Ground State of Liquid Helium (Mass 4)

The wave function describing the ground state of a boson system is approximated by the function $\Psi =\Pi \exp \left[\frac{1}{2}u\right(r_{\mathrm{ij}}\left)\right]$. The superposition approximation is then used to derive a linear, inhomogeneous integral equation for $\frac{\mathrm{du}}{\mathrm{dr}}$ in which the only other quantities occuring are the experimentally observed two-particle distribution function $g\left(r\right)\mathrm{}$ and its first derivative. A numerical solution for ${\mathrm{He}}^4$ is computed and compared with the explicit approximate solution derived by Abe. Using the computed $u\left(r\right)\mathrm{}$ and a proper smooth extrapolation of $g\left(r\right)\mathrm{}$ into the region below the apparent cutoff at $r=2.34\mathrm{}$ A, the kinetic energy of liquid ${\mathrm{He}}^4$ at absolute zero is estimated at 2.91×${10}^{-15\mathrm{}}$ ergs/atom.

A functional $J\left(\frac{\mathrm{du}}{\mathrm{dr}}\right)$ is constructed with the property that Abe's integral equation for $\frac{\mathrm{du}}{\mathrm{dr}}$ is just the Euler equation associated with the problem of finding a $u$ for which $J$ takes on an extreme value. The extreme value of $J$ (actually a maximum) is simply related to the expectation value of the kinetic energy. The variational property is used to determine the best $u\left(r\right)\mathrm{}$ from a family of trial functions.

The calculated value of the kinetic energy and the measured total energy are used, in conjunction with the virial theorem, to determine the coefficients of a $6-n$ Lennard-Jones potential. At $n=12\mathrm{}$, the calculation yields a deeper potential well and a slightly wider repulsive region than is calculated from the properties of the gas phase.