Barrier paradox in the Klein zone

We study the solutions for a one-dimensional electrostatic potential in the Dirac equation when the incoming wave packet exhibits the Klein paradox (pair production). With a barrier potential we demonstrate the existence of multiple reflections (and transmissions). The antiparticle solutions which are necessarily localized within the barrier region create new pairs with each reflection at the potential walls. Consequently we encounter a new “paradox” for the barrier because successive outgoing wave amplitudes grow geometrically.

Figure 1

Particle and antiparticle viewpoints. The energy of the “particle” is that of the incoming particle, i.e., $E$. Since the antiparticle sees a potential of $-V_0$ its energy, say $E_a$, must satisfy ${[E_a-(-V_0)]}^2={(E-V_0)}^2$. Consequently, $E_a=-E$.