  We use the density matrix renormalization group (DMRG) to perform a detailed
study of the critical properties of the two dimensional Q state Potts model,
including the magnetization and energy-density profiles, bulk and surface
critical exponents and the Casimir amplitudes. We apply symmetry breaking
boundary conditions to a $L \times \infty$ strip and diagonalize the
corresponding transfer matrix for a series of moderately large systems ($L \le
64$) by the DMRG method. The numerically very accurate finite lattice results
are then extrapolated by efficient sequence extrapolation techniques. The
critical density profiles and the Casimir amplitudes are found to follow
precisely the conformal predictions for Q=2 and 3. Similarly, the bulk and
surface critical exponents of the models are in very good agreement with the
conformal and exact values: their accuracy has reached or even exceeded the
accuracy of other available numerical methods. For the Q=4 model both the
profiles and the critical exponents show strong logarithmic corrections, which
are also studied.
