  I provide a transfer matrix method for the Foldy-Wouthuysen representation of
the Dirac equation. I develop the relationship between the reflection and
transmission coefficients of the Dirac spinors and the wavefunction in the
transformed representation. I develop a WKB approximation for Dirac fermions
which has the same elegant form as the WKB solution to Schr\"{o}dinger's
equation. My WKB approximation is to all orders and includes the semi-classical
turning point. I provide an extension to fully 2-dimensional periodic
structures by Fourier methods for band engineering. I verify my methods for all
energies by comparison with analytic solutions developed in the Dirac spinor
representation. I also include a rich appendix detailing my research into the
Green's functions of Dirac fermions.
