Seventy years of the Klein paradox

The Klein paradox is examined. Its explanation in terms of electron-positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can produce positron or electron emission spontaneously if the potential is strong enough. The vacuum charge and lifetime of the well/barrier are calculated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Furthermore, tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when it is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling. (C) 1999 Elsevier Science B.V. All rights reserved.