Universality in Uncertainty Relations for a Quantum Particle

TL;DR

This paper develops a comprehensive theory of uncertainty relations for a quantum particle, identifying universal states that saturate these relations and unifying existing inequalities within a geometric framework.

Contribution

It introduces a general method to determine bounds of uncertainty relations and shows that squeezed number states are universally optimal for these bounds.

Findings

Squeezed number states saturate all derived uncertainty relations.

A convex uncertainty region in second moments space is characterized.

New inequalities for second moments are established.

Abstract

A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle's variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schr\"{o}dinger. Our approach not only unifies existing uncertainty relations but also leads to new inequalities for second moments.