  In this paper, we have studied the one-dimensional commensurate quantum
Frenkel-Kontorova model by a density-matrix renormalization group (DMRG)
algorithm. The focus has been on its properties of the entanglement, the
coordinate correlation, the ground state energy and the energy gap between the
ground state and the first excited one. It is demonstrated that a quantum phase
transition (QPT) between the pinned and the sliding phases takes place as the
quantum fluctuation, measured by an effective Planck constant $\tilde\hbar$,
increases to a threshold value $\tilde \hbar_c$.
