Position-momentum uncertainty relations based on moments of arbitrary order

Abstract

The position-momentum uncertainty-like inequality based on moments of arbitrary order for $d$-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the R\'enyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in $d$-dimensional physics: the hydrogenic and oscillator-like systems.