  The $\Omega$ baryons with $J^P=3/2^\pm, 1/2^\pm$ are studied on the lattice
in the quenched approximation. Their mass levels are ordered as
$M_{3/2^+}<M_{3/2^-}\approx M_{1/2^-}<M_{1/2^+}$, as is expected from the
constituent quark model. The mass values are also close to those of the four
$\Omega$ states observed in experiments, respectively. We calculate the
Bethe-Salpeter amplitudes of $\Omega(3/2^+)$ and $\Omega(1/2^+)$ and find there
is a radial node for the $\Omega(1/2^+)$ Bethe-Salpeter amplitude, which may
imply that $\Omega(1/2^+)$ is an orbital excitation of $\Omega$ baryons as a
member of the $(D,L_N^P)=(70,0_2^+)$ supermultiplet in the $SU(6)\bigotimes
O(3)$ quark model description. Our results are helpful for identifying the
quantum number of experimentally observed $\Omega$ states.
