Hypervirial Theorems for Variational Wave Functions

Abstract

It is shown that a sufficient condition for an optimal energy variational wave function ${\psi }_0$ to satisfy the hypervirial relation $({\psi }_0, [H, W\left]{\psi }_0\right)=0\mathrm{}$ is for the trial function $\psi$ to admit variations of the form $\frac{\partial \psi }{\partial a}=\left(\frac{i}{\hslash }\right)W\psi$. Here $H$ is the Hamiltonian, $W$ is a Hermitian operator, and $a$ is a variational parameter. Explicit forms of such trial functions are exhibited for several $W\text{'}\mathrm{s}$. The case in which $W$ generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.