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Physical Review

Tensor Virial Theorems for Variational Wave Functions

Dave Pandres

Published Jul 15, 1963• 14 citations
Publisher

Abstract

It is shown that the tensor virial theorem and certain generalizations of it are satisfied by optimum energy variational wave functions in which the variational parameters are elements of a square matrix that scales the column matrix of particle coordinates.
DOI
10.1103/physrev.131.886

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It is shown that a sufficient condition for an optimal energy variational wave function ${\ensuremath{\psi}}_{0}$ to satisfy the hypervirial relation $({\ensuremath{\psi}}_{0}, [H, W]{\ensuremath{\psi}}_{0})=0$ is for the trial function $\ensuremath{\psi}$ to admit variations of the form $\frac{\ensuremath{\partial}\ensuremath{\psi}}{\ensuremath{\partial}a}=(\frac{i}{\ensuremath{\hbar}})W\ensuremath{\psi}$. Here HHH is the Hamiltonian, WWW is a Hermitian operator, and aaa is a variational parameter. Explicit forms of such trial functions are exhibited for several W′sW'\mathrm{s}W′s. The case in which WWW generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.
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