Lower bounds on the electronic charge and momentum densities of atomic systems at the origin

Rigorous lower bounds to the electronic charge and momentum densities of an atomic system at the origin, ρ(0) and γ(0), respectively, by the expectation values 〈$r^{\alpha }$〉 and 〈$p^{\alpha }$〉, respectively, are given for positive and negative values of α in both cases, provided that the charge and momentum densities have a monotone decrement behavior, respectively. They allow one to give rigorous bounds to ρ(0) and γ(0) in terms of experimentally measurable atomic quantities such as the diamagnetic susceptibility, the Compton profile peak, and the electronic energy. In particular it is shown that ρ(0)≥(4 √3 π${)}^{\mathrm{-}1}$N〈${\mathrm{r}}^{\mathrm{-}2}$ ${\mathrm{〉}}^{3/2}$ with the charge density normalized to N, the number of electrons of the system. The quality and the asymptotic Z behavior of the bounds for neutral atoms (N=Z) are studied.