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arXiv:quant-ph/9412009 · Physical Review Letters

Superconvergent Perturbation Method in Quantum Mechanics

Wolfgang Scherer

Published Feb 27, 1995• 27 citations
arXivHTMLPDF

Abstract

An analog of Kolmogorov's superconvergent perturbation theory in classical mechanics is constructed for self-adjoint operators. It is different from the usual Rayleigh-Schr"odinger perturbation theory and yields expansions for eigenvalues and eigenvectors in terms of functions of the perturbation parameter.
DOI
10.1103/physrevlett.74.1495

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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.
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.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.