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Journal of Physics A Mathematical and General

Zel'dovich's method of perturbation theory in quantum mechanics

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, Lauro Tomio

Published Dec 23, 1999• 1 citation
Publisher

Abstract

Many years ago Zel'dovich showed how the Lagrange condition in the theory of differential equations can be utilized in the perturbation theory of quantum mechanics. Zel'dovich's method enables us to circumvent the summation over intermediate states. As compared with other similar methods, in particular the logarithmic perturbation expansion method, we emphasize that this relatively unknown method of Zel'dovich has a remarkable advantage in dealing with excited states. That is, the ground and excited states can all be treated in the same way. The nodes of the unperturbed wavefunction do not give rise to any complication.

DOI
10.1088/0305-4470/33/2/305

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Many years ago Zel'dovich showed how the Lagrange condition in the theory of differential equations can be utilized in the perturbation theory of quantum mechanics. Zel'dovich's method enables us to circumvent the summation over intermediate states. As compared with other similar methods, in particular the logarithmic perturbation expansion method, we emphasize that this relatively unknown method of Zel'dovich has a remarkable advantage in dealing with excited states. That is, the ground and excited states can all be treated in the same way. The nodes of the unperturbed wavefunction do not give rise to any complication.
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.
We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.
The time independent, non-degenerate standard perturbation theory is compared the alternate treatment of perturbation theory called logarithmic perturbation theory (LPT). For determining the non-degenerate ground state the LPT is, in principle, easier to apply than standard perturbation theory. This is because, as opposed to the standard perturbation method which requires the knowledge of the complete set of eigenvalues and eigenfunctions of the unperturbed system, for the LPT one only needs to know the ground state wave function of the unperturbed system, the energy correction to the next lower order and some easily computable coupling constants. However, in reality, the LPT is a simpler method to apply when the ground state wavefunction is exponential in nature. But as shown here for trigonometric unperturbed wave function the LPT leads to integrals which have no analytical solutions thereby making LPT more difficult and less accurate method than the standard perturbation approach. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 595-600
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.
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.
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.