The minimal length and the Shannon entropic uncertainty relation

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.