Boundary conditions for one-dimensional Feshbach–Villars equation

Solution of the Feshbach–Villars equation for the step potential

“…Following the idea of a previous paper [7], we study the one dimensional F V 1 2 equation for a particle subjected to a step potential V (x). As we know the above problem is trivial in quantum mechanics but in relativistic quantum mechanics it emerges the famous Klein paradox.…”

“…(24), (25) using a symbolic software [16]. We have chosen the parameters of the hypergeometric function in order to have an analogy with the spin-0 solution [7]. An equivalent solution with others parameters is given by Eq.…”

“…We note also that the wave function of spin-0 particle [7,17] can be deduced from the wave function (59) of spin-1/2 particle if we take only the two components (ψ 1 , ψ 3 ) and remove the effect of the spin-1/2 term (eV 0 ). This analogy with spin-0 particle does not exist for the Dirac wave functions.…”

“…If we remove the effect of the spin-1/2 term (eV 0 ) from the above relations (60),(61)we find the reflection and the transmission coefficients of the spin-0 case [7].…”

“…Like in the spin-0 case [7], in the following we are going to look for boundary conditions using the Feshbach-Villars transformation. In the case of the step potential, the presence of the delta Dirac in Eq.…”