New Formulas for Lower Bounds to Expectation Values

Two new formulas are presented for rigorous lower bounds to the true quantum-mechanical expectation value of an arbitrary positive Hermitian operator. These formulas strengthen and extend previous results while requiring relatively simple additional matrix elements. Illustrative numerical applications are made at two levels of approximation to various one- and two-electron properties, powers of $r_1$ and $r_{12}$, in the normal helium atom.