History and physics of the Klein paradox

The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed t o derive the 'paradox' first obtained by Klein. The explanation of this eff ect in terms of electron-positron production is re-assessed. It is shown th at a potential well or barrier in the Dirac equation can become supercritic al and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant curren t is emitted. This phenomenon is transient whereas the tunnelling first cal culated by Klein is time-independent. It is shown that tunnelling without e xponential suppression occurs when an electron is incident on a high barrie r, even when the barrier is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and is not necessarily connected to particle emission. The Coulomb potential is investigated and it is shown that a heavy nucleus of sufficiently large Z will bind positron s. correspondingly, as Z increases the Coulomb barrier should become increa singly transparent to positrons. This is an example of Klein tunnelling. Ph enomena akin to supercritical positron emission may be studied experimental ly in superfluid He-3.