Computation of Bethe Logarithms and Other Matrix Elements of Analytic Functions of Operators

Abstract

A general method is given for the calculation of a matrix element 〈φ| ƒ(A) |ψ〉 of an analytic function ƒ of an operator A. The method begins by writing 〈φ| ƒ(A) |ψ〉 as a contour integral of the corresponding matrix element 〈φ| (ζ − A)−1 |ψ〉 of the resolvent (ζ − A)−1, where the contour surrounds the spectrum of A. The contour is then deformed to obtain 〈φ| ƒ(A) |ψ〉 as a sum of contributions from branch points and poles of f. The numerical evaluation of the Bethe logarithm, which is the dominant contribution to the Lamb shift, is used as an example. The difficulties which arise when the resolvent matrix element 〈φ| (ζ − A)−1 |ψ〉 must be evaluated by approximate methods are discussed.