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Journal of Mathematical Physics

Discrete state perturbation theory via Green’s functions

William Rubinson

Published Jan 1, 1975• 1 citation
Publisher

Abstract

The exposition of stationary state perturbation theory via the Green’s function method in Goldberger and Watson’s Collision Theory is reworked in a way that makes explicit its mathematical basis. It is stressed that the theory consists of the construction of, and manipulations on, a mathematical identity. The perturbation series fall out of the identity almost immediately. The logical status of the method is commented on.

DOI
10.1063/1.522413

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The exposition of stationary state perturbation theory via the Green’s function method in Goldberger and Watson’s Collision Theory is reworked in a way that makes explicit its mathematical basis. It is stressed that the theory consists of the construction of, and manipulations on, a mathematical identity. The perturbation series fall out of the identity almost immediately. The logical status of the method is commented on.
It is pointed out that the notation of the theory of functionals and the theorem in it analogous to Taylor's theorem facilitates a compact discussion of perturbation and variation methods in quantum mechanics, both for bound state and scattering problems. With this aid it is shown (i) how stationary expressions required for variation methods may be sought and found, (ii) that the perturbation and variation methods are virtually identical, (iii) that second order perturbations and expressions stationary to the second order may be found similarly, though these are complicated. The form found for the second order perturbation of bound states differs from that commonly used in that it does not explicitly involve the eigenfunctions of states other than that perturbed.
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the Correlated Iteration Theory, which has been developed by the author. This approach differs from all other known variants of perturbation theory for Green's functions by the combination of two factors: the systematic formulation of an algorithm for obtaining subsequent approximations and the consistent consideration of interparticle correlations at each step of the procedure.