Stationary and Quasistationary Bounds on Arbitrary Bound-State Matrix Elements

Rigorous upper and lower bounds on the diagonal and off-diagonal bound-state matrix elements of an arbitrary Hermitian operator $W$ are derived which represent a significant improvement over previous results in that all first-order error terms are eliminated. These bounds do, however, contain error terms of the three-halves power of the errors in the trial functions used; they are therefore termed "quasistationary bounds." In the particular but important case of the diagonal bound-state matrix elements of non-negative (nonpositive) Hermitian operators, true stationary lower (upper) bounds are obtained; the errors are of second order in the errors of the trial functions used. All trial functions can contain variational parameters. The results obtained are tested by computing upper and lower bounds on the expectation value of $r_1+r_2$ for the ground state of helium, and a substantial improvement over previous bounds is obtained.