Schaupp Thomas, Renziehausen Klaus, Barth Ingo, Engel Volker
Institut für Physikalische und Theoretische Chemie, Universität Würzberg, Emil-Fischer-Str. 42, 97074 Würzburg, Germany.
Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale), Germany.
J Chem Phys. 2021 Feb 14;154(6):064307. doi: 10.1063/5.0039466.
Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.
根据埃伦费斯特定理，动量算符的含时期望值可以通过两种等效方式来计算。表达式中出现的积分是对两个不同的函数进行的。在一种情况下，被积函数是量子力学通量密度(\underline{j})，而在另一种情况下，出现了一个不同的量(\underline{\tilde{j}})，它也具有通量密度的单位。量子通量密度(\underline{j})通过连续性方程与概率密度(\rho)相关，并且(\underline{\tilde{j}})也可用于定义满足连续性方程的密度(\tilde{\rho})。利用一个电子和一个质子的耦合动力学模型，我们记录了密度和通量密度的性质。结果表明，尽管从这两个量导出的平均动量是相同的，但各种函数表现出非常不同的坐标和时间依赖性。特别地，发现通量密度(\underline{\tilde{j}})直接监测概率密度的时间变化，而密度(\tilde{\rho})携带有关在不同空间方向发生的波包色散的信息。
