Variance ofHin the Bijl-Feynman Description of an Elementary Excitation

The trial function ${\rho }_{\mathrm{k}}{\psi }_0$ is a useful approximation to the lowest eigenstate of $H$ with momentum $\hslash \mathbf{k}$ in the theory of a uniform system of interacting bosons. The closeness of the approximation can be tested by studying the variance of $H$ with respect to ${\rho }_{\mathrm{k}}{\psi }_0$. The leading term in the variance at long wavelengths ($k\ll 2\pi {\rho }^{\frac{1}{3}}$) is found to vanish when ${\psi }_0$ is given the asymptotic properties implied by the identification of the elementary excitations described by ${\rho }_{\mathrm{k}}{\psi }_0$ with the phonons in the quantized version of the classical sound field.