### 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 ℏ^{2} 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(ℏ^{2}) 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 language | English (US) |
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Pages (from-to) | 157-167 |

Number of pages | 11 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 53 |

Issue number | 1 SUPPL. A |

State | Published - 1996 |

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### ASJC Scopus subject areas

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

### Cite this

**Smooth quantum potential for the hydrodynamic model.** / Gardner, Carl; Ringhofer, Christian.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 53, no. 1 SUPPL. A, pp. 157-167.

}

TY - JOUR

T1 - Smooth quantum potential for the hydrodynamic model

AU - Gardner, Carl

AU - Ringhofer, Christian

PY - 1996

Y1 - 1996

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=0001719509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001719509&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001719509

VL - 53

SP - 157

EP - 167

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 1 SUPPL. A

ER -