The Fisher information of single-particle systems with a central potential, which is a gradient functional of their quantum-mechanical probability density, is studied in detail in the position and momentum spaces. It is found that this local information-theoretic quantity can be expressed in a simple closed form via the radial expectation values (< p(2)>, < r(-2)>) in position space, and (< r(2)>,< p(-2)>) in momentum space. Applications to various prototype systems (hydrogen and harmonic oscillator) are shown. Furthermore, a new uncertainty relation which involves the Fisher information in the two complementary spaces, is proposed at the same level than the variance-based Heisenberg and entropic uncertainty relations. (c) 2005 Elsevier B.V. All rights reserved.