论文标题
自由粒子力矩的进化和不变
Evolution and invariants of free-particle moments
论文作者
论文摘要
矩是位置和动量力量产物的期望值，占量子状态（或一组经典粒子上的平均值）。对于游离粒子，量子情况下的演变与一组经典颗粒密切相关。在这里，我们考虑了在一个维度上自由颗粒对称矩的演变，首先检查了矩的进化的几何特性，直到第四阶，由其超级和弯曲确定。这些属性是通过{\ IT不变}的矩的组合来指定的，因为它们在自由进化下保持恒定。不等式限制了四阶矩，并表明量子粒子的某些几何类型的进化是可能的，但可能是经典的，并且检查了一些示例。就其初始值，不变组合以及这些不变的时刻而言，在任何顺序的时刻都可以找到明确的表达式。
Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a set of classical particles. Here we consider the evolution of symmetrized moments for free particles in one dimension, first examining the geometric properties of the evolution for moments up to the fourth order, as determined by their extrema and inflections. These properties are specified by combinations of the moments that are {\it invariant} in that they remain constant under free evolution. An inequality constrains the fourth-order moments and shows that some geometric types of evolution are possible for a quantum particle but not possible classically, and some examples are examined. Explicit expressions are found for the moments of any order in terms of their initial values, for the invariant combinations, and for the moments in terms of these invariants.
