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American Journal of Physics

Normalization of states in perturbation theories

Daniel R. Bes, G.G. Dussel, Hugo M. Sofia

Published Feb 1, 1977• 11 citations
Publisher

Abstract

Simple expressions are given for the normalization coefficients in Rayleigh–Schroedinger, Brillouin–Wigner, and Bloch–Horowitz perturbation methods. These expressions are derivatives of the energy, or of the matrix elements of the effective Hamiltonian.
DOI
10.1119/1.10653

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CRC Press eBooks•Mar 5, 2018•0 citations
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.
The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.