Local Measure of Quantum Effects in Quantum Dynamics

The Madelung–de Broglie–Bohm formulation of the Schrödinger equation casts the time-evolution of a wave function as dynamics of an ensemble of quantum, or Bohmian, trajectories, interacting via the nonlocal quantum potential. This trajectory perspective gives insight into the quantumness (or classicality) of a given system due to clear partitioning of the energy into classical and quantum components. Here, we propose a system-independent measure of the quantumness of dynamics, based on the energy time-change, referred to as “quantum power”. This measure is local in the coordinate space. Based on applications to model chemical systems, we argue that during the transition from the quantum to classical regime, defined as compression of quantization, the quantum features in dynamics do not “disappear” but are pushed forward in time. This feature may be used to gauge the validity of the semiclassical and other approximate dynamics approaches in applications to anharmonic systems.

马德隆-德布罗意-玻姆（Madelung–de Broglie–Bohm）对薛定谔方程（Schrödinger equation）的表述，将波函数的时间演化刻画为通过非局域量子势相互作用的量子轨迹（或称玻姆轨迹）系综的动力学过程。这种轨迹视角能够通过将能量清晰划分为经典与量子组分，为理解给定系统的量子性（或经典性）提供洞见。本文提出一种不依赖系统的动力学量子性度量方法，该方法基于能量时间变化率，被称为"量子功率（quantum power）"，其在坐标空间中具有局域性。基于对模型化学系统的应用研究，我们提出：在从量子到经典区域的过渡（定义为量子化压缩过程）中，动力学中的量子特征并未"消失"，而是随时间向前推移。该特性可用于评估半经典近似及其他近似动力学方法在应用于非简谐系统时的有效性。