Error bounds for quantum-mechanical perturbation theory

Abstract

A new method is described, which yields rigorous upper and lower bounds to the sums over excited states which occur in second-order quantum-mechanical perturbation theory. The only data required are values of certain expectation values over the wave function describing the ground state of the system, and the excitation energy of the first excited state which contributes to the sum. The error bounds are shown to be the most precise possible using only this information. As an example of the method, the optical polarizability of H and He atoms are calculated. The means of the upper and lower bounds in these cases have an accuracy of the order of 1%, although the rigorous error bounds can only guarantee an accuracy near 10%. A plausible, but non-rigorous argument is given for expecting this apparent greater accuracy of the mean value to hold in other cases as well. When wave functions of one or more excited states have been calculated, this additional information may be used to obtain more precise error limits for the perturbation sums. The method is applicable to a wide variety of problems which can be treated by perturbation theory.