  At sufficiently high temperature and density, quantum chromodynamics (QCD) is
expected to undergo a phase transition from the confined phase to the
quark-gluon plasma phase. In the Lagrangian lattice formulation the Monte Carlo
method works well for QCD at finite temperature, however, it breaks down at
finite chemical potential. We develop a Hamiltonian approach to lattice QCD at
finite chemical potential and solve it in the case of free quarks and in the
strong coupling limit. At zero temperature, we calculate the vacuum energy,
chiral condensate, quark number density and its susceptibility, as well as mass
of the pseudoscalar, vector mesons and nucleon. We find that the chiral phase
transition is of first order, and the critical chemical potential is $\mu_C
=m_{dyn}^{(0)}$ (dynamical quark mass at $\mu=0$). This is consistent with
$\mu_C \approx M_N^{(0)}/3$ (where $M_N^{(0)}$ is the nucleon mass at $\mu=0$).
