  The quantum theory of the Friedmann cosmological model with dust and
cosmological constant ($\Lambda$) is not exactly solvable analytically. We
apply Path Integral Monte Carlo (PIMC) techniques to study its quantum dynamics
using the physical Hamiltonian corresponding to the dust field as a clock. We
study (i) quantum fluctuations around classical paths and (ii) formulate the
analogues of the no-boundary and tunnelling proposals and simulate the ground
state wave functions. For $\Lambda < 0$ a unique ground state wave function
exists. For $\Lambda > 0$ the physical Hamiltonian is not bounded below, but
the path integral for the propagator is convergent over a range of Euclidean
time $T$. We investigate the properties of the convergent propagator. The path
integral can be made convergent for all values of $T$ by restricting the
integral over paths with action greater than equal to zero. We explore the
consequences of such a choice.
