  We investigate the behavior of vortex bound states in the quantum limit by
self-consistently solving the Bogoliubov-de Gennes equation. We find that the
energies of the vortex bound states deviates from the analytical result
$E_\mu=\mu\Delta^2/E_F$ with the half-integer angular momentum $\mu$ in the
extreme quantum limit. Specifically, the energy ratio for the first three
orders is more close to $1:2:3$ instead of $1:3:5$ at extremely low
temperature. The local density of states reveals an Friedel-like behavior
associated with that of the pair potential in the extreme quantum limit, which
will be smoothed out by thermal effect above a certain temperature even the
quantum limit condition, namely $T/T_c<\Delta/E_F$ is still satisfied. Our
studies show that the vortex bound states can exhibit very distinct features in
different temperature regimes, which provides a comprehensive understanding and
should stimulate more experimental efforts for verifications.
