  The mechanism of the Coulomb breakup reactions of the nuclei with
neutron-halo structure is investigated in detail. A time-dependent
Schr\"odinger equation for the halo neutron is numerically solved by treating
the Coulomb field of a target as an external field. The momentum distribution
and the post-acceleration effect of the final fragments are discussed in a
fully quantum mechanical way to clarify the limitation of the intuitive picture
based on the classical mechanics. The theory is applied to the Coulomb breakup
reaction of $^{11}$Be + $^{208}$Pb. The breakup mechanism is found to be
different between the channels of $j^{\pi}=\frac{1}{2}^{-}$ and
$\frac{3}{2}^{-}$, reflecting the underlying structure of $^{11}$Be. The
calculated result reproduces the energy spectrum of the breakup fragments
reasonably well, but explains only about a half of the observed longitudinal
momentum difference.
