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European Journal of Physics

A simple derivation of the series in perturbation theory

Luis Laín, Alicia Torre

Published Jul 1, 1987• 3 citations
Publisher

Abstract

A simple algorithm to derive the Brillouin-Wigner and the Rayleigh-Schrodinger series in time-independent perturbation theory is presented. The method provides a better understanding of the normalisation treatment of the perturbed wavefunctions and it establishes a clear relationship between the Brillouin-Wigner and Rayleigh-Schrodinger series.

DOI
10.1088/0143-0807/8/3/005

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A simple algorithm to derive the Brillouin-Wigner and the Rayleigh-Schrodinger series in time-independent perturbation theory is presented. The method provides a better understanding of the normalisation treatment of the perturbed wavefunctions and it establishes a clear relationship between the Brillouin-Wigner and Rayleigh-Schrodinger series.
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.
Rayleigh-Schrödinger time-independent perturbation theory is a key method in quantum mechanics for solving problems where the Hamiltonian of a quantum system cannot be exactly solved. By expanding the solutions in series, this method allows one to obtain approximations for quantum states and eigen-energies, especially for systems where the interactions or perturbing forces are small compared to the principal part of the Hamiltonian. This is crucial for understanding complex phenomena, predicting the behaviours of atomic and molecular systems, and calculating corrections to basic theoretical models, thereby improving the accuracy of quantum predictions. In this work, we will show that using the unitary operators, we arrive at the equations giving the different order of corrections of both energy as well as wave functions of the time-independent perturbation theory. Our approach closely mirrors the famous formalism of Rayleigh-Schrodinger perturbation theory (RSPT). The use of unitary operators simplifies the derivation of higher-order corrections, making the process more accurate and accessible to researchers. This technique harnesses the potential of unitary operators within a perturbation series framework. By combining these two powerful techniques, we offer a comprehensive framework for addressing normalization challenges in quantum mechanical calculations. This methodology not only facilitates a deeper understanding of perturbation theory but also serves as a pedagogical stepping stone, encouraging postgraduate researchers to apply their foundational knowledge and acquired skills in practical problem-solving situations with confidence. This is therefore a second proof or a new method to derive the fundamental equations of Rayleigh Schrodinger perturbation Theory RSPT using the unitary transformations.
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.