content_image/359887.jpg content_image/359888.jpg content_image/359889.jpg content_image/359890.jpg content_image/359891.jpg content_image/359892.jpg content_image/359893.jpg content_image/359894.jpg content_image/359895.jpg content_image/359896.jpg
## II. THE BASIC EQUATIONS 

We recall the free Dirac equation (using natural units, ¯ h = c = 1) 

<img src='content_image/359887.jpg'>

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: 

<img src='content_image/359888.jpg'>

Decomposing the Dirac spinor ψ ( x ) into upper ( u$_{1}$ ) and lower ( u$_{2}$ ) components: ψ = ( u$_{1}$ u$_{2}$ $^{)}$, gives the coupled equations: 

<img src='content_image/359889.jpg'>

<img src='content_image/359890.jpg'>

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]: 

<img src='content_image/359891.jpg'>

<img src='content_image/359892.jpg'>

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: 

<img src='content_image/359893.jpg'>

<img src='content_image/359894.jpg'>

<img src='content_image/359895.jpg'>

<img src='content_image/359896.jpg'>

