  Three analytic solutions to the Schr\"{o}dinger equation for the
time-dependent Landau-Zener Hamiltonian are presented. They correspond to
specific finite-time driving paths in a bounded parameter space of a two-level
system. Two of these paths go through the avoided crossing of levels, either
with a constant speed or with variable speed that decreases in the region of
reduced energy gap, the third path bypasses the crossing such that the energy
gap remains constant. The solutions yield exact time dependencies of the
excitation probability for the system evolving from the ground state of the
initial Hamiltonian. The Landau-Zener formula emerges as an approximation valid
within a certain interval of driving times for the constant-speed driving
through the avoided crossing. For long driving times, all solutions converge to
the prediction of the adiabatic perturbation theory. The excitation probability
vanishes at some discrete time instants.
