrodynamic equations
smoothing
Born approximation

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics

Cite this

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abstract = "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 ℏ2 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(ℏ2) 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.",
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AB - 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 ℏ2 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(ℏ2) 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.

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