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

TL;DR

This paper introduces a generalized method using the Miller-Good transformation to derive tighter bounds on quantum transmission and reflection probabilities, closely aligning with traditional WKB estimates.

Contribution

It develops a novel formalism that extends previous bounds on quantum transmission by applying the Miller-Good transformation, enhancing accuracy in barrier penetration estimates.

Findings

Derived a generalized bound on quantum transmission probabilities.

Applied the formalism to closely match WKB barrier penetration estimates.

Extended the applicability of bounds to a wider class of potentials.

Abstract

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.