Expectation values of operators in approximate two-component relativistic theories

Abstract

Two methods for the evaluation of expectation values with approximate two-component relativistic functions are analysed. The first of them is based on the change of picture for the operator whose expectation value is to be calculated and associated with approximations leading to the given two-component relativistic wave function. This method, though hardly used in numerical calculations, gives the expectation values that directly reflect the accuracy of the wave function used for their calculation. The second method, most commonly used in calculations, neglects the picture change and is shown always to introduce an error of the order of α2, where α is the fine structure constant. This error is present independently of the accuracy of the approximate two-component wave function. The perturbation formalism developed in this paper is illustrated by calculations of relativistic corrections to the expectation values of r−1 for arbitrary states of hydrogenic ions.