Klein paradox

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In relativistic quantum mechanics, the Klein paradox (also known as Klein tunneling) is a quantum phenomenon related to particles encountering high-energy potential barriers. It is named after physicist Oskar Klein who discovered in 1929.^[1] 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. The paradox presented a quantum mechanical objection to the notion of an electron confined within a nucleus.^[2] 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.^[2]