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Physical Review Letters

New Approach to Perturbation Theory

Yakir Aharonov, C. K. Au

Published Jun 11, 1979• 113 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.42.1582

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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 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.
A special nondegenerate perturbation theory is developed for solving the "nonlinear Schr\"odinger equation" of the type $[\stackrel{^}{H}+\stackrel{^}{V}(\ensuremath{\Psi})]\ensuremath{\Psi}=E\ensuremath{\Psi}$, which arises, e.g., from the study of the environmental effects on molecules. The interaction operator is taken as $\stackrel{^}{V}(\ensuremath{\Psi})=\stackrel{^}{A}〈\ensuremath{\Psi}|\stackrel{^}{B}|\ensuremath{\Psi}〉$. The in vacuo problem $\stackrel{^}{H}\ensuremath{\Phi}=\ensuremath{\epsilon}\ensuremath{\Phi}$ is considered the zeroth-order description. For the perturbed energies and wave functions recursive formulas are derived at any order of the perturbation. The formulas obtained are compared to the usual Rayleigh-Schr\"odinger perturbation theory where $\stackrel{^}{V}$ is supposed to be independent of $\ensuremath{\Psi}$. The formal results are applied to the case of the Onsager cavity model.