Reflection and transmission resonances and accuracy of the wkb method

TL;DR

This paper investigates wave transmission and reflection in various potentials, compares WKB approximation results with exact calculations, and assesses the accuracy of the WKB method through resonance analysis and transfer matrix bounds.

Contribution

It provides a detailed comparison of WKB and exact methods for calculating wave amplitudes across different potentials, highlighting the method's accuracy and resonance behavior.

Findings

WKB results closely match exact calculations for certain potentials

Identification of reflection resonances specific to each potential

Transfer matrix bounds offer rigorous limits on transmission and reflection probabilities

Abstract

In this paper, we calculate the transmission and reflection amplitudes of wave functions for different potentials such as the delta function, the rectangular barrier, the Eckart potential, and the Hulthen potential. We describe the relationship between these amplitudes and compute the reflection resonances between each potential. We describe the transmission and reflection probabilities using the WKB formula and compare the results with ones obtained from matching the boundary conditions. Furthermore, we use a two by two transfer matrix to calculate a rigorous bound on the transmission and reflection probabilities.