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Canadian Journal of Physics

Lower bounds to eigenvalues

Maurice Cohen, Tova Feldmann

Published Sep 1, 1969• 21 citations
Publisher

Abstract

The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.

DOI
10.1139/p69-235

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The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.
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.
Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where WWW is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the nnnth and mmmth states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for WWW and only for real wave functions) or have been very conservative. We remove most of these restrictions (WWW need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on Wnm{W}_{\mathrm{nm}}Wnm​ in terms of "simple" upper bounds (which may be poor) on W2nn{{W}^{2}}_{\mathrm{nn}}W2nn​. Here again, only for a very few particular choices of WWW have such simple bounds on W2nn{{W}^{2}}_{\mathrm{nn}}W2nn​ been given previously (for example, nnn restricted to be the ground state, and WWW the operator zi{z}_{i}zi​, the coordinate of the iiith electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators WWW in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on W2nn{{W}^{2}}_{\mathrm{nn}}W2nn​ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of r1{r}_{1}r1​ and r12{r}_{1}^{2}r12​---we, therefore, require simple bounds on r12{r}_{1}^{2}r12​ and r14{r}_{1}^{4}r14​---for the states $1{s}^{2} ^{1}S$ and 1s2s3S1s2s ^{3}S1s2s3S of the helium atom, with rather good results.