  We derive the Gross-Pitaevskii equation in two-dimension from the first
principles of two-dimensional scattering theory. Numerical calculation of the
condensate wave function shows that atoms in a two-dimensional harmonic trap
can be condensed into its ground state. Moreover, the ground state energy and
the wave function of two-dimensional vortex state are also obtained.
Quantitative comparisons between 2D results and 3D ones are made in detail.
