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Journal of Physics A Mathematical and General

An examination of the accuracy of truncated perturbation theory using bounds for the exact energy

B. L. Burrows, M. Cohen, Tova Feldmann

Published Mar 11, 1987• 3 citations
Publisher

Abstract

A procedure is developed for assessing the accuracy of a truncated perturbation expansion, which leads to upper and lower bounds to some eigenvalue. Improved bounds, which require a lower bound to the overlap of the approximate and exact wavefunctions, are also considered and a lower bound to the overlap is derived. Two examples are presented, using model problems.

DOI
10.1088/0305-4470/20/4/023

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A procedure is developed for assessing the accuracy of a truncated perturbation expansion, which leads to upper and lower bounds to some eigenvalue. Improved bounds, which require a lower bound to the overlap of the approximate and exact wavefunctions, are also considered and a lower bound to the overlap is derived. Two examples are presented, using model problems.
The accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ | ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.
With a limit on the small number of problems that admit exact solutions, we turn in this chapter to approximation methods that can be used for bound state problems. The most commonly used method is the perturbation theory for which one use a small parameter to form the basis of a systematic expansion, or a variational method in which one hopes to set the best limit on the energy of a system. We will consider both of these approximation methods with applications to problems in atomic and molecular physics.
For scattering in nonrelativistic quantum mechanics, the range of validity of the perturbation expansion is studied by viewing the expansion as being generated by an iteration procedure, and applying to it a fundamental theorem for iteration processes. The theorem provides a sufficient condition for convergence and, at the same time, gives upper bounds on the error generated in truncating the expansion at a given number of terms. These error bounds for the wave function are in turn used to find upper bounds on the truncation error for the SSS-matrix expansion. Bounds for the exact SSS matrix are also given. These results are illustrated by applying them to the simple case of one-particle potential scattering, for both the plane-wave and partial-wave analyses.
The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.