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 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.

矩是位置和动量的乘积的期望值，采用量子态（或一组经典粒子的平均值）。对于自由粒子，量子情况下的演化与一组经典粒子的演化密切相关。在这里，我们考虑一维自由粒子对称矩的演化，首先检查四阶矩演化的几何特性，由它们的极值和拐点决定。这些属性由不变矩的组合指定因为它们在自由进化下保持不变。不等式约束了四阶矩，并表明对于量子粒子，某些几何类型的演化是可能的，但经典上是不可能的，并检查了一些示例。对于任何阶矩的初始值、不变量组合以及这些不变量的矩，都可以找到显式表达式。