  We discuss several problems concerning domain walls in the spin $S$ Ising
model at zero temperature in a magnetic field, $H/(2S)$, applied in the $x$
direction. Some results are also given for the planar ($y$-$z$) model in a
transverse field. We treat the quantum problem in one dimension by perturbation
theory at small $H$ and numerically over a large range of $H$. We obtain the
spin density profile by fixing the spins at opposite ends of the chain to have
opposite signs of $S_z$. One dimension is special in that there the quantum
width of the wall is proportional to the size $L$ of the system. We also study
the quantitative features of the `particle' band which extends up to energies
of order $H$ above the ground state. Except for the planar limit, this particle
band is well separated from excitations having energy $J/S$ involving creation
of more walls. At large $S$ this particle band develops energy gaps and the
lowest sub-band has tunnel splittings of order $H2^{1-2S}$. This scale of
energy gives rise to anomalous scaling with respect to a) finite size, b)
temperature, or c) random potentials. The intrinsic width of the domain wall
and the pinning energy are also defined and calculated in certain limiting
cases. The general conclusion is that quantum effects prevent the wall from
being sharp and in higher dimension would prevent sudden excursions in the
configuration of the wall.
