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Transmission through and reflection from a potential barrier, and the veryclosely related issue of particle production from a parametric resonance, aretopics of considerable general interest in quantum physics. We have developed arather general bound on quantum transmission probabilities, and recentlyapplied it to bounding the greybody factors of a Schwarzschild black hole. Inthis current paper, we take a different tack -- we report a way of using theMiller-Good transformation (which maps an initial Schrodinger equation to afinal Schrodinger equation for a different potential) to significantlygeneralize the previous bound. We then apply this general formalism in a veryspecific manner to derive a rigorous bound that is "as close as possible" tothe usual WKB estimate for barrier penetration.

Constraining transmission and reflection probabilities by using the Miller-Good transformation