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College Physics

Perturbation methods in quantum mechanics

Zhi Chen

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Three kinds of perturbation methods for nondegenerate quantum system are briefly introduced and compared applying to a simple model.The first one is Rayleigh-Schrdinger (R-S) method which is widely discussed in text book,a very useful recurrence relation is also given.The second one is iterative perturbation method which is formulated based on Brillouin-Wigner (B-W) perturbation,it can be used to make a program to evaluate the eigensolutions by computer. The third one is superconvergent method proposed by Scherer,which is important to determine the existence of quantum Lie transformation.

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American Journal of Physics•Mar 1, 2000•20 citations
Three kinds of perturbation methods for nondegenerate quantum system are briefly introduced and compared applying to a simple model.The first one is Rayleigh-Schrdinger (R-S) method which is widely discussed in text book,a very useful recurrence relation is also given.The second one is iterative perturbation method which is formulated based on Brillouin-Wigner (B-W) perturbation,it can be used to make a program to evaluate the eigensolutions by computer. The third one is superconvergent method proposed by Scherer,which is important to determine the existence of quantum Lie transformation.
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.
This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.
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.