The microscopic picture for fractional quantum Hall effect (FQHE) is difficult
to work with analytically for a large number of electrons. Therefore to make
predictions and attempt to describe experimental measurements on quantum Hall systems, effective theories are usually employed such as the chiral Luttinger liquid system. In this thesis the Monte Carlo method is used for Laughlin-type quantum Hall systems to compute microscopic observables. In particular such computations are carried out for the large system size expansion of the free energy. This work was motivated by some disagreement in the literature about the form of the free energy expansion and is still an ongoing project. Tunnelling in the FQHE is an interesting problem since the tunnelling operators are derived from an effective theory which has not yet been checked microscopically. To perform a test for the effective tunnelling Hamiltonian, microscopic calculations were performed numerically for charges tunnelling across the bulk states of a FQH device. To compute these matrix elements, two methods were found to overcome a phase problem encountered in the Monte Carlo simulations. The Monte Carlo results were compared to the matrix elements predicted by the effective tunnelling Hamiltonian and there was a good match between the data. Performing this comparison enabled the operator ordering in the effective tunnelling Hamiltonian to be deduced and the data also showed that the quasiparticle tunnelling processes were more relevant than the electron tunnelling processes for all system sizes, supporting the idea that when
tunnelling is considered at a weak barrier, the electron tunnelling process can be
neglected.