Generalized Cramér–Rao relations for non-relativistic quantum systems

The Cramér–Rao product of the Fisher information F [ ρ ] and the variance 〈 x 2 〉 ≡ ∫ x 2 ρ ( x ) d x of a probability density ρ ( x ) , defined on a domain Ω ⊂ R D , 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.