The Fisher information of single-particle systems with a central potential

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 (〈p2〉, 〈r−2〉) in position space, and (〈r2〉, 〈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.