  Motivated by a desire to understand quantum fluctuation energy densities and
stress within a spatially varying dielectric medium, we examine the vacuum
expectation value for the stress tensor of a scalar field with arbitrary
conformal parameter, in the background of a given potential that depends on
only one spatial coordinate. We regulate the expressions by incorporating a
temporal-spatial cutoff in the (imaginary) time and transverse-spatial
directions. The divergences are captured by the zeroth- and second-order WKB
approximations. Then the stress tensor is "renormalized" by omitting the terms
that depend on the cutoff. The ambiguities that inevitably arise in this
procedure are both duly noted and restricted by imposing certain physical
conditions; one result is that the renormalized stress tensor exhibits the
expected trace anomaly. The renormalized stress tensor exhibits no pressure
anomaly, in that the principle of virtual work is satisfied for motions in a
transverse direction. We then consider a potential that defines a wall, a
one-dimensional potential that vanishes for $z<0$ and rises like $z^\alpha$,
$\alpha>0$, for $z>0$. The full finite stress tensor is computed numerically
for the two cases where explicit solutions to the differential equation are
available, $\alpha=1$ and 2. The energy density exhibits an inverse linear
divergence as the boundary is approached from the inside for a linear
potential, and a logarithmic divergence for a quadratic potential. Finally, the
interaction between two such walls is computed, and it is shown that the
attractive Casimir pressure between the two walls also satisfies the principle
of virtual work (i.e., the pressure equals the negative derivative of the
energy with respect to the distance between the walls).
