Generalized Cramér-Rao relations for non-relativistic quantum systems

The Cramér-Rao product of the Fisher information F[ρ] and the variance 〈x2〉≡∫x2ρ(x)dx of a probability density ρ(x), defined on a domain Ω⊂RD, is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in Ω, when Ω is an unbounded domain. If Ω is bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.