Definitive variational principles for physical quantities

Abstract

New variational principles are proposed that provide the upper and lower limits to the square of the modulus of the matrix element for a transition. The corresponding variational problems for maximum and minimum are formulated via functional of general form that are bilinear with respect to the hamiltonian. An analogous variational principle may be derived via a functional linear with respect to the hamiltonian for a transition to a state of lowest energy of the given symmetry type. In some cases the variational estimates of the square of the modulus involve passage to the limit. Errors due to inexact eigenfunctions and replacement of limiting values by specified values for the parameters are discussed.