  Quantum Monte Carlo (QMC) simulations and the Local Density Approximation
(LDA) are used to map the constant particle number (canonical) trajectories of
the Bose Hubbard Hamiltonian confined in a harmonic trap onto the $(\mu/U,t/U)$
phase diagram of the uniform system. Generically, these curves do not intercept
the tips of the Mott insulator (MI) lobes of the uniform system. This
observation necessitates a clarification of the appropriate comparison between
critical couplings obtained in experiments on trapped systems with those
obtained in QMC simulations. The density profiles and visibility are also
obtained along these trajectories. Density profiles from QMC in the confined
case are compared with LDA results.
