## 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 language | English (US) |
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Pages (from-to) | 88-157 |

Number of pages | 70 |

Journal | Annals of Physics |

Volume | 226 |

Issue number | 1 |

DOIs | |

State | Published - Aug 15 1993 |

## All Science Journal Classification (ASJC) codes

- General Physics and Astronomy