  We study the Bose-Hubbard model using the finite size density matrix
renormalization group method. We obtain for the first time a complete phase
diagram for a system in the presence of a harmonic trap and compare it with
that of the homogeneous system. To realize the transition from the superfluid
to the Mott insulator phase we investigate different experimental signatures of
these phases in quantities such as momentum distribution, visibility,
condensate fraction and the total number of bosons at a particular density. The
relationships between the various experimental signatures and the phase diagram
are highlighted.
