II. THE BASIC EQUATIONS We recall the free Dirac equation (using natural units, ¯ h = c = 1) and µ = 0 , 1 , 2 , 3 is a space-time index. Considering instead the case of one space dimension it is possible to choose the gamma matrices γ x and γ 0 of dimension two and it is customary to set them respectively to the Pauli matrices iσ$_{x}$ and σ$_{z}$ [9]. Considering a charge particle minimally coupled to an electromagnetic potential, in the absence of the space component of a vector potential, and setting V ( x ) = eA$_{0}$ ( x ) the one-dimensional Dirac equation for a stationary state Ψ( x, t ) = e − $^{iEt}$ψ ( x ) becomes: Decomposing the Dirac spinor ψ ( x ) into upper ( u$_{1}$ ) and lower ( u$_{2}$ ) components: ψ = ( u$_{1}$ u$_{2}$ $^{)}$, gives the coupled equations: It turns out to be convenient to define two auxiliary components φ ( x ) and χ ( x ) in terms of u$_{1}$ ( x ) and u$_{2}$ ( x ) as in [9]: Using the above definitions and Eqs. (3a,3b) we find the first order coupled equations for the components φ ( x ) and χ ( x ): which give the second order equations: 4 ( iγ µ ∂ ∂x µ − m$^{)}$ Ψ( x ) = 0 , (1) [ σ$_{x}$ d dx − ( E − V ( x )) σ$_{z}$ + m$^{]}$ ψ ( x ) = 0 . (2) u ′ $_{1}$( x ) = − [ m + E − V ( x )] u$_{2}$ ( x ) , (3a) u ′ $_{2}$( x ) = − [ m − E + V ( x )] u$_{1}$ ( x ) , (3b) φ ( x ) = u$_{1}$ ( x ) + iu$_{2}$ ( x ) , (4a) χ ( x ) = u$_{1}$ ( x ) − iu$_{2}$ ( x ) , (4b) φ $^{′}$( x ) = − imχ ( x ) + i [ E − V ( x )] φ ( x ) , (5a) χ $^{′}$( x ) = + imφ ( x ) − i [ E − V ( x )] χ ( x ) , (5b) φ $^{′′}$( x ) + $^{[}$( E − V ( x )) 2 − m 2 + iV $^{′}$( x ) ] φ ( x ) = − im $^{′}$χ ( x ) , (6a) χ $^{′′}$( x ) + $^{[}$( E − V ( x )) 2 − m 2 − iV $^{′}$( x ) ] χ ( x ) = + im $^{′}$φ ( x ) , (6b)