Bounds for kinetic and exchange energies of fermion systems

We show that the physical quantities of an A-fermion system expressed by means of integrals of powers of the single-particle density {ρ}(r) are bounded by any kth moment around the origin 〈rᵏ〉 of {ρ} and A in a simple manner. As applications, the kinetic and semiclassical exchange energies of a neutral atom (A=Z) turn out to be bounded from below by 0.2184Z5/3〈r^-1{〉}²$ and from above by 0.3046${Z}4/3$〈r〉, respectively. Comparison of these bounds with the microscopic values of the corresponding quantities is made. Similar bounds for the same quantities in nuclei are also given and analyzed.