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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Perturbation theory for projected states II. Convergence criteria and a soluble model

B. Atalay, A. Mann, Ronald F. Peierls

Published Nov 27, 1973• 14 citations
Publisher

Abstract

Abstract It is shown that a perturbation expansion for projected states proposed earlier satisfies the Schrödinger equation. Modified forms of the series are suggested, and criteria for their convergence are discussed. The method is applied to the ‘Lipkin model' for a two-level many-body problem, which is exactly soluble. The second-order term in the perturbation series is shown to give a good approximation, and is compared with alternative methods. The convergence factor of the series is shown to be better than that of ordinary perturbation theory, which in most cases of this model is divergent.

DOI
10.1098/rspa.1973.0124

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Abstract It is shown that a perturbation expansion for projected states proposed earlier satisfies the Schrödinger equation. Modified forms of the series are suggested, and criteria for their convergence are discussed. The method is applied to the ‘Lipkin model' for a two-level many-body problem, which is exactly soluble. The second-order term in the perturbation series is shown to give a good approximation, and is compared with alternative methods. The convergence factor of the series is shown to be better than that of ordinary perturbation theory, which in most cases of this model is divergent.
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.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.
Abstract The theory of perturbations of electronic systems has been considered by various writers, and the method is the basis of the solution of most quantum problems. It therefore seems desirable that the theory should be placed on a firm footing, and this necessitates the discussion of the convergence of the series of perturbations, which up to the present has been avoided. This investigation is all the more necessary since the wave equation of the Stark Effect has no solution of the form assumed by Schrödinger in his original discussion of the perturbation theory. Dirac’s theory leads to a system of linear differential equations in an infinite number of variables, and 2 is devoted to the discussion of the existence theorem for such a system by a matrix method. In 3 the application is made to the perturbation theory in the case of perturbations satisfying the conditions assumed in 2. In 4 the theory is extended so as to include more general types of disturbances. It is shown that, though the series of perturbations does not in general converge, yet it usually possesses the same asymptotic character as in the classical theory, and its use can therefore be justified.
Abstract In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four‐electron models, in which the degree of quasi‐degeneracy can be continuously varied, by the finite‐order nondegenerate many‐body perturbation theory, we examine in more detail a simple two‐state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Padè approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi‐reference perturbation approaches, which Kaldor advocates, are pointed out.
A time-dependent perturbation theory for the solution of the time-dependent Schrödinger equation is developed and is applied to systems having two states with pulse and Lorentzian types of perturbations. It is shown in the case of the pulse perturbation that the expression corresponding to the Nth order of the perturbation theory converges to the exact solution when N tends to infinity. The results obtained from this perturbation theory, the Magnus approximation, and the time-dependent perturbation theory developed by Dirac are compared with the exact results. This theory is found to be good for larger values of the perturbations and for larger intervals of time.