Generalized Cramer-Rao relations for non-relativistic quantum systems

The Cramer-Rao product of the Fisher information and the variance of a probability density \rho(x), defined on a domain \Delta \in R^D, is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in \Delta, when \Delta is an unbounded domain. If \Delta is bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.