The minimal length and the Shannon entropic uncertainty relation

TL;DR

This paper investigates the Shannon entropic uncertainty relation within a generalized uncertainty principle framework, demonstrating its validity with a specific representation and analyzing the harmonic oscillator with a minimal length.

Contribution

It shows that the BBM entropic uncertainty relation holds under a modified commutation relation using a self-adjoint representation, extending its applicability.

Findings

The BBM inequality remains valid in the generalized framework.

Explicit analysis for the harmonic oscillator with minimal length.

Validation of the entropic uncertainty relation in a new representation.

Abstract

In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation $[X,P]=i\hbar\left(1+\beta P^2\right)$ where $\beta$ is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycieslki (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, i.e., $X=x$ and $P=\tan\left(\sqrt{\beta}p\right)/\sqrt{\beta}$ where $[x,p]=i\hbar$, the BBM inequality is still valid in the form $S_x+S_p\geq1+\ln\pi$ as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.