  We introduce the study of dynamical quantum noise in Bose-Einstein
condensates through numerical simulation of stochastic partial differential
equations obtained using phase space representations. We derive evolution
equations for a single trapped condensate in both the positive-$P$ and Wigner
representations, and perform simulations to compare the predictions of the two
methods. The positive-$P$ approach is found to be highly susceptible to the
stability problems that have been observed in other strongly nonlinear, weakly
damped systems. Using the Wigner representation, we examine the evolution of
several quantities of interest using from a variety of choices of initial state
for the condensate, and compare results to those for single-mode models.
