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CRC Press eBooks

Stationary State Perturbation Theory

Gordon Baym

Published Mar 5, 2018• 0 citations
Publisher

Abstract

The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.

DOI
10.1201/9780429499265-11

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The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.
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.
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.
Exactly solvable models in quantum mechanics are very rare.However, perturbation theory is a powerful tool to research the complex quantum systems and is widely used in theoretical physics and chemistry.For the particle weakly perturbed by the surrounding environment, its energy eigenstate and eigenvalue are separately calculated up to the first-order and second-order perturbation approximations in the undergraduate textbook of quantum mechanics.Whereas in reality, we have to calculate the high-order perturbation correction terms occasionally.Along the same lines of the low-order perturbation correction terms of energy eigenvalues and eigenstate for stationary state perturbation theory in the non-degenerate case, the third-order and fourth-order perturbation correction terms are strictly attained in this paper.Moreover, it points out the essence of perturbation theory and provides its derivation ideas.Students are no longer afraid of tedious derivation and can derive any-order perturbation correction.These results can serve as an extension and supplement for the quantum mechanics textbooks.Furthermore, it is very helpful for students to deeply understand quantum perturbation theory.
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). […]
The partial-wave Schrödinger equation with real boundary conditions is recast as an equation for the probability density. When a small additional potential is included, the changes in the bound-state energy eigenvalues are obtained, up to third order in the perturbation, purely in terms of the perturbing potential and the unperturbed probability density. Although the approach is different, our results are equivalent to those derived by Bender (Bender C M 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p 330). Knowledge of neither the unperturbed energy spectrum nor the wavefunctions of excited states is required. Evaluations of the second-order energy shift are given for some soluble S -wave problems.