Applying the Dirac equation to derive the transfer matrix for piecewise constant potentials

TL;DR

This paper develops a relativistic transfer matrix method based on the Dirac equation for electrons traversing multiple rectangular barriers, extending traditional non-relativistic approaches to relativistic regimes.

Contribution

It introduces a relativistic transfer matrix derived from the Dirac equation, applicable to arbitrary-shaped barriers, bridging non-relativistic and relativistic quantum mechanics.

Findings

Relativistic transfer matrix reduces to traditional form in the non-relativistic limit

Method applicable to barriers of arbitrary shape

Provides a consistent relativistic framework for quantum tunneling

Abstract

One propose a relativistic version of the transfer matrix method for an electron moving through a given number of rectangular barriers of arbitrary shape. It is shown that starting with the Dirac equation depending on the effective mass and a suitably chosen relativistic potential, one obtains a relativistic transfer matrix which takes the correct traditional form in the non-relativistic limit.