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The Physics Thinking and Mathematics Approximate on the Adaptable Condition of the Perturbation Theory

Min Liu

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Through the discussion of the adaptable condition about non-degeneration stationary state perturbation theroy. We know that this conditions has not only made use of physical model, but also series convergence theory, which is different from series method of pure mathmathics.This paper has discussed the problem and has the following credible result.

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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.

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Through the discussion of the adaptable condition about non-degeneration stationary state perturbation theroy. We know that this conditions has not only made use of physical model, but also series convergence theory, which is different from series method of pure mathmathics.This paper has discussed the problem and has the following credible result.
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.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]
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.
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.