The least-squares calculation of the expectation values of arbitrary operators

The least-squares method for finding the eigenvalues of the Schrödinger equation has the conceptual and practical advantage that it does not require the evaluation of integrals such as (ψ, H ψ) as does the variational method. In this paper we consider the evaluation of the expected value < Q > = (ψ, Q ψ)/(ψ, ψ) of an arbitrary Hermitian operator Q . We give a least-squares method for the evaluation of < Q > which also does not involve the evaluation of any integrals. The method is illustrated with calculations for the expectation values of a number of operators over the ground state of the helium atom, using a mesh containing only 126 points. The relative accuracy obtained varies between 10 -4 and 10 -6 .