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Successive approximation; perturbation theory in quantum mechanics

Geoffrey Brooker

Published Jul 16, 2021• 0 citations
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Abstract

Abstract “Successive approximation; perturbation theory in quantum mechanics” introduces a toolbox for handling successive-approximation problems in any context. An iterative procedure is presented with examples. Newton's approximation is also an iterative procedure, but often other methods are better. Perturbation theory is presented, organized as an application of the toolbox.

DOI
10.1093/oso/9780198857242.003.0012

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Abstract “Successive approximation; perturbation theory in quantum mechanics” introduces a toolbox for handling successive-approximation problems in any context. An iterative procedure is presented with examples. Newton's approximation is also an iterative procedure, but often other methods are better. Perturbation theory is presented, organized as an application of the toolbox.
Abstract One of the great achievements of the Schrödinger wave-mechanics is the elegance of its perturbation theory, which has brought many problems, formerly considered intractable, within the range of a highly-developed mathematical technique. It is not necessary at this stage to review the numerous applications which have been made of this perturbation theory or to dwell upon its many advantages. The important advance towards an understanding of chemical forces which it has made possible is in itself a considerable achievement. There are, however, certain disadvantages in the perturbation theory in its present form, which limit the extent of its applications to complex problems of atomic and molecular structure. If the interaction of atoms, for instance, is to be calculated, as it most desirable, improved methods will have to be found.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.