M. H. Kalos

The Metropolis Monte Carlo method is used to sample the square of an antisymmetric wave function composed of a product of a Jastrow wave function and a number of Slater determinants. We calculate variational energies for $^{3}\mathrm{He}$ and several models of neutron matter. The first-order Wu-Feenberg expansion is shown always to underestimate the energy, sometimes seriously. The phase diagram for ground-state Yukawa matter is determined. There is a class of Yukawa potentials which do not lead to a crystal phase at any density.

Various theoretical and numerical problems relating to heliumlike systems in their ground states are treated. New developments in the numerical solution of the Schr\"odinger equation permit the solution of 256-body systems with hard-sphere forces. Using periodic boundary conditions, fluid and crystal states can be described; results for the energy and radial-distribution functions are given. A new method of correcting for low-lying phonon excitations so as to extrapolate the energy of fluids to an infinite system is described. A perturbation theory relating the properties of the system with pure hard-sphere forces to those with smoother, more realistic two-body forces is introduced. As in recent work on classical systems the potential is divided into two continuous parts: One is repulsive, one attractive, the latter being treated as a perturbation. The solution for the repulsive part is taken directly from the hard-sphere problem when the radius is identified as the scattering length of the repulsive part of the smooth potential. The convergence for the Lennard-Jones potential is very good. Using our numerical results for the hard-sphere problem, with phonon corrections, together with this perturbation theory, results for energy versus density agree with experiment within our error of (3-10)% except at high crystal densities. We carry further Schiff's recent application of this perturbation theory to ${\mathrm{He}}^{3}$ and conclude that antisymmetrization by the method of Wu and Feenberg is the reason for lack of agreement with experiment in that system.

By means of a suitable Green's function, the Schr\"odinger equation for the few-body nuclear problem is written as an integral equation. In this equation, the binding energy of the ground state is assumed known and the strength of potential required to give this energy is an eigenvalue to be determined. A random walk can be devised whose collision density satisfies the same integral equation. The simulation of this random walk therefore permits an exact numerical solution by Monte Carlo methods. The calculation has been carried out with pairwise potentials of square, Gauss, and exponential shape.