Klein paradox

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. 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. 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.

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. One solution applies when the particles approaching the barrier have less kinetic energy than the step: the particles are reflected. If the particles have more energy than the step, some are transmitted past the step, while some are reflected. The ratio of reflection to transmission depends on the energy difference. 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. Before antiparticles were discovered and quantum field theory developed, this third solution was not understood. The puzzle came to be called the Klein paradox.

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, ${\displaystyle \hbar m/c}$, which works out to 10¹⁶ V/cm for electrons. 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.

Oskar Klein published the paper describing what later came to be called the Klein paradox in 1929, just as physicists were grappling with two problems: how to combine the theories of relativity and quantum mechanics and how to understand the coupling of matter and light known as electrodynamics. The paradox raised questions about how relativity was added to quantum mechanics in Dirac's first attempt. It would take the development of the new quantum field theory developed for electrodynamics to resolve the paradox. Thus the background of the paradox has two threads: the development of quantum mechanics and of quantum electrodynamics.

The Bohr model of the atom published in 1913 assumed electrons in motion around a compact positive nucleus. An atomic electron obeying classical mechanics in the presence of a positive charged nucleus experiences a Lorentz force: they should radiate energy and accelerate in to the atomic core. The success of the Bohr model in predicting atomic spectra suggested that the classical mechanics could not be correct.

In 1926 Erwin Schrödinger developed a new mechanics for the electron, a quantum mechanics that reproduced Bohr's results. Schrödinger and other physicists knew this mechanics was incomplete: it did not include effects of special relativity nor the interaction of matter and radiation.

Paul Dirac solved the first issue in 1928 with his relativistic quantum theory of the electron. The combination was more accurate and also predicted electron spin. However, it also included twice as many states as expected, all with lower energy than the ones involved in atomic physics.