Moments and saturation properties of eigenstates

TL;DR

This paper introduces an algebraic method using moments to analyze eigenstates of harmonic and anharmonic oscillators, revealing saturation of uncertainty relations for higher-order moments and uncovering new physical features.

Contribution

It presents a systematic algebraic approach based on moments and inequalities to study eigenstates, extending to anharmonic systems with perturbation theory.

Findings

Uncertainty relations for higher-order moments are saturated by harmonic oscillator states.

Saturation properties extend to anharmonic systems perturbatively.

The method uncovers new physical features of eigenstates.

Abstract

Eigenvalues are defined for any element of an algebra of observables and do not require a representation in terms of wave functions or density matrices. A systematic algebraic derivation based on moments is presented here for the harmonic oscillator, together with a perturbative treatment of anharmonic systems. In this process, a collection of inequalities is uncovered which amount to uncertainty relations for higher-order moments saturated by the harmonic-oscillator excited states. Similar saturation properties hold for anharmonic systems order by order in perturbation theory. The new method, based on recurrence relations for moments of a state combined with positivity conditions, is therefore able to show new physical features.