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American Journal of Physics

Extensions of the Feynman–Hellman theorem and applications

Sukhbinder Singh, Chetan Singh

Published Oct 1, 1989• 22 citations
Publisher

Abstract

Epstein’s [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman–Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein’s rederivation of the Rayleigh–Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones–Brillouin–Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.

DOI
10.1119/1.15842

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Epstein’s [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman–Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein’s rederivation of the Rayleigh–Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones–Brillouin–Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
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.
If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ 〈 ∂ ∂λ ∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈 ∣∣∣∣∣ ∂(Hˆ − E) ∂λ ∣∣∣∣∣ 〉 + 〈 ∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ 〉 = 0 (3.5) and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E ∂λ < | >= 〈 ∣∣∣∣∣ ∂Hˆ ∂λ ∣∣∣∣∣ 〉 , (3.6) or ∂E ∂λ = 〈 ∂Hˆ ∂λ 〉 (3.7) if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur- bation theory.
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.