A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets

TL;DR

This paper introduces a new uncertainty product for complex Gaussian wave packets in one dimension, showing it is minimized by certain phase space rotated variables, extending the understanding of uncertainty relations.

Contribution

It establishes a minimal uncertainty product for complex Gaussian wave packets involving phase space rotations, broadening the classical uncertainty principle.

Findings

Complex Gaussian wave packets do not minimize the standard uncertainty product.

An alternative uncertainty product involving phase space rotations is minimized by these wave packets.

The result generalizes the minimal uncertainty condition to a broader class of Gaussian states.

Abstract

Although real, normalized Gaussian wave packets minimize the product of position and momentum uncertainties, generic complex normalized Gaussian wave packets do not. We prove they minimize an alternative product of uncertainties that correspond to variables that are phase space rotations of position and momentum.