  We consider the three-dimensional electron gas confined by a strictly
two-dimensional homogeneous positive charge density at $z=0$. Within the
Hartree-Fock approximation, we study the mode structure in the confined
direction in the metallic regime. We find, that for $r_s<1.3$ ($r_s<2.5$) the
unpolarized (polarized) electron gas starts to populate also the first excited
state in the $z$-direction.
