  For a nanowire quantum dot with the confining potential modeled by both the
infinite and the finite square wells, we obtain exactly the energy spectrum and
the wave functions in the strong spin-orbit coupling regime. We find that
regardless of how small the well height is, there are at least two bound states
in the finite square well: one has the $\sigma^{x}\mathcal{P}=-1$ symmetry and
the other has the $\sigma^{x}\mathcal{P}=1$ symmetry. When the well height is
slowly tuned from large to small, the position of the maximal probability
density of the first excited state moves from the center to $x\ne0$, while the
position of the maximal probability density of the ground state is always at
the center. A strong enhancement of the spin-orbit effect is demonstrated by
tuning the well height. In particular, there exists a critical height
$V^{c}_{0}$, at which the spin-orbit effect is enhanced to maximal.
