  Transmission through and reflection from a potential barrier, and the very
closely related issue of particle production from a parametric resonance, are
topics of considerable general interest in quantum physics. We have developed a
rather general bound on quantum transmission probabilities, and recently
applied it to bounding the greybody factors of a Schwarzschild black hole. In
this current paper, we take a different tack -- we report a way of using the
Miller-Good transformation (which maps an initial Schrodinger equation to a
final Schrodinger equation for a different potential) to significantly
generalize the previous bound. We then apply this general formalism in a very
specific manner to derive a rigorous bound that is "as close as possible" to
the usual WKB estimate for barrier penetration.
