  A full quantum-mechanical treatment of the Bogoliubov-de Gennes equation for
a single vortex in a d-wave superconductor is presented. First, we find
low-energy states extended in four diagonal directions, which have no
counterpart in a vortex of s-wave superconductors. The four-fold symmetry is
due to 'quantum effect', which is enhanced when $p_{F}\xi $ is small. Second,
for $p_{F}\xi \sim 1$, a peak with a large energy gap $E_{0}\sim \Delta $ is
found in the density of states, which is due to the formation of the lowest
bound states.
