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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Complementary variational principles in perturbation theory

A. M. Arthurs, Peter D. Robinson

Published Mar 19, 1968• 9 citations
Publisher

Abstract

Abstract Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

DOI
10.1098/rspa.1968.0065

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American Journal of Physics•Dec 12, 2008•1 citation
Abstract Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.
Variational principles for upper and lower bounds to all even order Rayleigh-Schrodinger perturbation energies are given. The upper bound for the second order energy is just the Hylleraas principle and the upper bounds for energies through 12th order are the same as those used by Knight and Scherr. The general upper bound principle for the (2n)th order energy is equivalent to the variational principle obtained by Sinanoglu for the n-th order wave function, but our principle gives the exact (2n)th order energy when the trial wave function is exact. The lower bounds for the even order energies are generalizations of the lower bound obtained by Prager and Hirschfelder for the second order. (Author)
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. ƒree time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. ƒe WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. ƒe third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
The variational principle is revisited in the context of finding the upper and lower bounds to the ground state energy. It is shown how the variational principle can be employed to find the lower bound by partitioning the Hamiltonian into several parts. We demonstrate how the variational principle can be used to find the exact ground states in some special cases. We consider the harmonic oscillator and the Ising Hamiltonian on a bipartite lattice which are familiar yet instructive examples for students.