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Studies in perturbation theory

1963; Elsevier BV; Volume: 10; Issue: 1-6 Linguagem: Inglês

10.1016/0022-2852(63)90151-6

1096-083X

Per‐Olov Löwdin,

Numerical methods for differential equations

The fundamental Schrödinger equation in quantum mechanics may be transformed to a discrete representation by introducing a complete set of basis functions in which the eigenfunctions may be developed. The eigenvalues are then determined by solving a secular equation, and this problem is here attacked by a partitioning which leads to an implicit relation for the energy E of the form E = f(E), which corresponds to the Schrödinger-Brillouin perturbation formula but has a more condensed form. The first-order iteration process based on the relation E(k+1) = f{E(k)} is studied, and it is shown that, independent of whether this process is convergent or not, one can go over to a second-order process which appears to be closely connected with the variational expression. This second-order iterative procedure turns out to be very convenient for numerical work, particularly since it treats degenerate eigenvalues just as easily as the single eigenvalues. The general behavior of the curve y = E − f(E) is discussed, and the method is illustrated by a few numerical examples. In an appendix, a brief survey of the classification of iteration processes is also given.