  We solve the Dirac radial equation for a nucleon in a scalar Woods-Saxon
potential well of depth $V_0$ and radius $r_0$. A sequence of values for the
depth and radius are considered. For shallow potentials with $-1000 MeV\lesssim
V_0 < 0$ the wave functions for the positive-energy states $\Psi _+(r)$ are
dominated by their nucleon component $g(r)$. But for deeper potentials with
$V_0 \lesssim -1500 MeV $ the $\Psi_+(r)$s begin to have dominant anti-nucleon
component $f(r)$. In particular, a special intruder state enters with wave
function $\Psi_{1/2}(r)$ and energy $E_{1/2}$. We have considered several $r_0$
values between 2 and 8 fm. For $V_0 \lesssim -2000 MeV$ and the above $r_0$
values, $\Psi _{1/2}$ is the only bound positive-energy state and has its
$g(r)$ closely equal to $-f(r)$, both having a narrow wave-packet shape
centered around $r_0$. The $E_{1/2}$ of this state is practically independent
of $V_0$ for the above $V_0$ range and obeys closely the relation
$E_{1/2}=\frac{\hbar c}{r_0}$.
