Solving Klein's paradox

Abstract: We figure out the famous Klein's paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle's energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein-Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials are considered. The reflection coefficient never exceeds 1. The results are applied to discuss the transmissions of particles with no mass or with very small mass.