Computation of bethe logarithms and other matrix elements of analytic functions of operators

Robert C. Forrey, Robert Nyden Hill

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Abstract

A general method is given for the calculation of a matrix element <φ| f(A) |ψ> of an analytic function f of an operator A. The method begins by writing <φ| f(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| f(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.

Original languageEnglish (US)
Pages (from-to)88-157
Number of pages70
JournalAnnals of Physics
Volume226
Issue number1
DOIs
StatePublished - Aug 15 1993

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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