Monte Carlo studies in weakly interacting quantum systems

This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have<br>employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy<br>curve, while the Slater-Jastrow wave function fails to bind the dimer at all.<br><br>In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.<br>