Lagrange multipliers and bounds to quantum mechanical properties

Variational lower and upper bounds to energy levels of a quantum mechanical system with Hamiltonian H as well as to overlap integrals between approximate and exact wavefunctions, have been obtained by a unified treatment. This is based on Lagrange's method of undetermined multipliers, in which a number of calculated moments of H play the role of constraints. New lower and upper bounds to the energy of the first excited state of the system are derived.