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
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