  The $S=1$ antiferromagnetic Heisenberg chain both with and without single ion
anisotropy is studied. Using the recently proposed density matrix
renormalization group technique we calculate the energy gaps as well as several
different correlation functions. The two gaps, $\Delta_{||}, \Delta_\perp$,
along with associated correlation lengths and velocities are determined. The
numerical results are shown to be in good agreement with theoretical
predictions derived from the nonlinear sigma model and a free boson model. We
also study the $S=1/2$ excitations that occur at the ends of open chains; in
particular we study the behavior associated with open boundary conditions,
using a model of $S=1/2$ spins coupled to the free bosons.

Condensed Matter