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Proceedings of the National Academy of Sciences

Upper and Lower Bounds for Rayleigh-Schrödinger Perturbation Energies

Sando,Kenneth M., Hirschfelder,J. O.

Published Sep 1, 1970• 0 citations
Publisher

Abstract

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)

DOI
10.1073/pnas.67.1.356

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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)
The scattering length AAA characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on AAA could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on AAA. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on AAA is rigorously established provided a certain modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on AAA can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on AAA. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.
Weinstein's modification of the Ritz principle is used to derive (1) a lower bound for the nnn-th energy-level of a quantum mechanical system if a lower bound for the $(n+1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known; and (2) an upper bound for the nnn-th level if an upper bound for the $(n\ensuremath{-}1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known.
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.