  Caroli-de Gennes-Martricon (CdGM) states were predicted in 1964 as low energy
excitations within vortex cores of type-II superconductors. In the quantum
limit, namely $T/T_\mathrm{c} \ll \Delta/E_\mathrm{F}$, the energy levels of
these states were predicted to be discrete with the basic levels at $E_\mu =
\pm \mu \Delta^2/E_\mathrm{F}$ ($\mu = 1/2$, $3/2$, $5/2$, ...). However, due
to the small ratio of $\Delta/E_\mathrm{F}$ in most type-II superconductors, it
is very difficult to observe the discrete CdGM states, but rather a symmetric
peak appears at zero-bias at the vortex center. Here we report the clear
observation of these discrete energy levels of CdGM states in
FeTe$_{0.55}$Se$_{0.45}$. The rather stable energies of these states versus
space clearly validates our conclusion. Analysis based on the energies of these
CdGM states indicates that the Fermi energy in the present system is very
small.
