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Physical Review A

Upper and Lower Bounds on Quantum-Mechanical Matrix Elements

Robert Blau, A. R. P. Rau, Larry Spruch

Published Jul 1, 1973• 33 citations
Publisher

Abstract

Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where $W$ is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the $n$th and $m$th 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 $W$ and only for real wave functions) or have been very conservative. We remove most of these restrictions ($W$ 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 ${W}_{\mathrm{nm}}$ in terms of "simple" upper bounds (which may be poor) on ${{W}^{2}}_{\mathrm{nn}}$. Here again, only for a very few particular choices of $W$ have such simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ been given previously (for example, $n$ restricted to be the ground state, and $W$ the operator ${z}_{i}$, the coordinate of the $i$th 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 $W$ 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 ${{W}^{2}}_{\mathrm{nn}}$ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of ${r}_{1}$ and ${r}_{1}^{2}$---we, therefore, require simple bounds on ${r}_{1}^{2}$ and ${r}_{1}^{4}$---for the states $1{s}^{2} ^{1}S$ and $1s2s ^{3}S$ of the helium atom, with rather good results.

DOI
10.1103/physreva.8.119

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Upper and lower bounds were obtained previously on matrix elements of the form Wnm=(\ensuremathψn,W\ensuremathψm){W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})Wnm​=(\ensuremathψn​,W\ensuremathψm​), where WWW is a Hermitian operator and \ensuremathψn{\ensuremath{\psi}}_{n}\ensuremathψn​ and \ensuremathψm{\ensuremath{\psi}}_{m}\ensuremathψm​ 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 \ensuremathΨnt{\ensuremath{\Psi}}_{\mathrm{nt}}\ensuremathΨnt​ and \ensuremathΨmt{\ensuremath{\Psi}}_{\mathrm{mt}}\ensuremathΨmt​ containing variational parameters, but the error in the bound is of first order in the errors in the \ensuremathΨnt{\ensuremath{\Psi}}_{\mathrm{nt}}\ensuremathΨnt​ and \ensuremathΨmt{\ensuremath{\Psi}}_{\mathrm{mt}}\ensuremathΨmt​. 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.
The second order error δF(2), made by employing the Schwartz−Dalgarno−Delves variational principle for the diagonal matrix element φ†Wφ of an arbitrary Hermitian operator W, is examined in the case that φ is the bound ground state eigenfunction of some given Hamiltonian H. This variational principle characteristically involves not only a trial estimate φt of φ, but also a trial estimate Lt(φt) Lt(φt) of a well−defined but generally not exactly known auxiliary function L (φ). It has previously been shown that, for certain special choices of a trial Hamiltonian Ht, the trial Lt(φt) can be found from a minimum principle. The present work finds that for these same special Ht it is possible to express δF(2) in comparatively simple closed form, depending only on known quantities, so that δF(2) should be calculable when the system described by H is not too complicated. However, these results for δF(2) are obtained on the assumption that Lt(φt) is known essentially exactly for any given φt; the practical utility of the formulas derived still must be tested, therefore. If Lt(φt) can be determined to this necessary accuracy, one expects that combining the computed δF(2) with the usual variational estimate of φ†Wφ will be a significant improvement over the usual variational estimate alone. Under the same circumstances, when W is a positive definite operator, the expression for δF(2) can provide nonrigorous but nonetheless potentially useful second order (variational) upper and lower bounds on the exact φ†Wφ.
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.
We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.