Klein paradox
[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.
(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.
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] The Klein paradox is an unexpected consequence of relativity on the interaction of quantum particles with electrostatic potentials.
The quantum mechanical problem of free particles striking an electrostatic step potential has two solutions when relativity is ignored.
Relativity adds a third solution: very steep potential steps appear to create particles and antiparticles that then change the calculated ratio of transmission and reflection.
The theoretical tools called quantum mechanics cannot handle the creation of particles, making any analysis of the relativistic case suspect.
[4] For massive particles, the electric field strength required to observe the effect is enormous.
[5] For electrons, such extreme fields might only be relevant in Z>170 nuclei or evaporation at the event horizon of black holes, but for 2-D quasiparticles at graphene p-n junctions the effect can be studied experimentally.
The paradox raised questions about how relativity was added to quantum mechanics in Dirac's first attempt.
[7]: 350 The Bohr model of the atom published in 1913 assumed electrons in motion around a compact positive nucleus.
The success of the Bohr model in predicting atomic spectra suggested that the classical mechanics could not be correct.
Schrodinger and other physicists knew this mechanics was incomplete: it did not include effects of special relativity nor the interaction of matter and radiation.
Klein found that these extra states caused absurd results from models for electrons striking a large, sharp change in electrostatic potential: a negative current appeared beyond the barrier.
[7]: 351 Hermann Weyl suggested they corresponded to recently discovered protons, the only elementary particle other than the electron known at the time.
Robert Oppenheimer and separately Igor Tamm showed that this would make atoms unstable.
The concept goes back to Max Planck's demonstration that Maxwell's classical electrodynamics so successful in many applications, fails to predict the blackbody spectrum.
With this foundation, Heisenberg, Jordan, and Pauli incorporated relativistic invariance in quantized Maxwell's equations in 1928 and 1929.
In 1941 Friedrich Hund showed that,[8] under the conditions of the paradox, two currents of opposite charge are spontaneously generated at the step.
In modern terminology pairs of electrons and positrons are spontaneously created the step potential.
The definition of the probability current associated with the Dirac equation is: In this case: The transmission and reflection coefficients are: Continuity of the wave function at
One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle.
Other, more complex interpretations are suggested in literature, in the context of quantum field theory where the unrestrained tunnelling is shown to occur due to the existence of particle–antiparticle pairs at the potential.
Many experiments in electron transport in graphene rely on the Klein paradox for massless particles.
