Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order $\hslash$ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.

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期望值的半经典评估

半经典力学允许描述量子系统，该系统保留其相位信息，从而保留干涉效应，同时仅使用系统的经典动力学作为输入。特别地，半经典描述的优势之一是呈现出连贯的图片，该图片可以忽略不计。$\hslash$修正）与所使用的特定规范坐标无关。但是，这种相干性在很大程度上依赖于固定相位近似的使用。然而，事实证明，在某些重要情况下，对固定相位近似的残酷应用会消除所有干扰，从而消除量子效应。在本文中，我们将以其最简单的实例之一详细讨论此问题，即对算子期望值的时间演化进行评估。我们解释了为什么有必要包括不在固定点附近的贡献，并为期望值的演化提供新的半经典表达式。我们方法的效率基于以下事实：我们分析性地处理了在固定相近似内可以执行的所有积分，简单积分，从某种意义上说，被积物在量子尺度上没有显着变化（因此，它们在数值上非常容易执行）。这与其他方法形成对比，例如基于初始值表示法的方法，该方法在化学和分子物理学中很流行，该方法避免了对经典动力学的根本搜索，但以执行数值积分为代价，其计算需要对量子进行采样规模，因此不能很好地设计来解决深层的半古典制度。一路走来，我们对这些干扰效应的起源有了更深入的了解，并获得了与之相关的直观几何图形。