Evolution and invariants of free-particle moments

TL;DR

This paper investigates the geometric evolution of symmetrized moments for free particles in one dimension, identifying invariants and inequalities that distinguish quantum from classical behavior.

Contribution

It introduces a detailed analysis of invariant combinations of moments and provides explicit formulas for moments of any order based on initial conditions.

Findings

Invariant moment combinations remain constant during free evolution.

Certain geometric evolution types are exclusive to quantum particles.

Explicit formulas relate moments to initial values and invariants.

Abstract

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.