Klein Paradox for Spin-0 Particles by Douglas M. Gingrich (University of Alberta lecture notes, page dated 18 March 2004)

The institutional lecture derives the outgoing branch, boundary amplitudes and signed charge-current balance for a scalar particle at an electrostatic step. Equations 4.142–4.155 distinguish ordinary transmission, evanescence and the negative-kinetic-energy regime; the following discussion interprets the latter through pair creation.