The Fisher-information-based uncertainty relation, Cramer–Rao inequality and kinetic energy for the<i>D</i>-dimensional central problem

Abstract

The inequality , with L being the grand orbital quantum number, and its conjugate relation for (⟨r2⟩, ⟨p−2⟩) are shown to be fulfilled in the D-dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation (IρIγ⩾ const) and the Cramer–Rao inequalities (⟨r2⟩Iρ ⩾ D2; ⟨p2⟩Iγ ⩾ D2). In addition, the kinetic energy and the radial expectation value ⟨r2⟩ are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by Iρ and Iγ, respectively.