Upper and Lower Bounds on Quantum-Mechanical Matrix Elements

Upper and lower bounds were obtained previously on matrix elements of the form $W_{\mathrm{nm}}=({\psi }_n,W{\psi }_m)$, where $W$ is a Hermitian operator and ${\psi }_n$ and ${\psi }_m$ 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 ${\Psi }_{\mathrm{nt}}$ and ${\Psi }_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\Psi }_{\mathrm{nt}}$ and ${\Psi }_{\mathrm{mt}}$. 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 $1s^2 {}_{}{}^1{}_{}{}^{}S$ and $1s2s {}_{}{}^3{}_{}{}^{}S$ of the helium atom, with rather good results.