Klein paradox

In relativistic quantum mechanics, the Klein paradox was an unexpected effect of relativity on the predictions of quantum tunneling theory for particles encountering high potential-energy barriers. The phenomenon itself is now known as Klein tunneling. It is named after physicist Oskar Klein who discovered it in 1929. Originally, Klein obtained a paradoxical result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. However, Klein's result showed that if the potential is at least of the order of the electron mass ${\displaystyle Ve\approx mc^{2}}$ (where V is the electric potential, e is the elementary charge, m is the electron mass and c is the speed of light), the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.

The immediate application of the paradox was to Rutherford's proton–electron model for neutral particles within the nucleus, before the discovery of the neutron, which made the model obsolete. The paradox presented a quantum mechanical objection to the notion of an electron confined within a nucleus. This clear and precise paradox suggested that an electron could not be confined within a nucleus by any potential well. The meaning of this paradox was intensely debated by Niels Bohr and others at the time.

For massive particles, the electric field strength required to observe the effect is enormous. The electric potential energy change similar to the rest energy of the incoming particle, ${\displaystyle mc^{2}}$, would need to occur over the Compton wavelength of the particle, ${\displaystyle \hbar /mc}$, which works out to ${\displaystyle 10^{16}~{\text{V}}/{\text{cm}}}$ for electrons. For electrons, such extreme fields might only be relevant in ${\displaystyle Z>170}$ nuclei or evaporation at the event horizon of black holes, but for 2-D quasiparticles at graphene p-n junctions, for which the required electric fields are on the order of ${\displaystyle 10^{5}~{\text{V}}/{\text{cm}}}$, the effect has been studied both theoretically and experimentally.