Klein paradox

Last updated September 01, 2024

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 $Ve\approx mc^{2}$ (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]

Physics overview

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.^[3] Before antiparticles where discovered and quantum field theory developed, this third solution was not understood. The puzzle came to be called the Klein paradox.^[4]

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, $mc^{2}$ , would need to occur over the Compton wavelength of the particle, $\hbar m/c$ , which works out to 10¹⁶ V/cm for electrons.^[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.^[5]^[6]^: 421

History

Oscar Klein published the paper describing what later came to be called the Klein paradox in 1929,^[1] 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.^[7]^: 350

Dirac equation mysteries

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 Edwin Schrodinger developed a new mechanics for the electron, a quantum mechanics that reproduced Bohr's results. Schrodinger 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.

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. Significantly Dirac's theory only predicted single-particle states. Creation or annihilation of particles could not be correctly analyzed in the single particle theory.

The Klein result was widely discussed immediately after it publication. Niels Bohr thought the result was related to the abrupt step and as a result Arnold Sommerfeld asked Fritz Sauter to investigate sloped steps. Sauter was able to confirm Bohr's conjecture: the paradoxical result only appeared for a step of $\Delta V>2mc^{2}$ over a distance similar to the electrons Compton wavelength, $\lambda =h/mc$ , about 2 x 10^-12m.^[8]

Throughout 1929 and 1930, a series of papers by different physicists attempted to understand Dirac's extra states.^[7]^: 351 Hermann Weyl suggested they corresponded to recently discovered protons, the only elementary particle other than the electron known at the time. Dirac pointed out Klein's negative electrons could not convert themselves to positive protons and suggested that the extra states were all filled with electrons already. Then a proton would amount to a missing electron in these lower states. Robert Oppenheimer and separately Igor Tamm showed that this would make atoms unstable. Finally in 1931 Dirac concluded that these states must correspond to a new "anti-electron" particle. In 1932 Carl Anderson experimentally observed these particles, renamed positrons.

Positron-electron creation

Resolution of the paradox would require quantum field theory which developed alongside quantum mechanics but at a slower pace due its many complexities. 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. Planck showed that the blackbody oscillators must be restricted to quantum transitions.^[7]^: 332 In 1927, Dirac published his first work on quantum electrodynamics by using quantum field theory. With this foundation, Heisenberg, Jordan, and Pauli incorporated relativistic invariance in quantized Maxwell's equations in 1928 and 1929.^[7]^: 341

However it took another 10 years before the theory could be applied to the problem of the Klein paradox. In 1941 Friedrich Hund showed that,^[9] 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. These results were confirmed in 1981 by Hansen and Ravndal using a more general treatment.^[10]^[8]^: 316

Massless particles

Consider a massless relativistic particle approaching a potential step of height $V_{0}$ with energy $E_{0}<V_{0}e$ and momentum $p$ .

The particle's wave function, $\psi$ , follows the time-independent Dirac equation:

\left(\sigma _{x}p+V\right)\psi =E_{0}\psi ,\quad V={\begin{cases}0,&x<0\\V_{0},&x>0\end{cases}}

And $\sigma _{x}$ is the Pauli matrix:

\sigma _{x}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right)

Fig. 1 A depiction of the dispersion relation, the x-axis represents momentum while the y-axis represents energy. Dispersion1.png — Fig. 1 A depiction of the dispersion relation, the x-axis represents momentum while the y-axis represents energy.

Assuming the particle is propagating from the left, we obtain two solutions — one before the step, in region (1) and one under the potential, in region (2):

\psi _{1}=Ae^{ipx}\left({\begin{matrix}1\\1\end{matrix}}\right)+A'e^{-ipx}\left({\begin{matrix}-1\\1\end{matrix}}\right),\quad p=E_{0}\,

\psi _{2}=Be^{ikx}\left({\begin{matrix}1\\1\end{matrix}}\right),\quad \left|k\right|=V_{0}-E_{0}\,

where the coefficients $A$ , $A'$ and $B$ are complex numbers. Both the incoming and transmitted wave functions are associated with positive group velocity (Blue lines in Fig.1), whereas the reflected wave function is associated with negative group velocity. (Green lines in Fig.1)

We now want to calculate the transmission and reflection coefficients, $T,R.$ They are derived from the probability amplitude currents.

The definition of the probability current associated with the Dirac equation is:

J_{i}=\psi _{i}^{\dagger }\sigma _{x}\psi _{i},\ i=1,2\,

In this case:

J_{1}=2\left[\left|A\right|^{2}-\left|A'\right|^{2}\right],\quad J_{2}=2\left|B\right|^{2}\,

The transmission and reflection coefficients are:

R={\frac {\left|A'\right|^{2}}{\left|A\right|^{2}}},\quad T={\frac {\left|B\right|^{2}}{\left|A\right|^{2}}}\,

Continuity of the wave function at $x=0$ , yields:

\left|A\right|^{2}=\left|B\right|^{2}\,

\left|A'\right|^{2}=0\,

And so the transmission coefficient is 1 and there is no reflection.

One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle. This explanation best suits the single particle solution cited above. 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.

Massive case

For the massive case, the calculations are similar to the above. The results are as surprising as in the massless case. The transmission coefficient is always larger than zero, and approaches 1 as the potential step goes to infinity.

The Klein zone

If the energy of the particle is in the range $mc^{2}<E<Ve-mc^{2}$ , then partial reflection rather than total reflection will result.

Resolutions for the massive case

The traditional resolution uses particle–anti-particle pair production in the context of quantum field theory.^[10]

Other cases

These results were expanded to higher dimensions, and to other types of potentials, such as a linear step, a square barrier, a smooth potential, etc. Many experiments in electron transport in graphene rely on the Klein paradox for massless particles.^[5]^[11]

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References

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↑ Hund, Fritz. "Materieerzeugung im anschaulichen und im gequantelten Wellenbild der Materie." Zeitschrift für Physik 117.1 (1941): 1-17.
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