Klein paradox

In <a href="/facts/Relativistic_quantum_mechanics/DhkDNnHf">relativistic quantum mechanics</a>, 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 <a href="/facts/Oskar_Klein/xMUugBRR">Oskar Klein</a> who discovered in 1929. Originally, Klein obtained a paradoxical result by applying the <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> to the familiar problem of <a href="/facts/Electron_scattering/wA7Eh2oz">electron scattering</a> from a <a href="/facts/Potential_barrier/Cer7zMUn">potential barrier</a>. In nonrelativistic quantum mechanics, <a href="/facts/Electron_tunneling/WU4NMrKx">electron tunneling</a> into a barrier is observed, with exponential <a href="/facts/Damping_ratio/UW3RsjEd">damping</a>. However, Klein's result showed that if the potential is at least of the order of the <a href="/facts/Electron_mass/L0KBo5cW">electron mass</a> 
 
 
 
 V
 e
 ≈
 m
 
 c
 
 2
 
 
 
 
 {\displaystyle Ve\approx mc^{2}}
 
 (where V is the <a href="/facts/Electric_potential/Nmqm6hm5">electric potential</a>, e is the <a href="/facts/Elementary_charge/lPcj6CoQ">elementary charge</a>, m is the <a href="/facts/Electron_mass/L0KBo5cW">electron mass</a> and c is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>), 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 <a href="/facts/Neutron/oRCM1geb">proton–electron</a> model for neutral particles within the nucleus, before the discovery of the <a href="/facts/Neutron/oRCM1geb">neutron</a>. 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 <a href="/facts/Niels_Bohr/ydLPvQXT">Niels Bohr</a> and others at the time.

Klein paradox open-in-new

Klein paradox