Improved transfer matrix methods for calculating quantum transmission-coefficient

TL;DR

This paper introduces improved transfer matrix methods with advanced boundary conditions for more accurate calculation of quantum transmission coefficients in one-dimensional systems, significantly reducing errors especially with third order WKB conditions.

Contribution

It develops novel transfer matrix algorithms incorporating improved boundary conditions, notably achieving substantial error reduction in quantum transmission calculations.

Findings

Third order WKB boundary conditions reduce error by a factor of 4100.

Modified transfer matrix methods outperform traditional approaches.

Methods show excellent accuracy with first order WKB boundary conditions.

Abstract

Methods for calculating the transmission coefficient are proposed, all of which arise from improved non-reflecting WKB boundary conditions at the edge of the computational domain in 1-dimensional geometries. In the first, the Schr\"{o}dinger equation is solved numerically while the second is a transfer matrix (TM) algorithm where the potential is approximated by steps, but with the first and last matrix modified to reflect the new boundary condition. Both methods give excellent results with first order WKB boundary conditions. The third uses the transfer matrix method with third order WKB boundary conditions. For the the parabolic potential, the average error for the modified third order TM method reduces by factor of 4100 over the unmodified TM method.