  The Markov chain Monte Carlo (MCMC) method is used to evaluate the
imaginary-time path integral of a quantum oscillator with a potential that
includes both a quadratic term and a quartic term whose coupling is varied by
several orders of magnitude. This path integral is discretized on a time
lattice and calculations for the energy and probability density of the ground
state and energies of the first few excited states are carried out on lattices
with decreasing spacing to estimate these quantities in the continuum limit.
The variation of the quartic coupling constant produces corresponding
variations in the optimum simulation parameters for the MCMC method and in the
statistical uncertainty for a fixed number of paths used for measurement. The
energies and probability densities are in excellent agreement with those
obtained from numerical solutions of Schr\"odinger's equation.
