  We demonstrate a spatially resolved autocorrelation measurement with a
Bose-Einstein condensate (BEC) and measure the evolution of the spatial profile
of its quantum mechanical phase. Upon release of the BEC from the magnetic
trap, its phase develops a form that we measure to be quadratic in the spatial
coordinate. Our experiments also reveal the effects of the repulsive
interaction between two overlapping BEC wavepackets and we measure the small
momentum they impart to each other.
