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Physical Review Letters

New Approach to Perturbation Theory.

Yakir Aharonov, C. K. Au

Published Jul 9, 1979• 21 citations
Publisher

Abstract

We present here a new approach to nonrelativistic perturbation theory. We show that in problems reducible to one dimension, the energy shifts and wave-function corrections, including corrections to the position of the nodes, to any order, can be expressed in quadrature in a hierarchy scheme. The second-order energy shift calculated is explicitly shown to be equivalent to that in the ordinary Rayleigh-Schr\odinger theory.

DOI
10.1103/physrevlett.43.176

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We present here a new approach to nonrelativistic perturbation theory. We show that in problems reducible to one dimension, the energy shifts and wave-function corrections, including corrections to the position of the nodes, to any order, can be expressed in quadrature in a hierarchy scheme. The second-order energy shift calculated is explicitly shown to be equivalent to that in the ordinary Rayleigh-Schr\odinger theory.
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 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.
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.
A modified version of Rayleigh-Schr\"odinger (RS) perturbation theory, which has been developed previously, can dispense with the usual RS restriction that the unperturbed Hamiltonian should be diagonal in the chosen basis. This modified RS (MRS) perturbation scheme is discussed in some detail and extended to the degenerate case. The new degenerate MRS perturbation theory remains based upon a rearrangement of the unperturbed and perturbed parts of the Hamiltonian. Basically, an unperturbed Schr\"odinger equation is thereby produced which may be satisfied trivially and identically for an arbitrary Hamiltonian. The zeroth-order approximation to the energy remains a free parameter of the degenerate version of the MRS formalism. The explicit MRS expressions for the higher-order correction terms also preserve a strict analogy with their counterparts in the nondegenerate formalism. What is new is an enormous increase in the applicability and universality of the whole approach. Thus, a priori, the unperturbed Hamiltonian may now be chosen so as to have all of its larger off-diagonal elements much closer to those of the full original Hamiltonian than is permitted in the standard RS approach. This is a particular advantage of the whole MRS approach. By testing the method on two nontrivial model Hamiltonians, the method is demonstrated to have considerably improved convergence properties over the standard RS scheme, and at relatively little extra cost.