# Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems

## TL;DR

This paper analyzes the leading behavior of Heisenberg and entropic uncertainty measures for large-dimensional harmonic oscillators, revealing saturation of uncertainty relations and similarities with hydrogenic systems in high dimensions.

## Contribution

It rigorously determines the asymptotic forms of uncertainty measures for high-dimensional harmonic systems and compares them with hydrogenic systems, highlighting universal features.

## Key findings

- Uncertainty measures saturate the multidimensional uncertainty relations.
- Heisenberg and Renyi entropy relations coincide with hydrogenic systems in the limit Dd0.
- A conjecture is proposed for the Shannon-like uncertainty relation.

## Abstract

The $D$-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the R\'enyi entropies, respectively, at the limit of large $D$. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given and an interesting phenomenon is observed: the Heisenberg-like and R\'enyi-entropy-based equality-type uncertainty relations for all the $D$-dimensional harmonic oscillator states in the pseudoclassical ($D \to \infty$) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.