Entropic analysis of the quantum oscillator with a minimal length

TL;DR

This paper investigates how a minimal length, arising from modified quantum commutation relations, affects the entropic uncertainty relations in a quantum harmonic oscillator, providing numerical analysis of the resulting entropy measures.

Contribution

It introduces a numerical study of entropic uncertainty relations in a deformed quantum harmonic oscillator with minimal length effects.

Findings

Deformation of entropic uncertainty relations due to minimal length.

Numerical results for Rènyi entropies in the lowest energy states.

Analysis of position and momentum distributions under deformation.

Abstract

The well-known Heisenberg--Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting $[X,P] = i\hbar (1+\beta P^2)$ implies the existence of a minimal length proportional to $\sqrt\beta$. The Bialynicki-Birula--Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter $\beta$. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum R\'enyi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.