Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

TL;DR

This paper derives a new class of uncertainty relations for quantum position and momentum, showing they are saturated by all harmonic oscillator eigenstates, extending beyond the ground state.

Contribution

It introduces a novel derivation of uncertainty relations using eigenvalue problems related to the harmonic oscillator, revealing saturation by all eigenstates.

Findings

Uncertainty relations are saturated by all harmonic oscillator eigenstates.

The derivation does not rely directly on commutation relations.

A broader class of constrained uncertainty relations is established.

Abstract

We re-derive the Schr\"{o}dinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.