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Birkhäuser Basel eBooks

The Quantum Hydrodynamic Model

Ansgar Jüngel

Published Jan 1, 2001• 0 citations
Publisher

Abstract

In this chapter we are concerned with the quantum hydrodynamic model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
DOI
10.1007/978-3-0348-8334-4_5

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Direct Numerical Solution of the Quantum Hydrodynamic Equations of Motion

Systems .........................................................................................337 21.5 The Iterative Finite Difference Method ......................................................339 21.6 Concluding Remarks and Future Research.................................................342 Acknowledgments..................................................................................................343 Bibliography ..........................................................................................................343 The quantum hydrodynamic formulation of quantum mechanics is intuitively appealing [1-7]. The quantum potential (Q) and its associated force fq = −∇Q appear on an equal footing with the classical potential and force in the equations of motion. Thus, this approach provides a unified computational method for including both classical and quantum mechanical effects in a molecular dynamics (trajectory based) calculation. However, despite their deceptively simple form, the quantum hydrodynamic equations of motion are very challenging to solve numerically. The quantum potential is non-local (i.e., the trajectories are coupled) and it is a non-linear function of the density (i.e., it depends upon the solution itself). Thus, unlike a typical classical potential, the quantum potential is manifestly time dependent and evolves in “lock-step” fashion with the dynamics. Furthermore, the quantum potential can

CRC Press eBooks•Apr 19, 2016•0 citations
Systems .........................................................................................337 21.5 The Iterative Finite Difference Method ......................................................339 21.6 Concluding Remarks and Future Research.................................................342 Acknowledgments..................................................................................................343 Bibliography ..........................................................................................................343 The quantum hydrodynamic formulation of quantum mechanics is intuitively appealing [1-7]. The quantum potential (Q) and its associated force fq = −∇Q appear on an equal footing with the classical potential and force in the equations of motion. Thus, this approach provides a unified computational method for including both classical and quantum mechanical effects in a molecular dynamics (trajectory based) calculation. However, despite their deceptively simple form, the quantum hydrodynamic equations of motion are very challenging to solve numerically. The quantum potential is non-local (i.e., the trajectories are coupled) and it is a non-linear function of the density (i.e., it depends upon the solution itself). Thus, unlike a typical classical potential, the quantum potential is manifestly time dependent and evolves in “lock-step” fashion with the dynamics. Furthermore, the quantum potential can