Introduction to relativistic quantum mechanics smokey. The behavior of a beam of spinzero particles incident on a region of large potential increase is examined. Using the complex wkb method, the timeindependent scattering theory in terms of incoming. The discovery of dirac equation and its impact on present. The rst point to be elucidated is that the potential in the extended form of the dirac equation is not a scalar potential as stated by nitta et al. Pdf history and physics of the klein paradox semantic. This demonstration shows the reflection and transmission coefficients for a dirac particle with spinup incident on a square barrier of variable height the energy of the particle is fixed at 1 unit but its mass is allowed to vary from 0 to. In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. The dirac equation, proposed by paul dirac in 1928 to describe the behaviour of relativistic quantum particles, merges quantum mechanics with special relativity. Perhaps, the most paradoxical implications of the dirac equation are the klein tunneling3456 7 8910 and the socalled zitterbewegung phenomenon 3, 11, 12. Solution of dirac equation for a step potential and the. First, the lorentz structure of the potential and its connection with the klein paradox. The discovery of dirac equation and its impact on presentday physics reproduced with permission from dirac cmd feynman. However, the relevant equation of motion is the dirac equation, not the klein gordon equation, though the difference is not large insofar as the paradox.
The klein paradox was first described by oskar klein in 1928 when investigating scattering of relativistic particles described by the dirac equation. The quantum theory of the electron 1928 describing electrons, protons, quarks,neutrinos. The results are compared with those obtained from similar computations employing the dirac equation. Kleins paradox in a fourspace formulation of diracs. This comparison yields an instructive illustration of the difference between particles and antiparticles in spin zero and spin onehalf singleparticle theory. It is shown that a potential well or barrier in the dirac equation can become supercritical and emit positrons. Understanding the dirac equation and the electronvacuum. Some were even bold enough to seek consistency with gen.
The klein paradox was a generic problem for the dirac equation that was resolved very soon after its inception. Lecture 15 page 2 of 6 klein gordon equation 1926 schrodinger. Following recent results on the dirac equation, we propose a solution to this paradox for the klein gordon case by introducing virtual beams in a natural wellposed generalization of the method of images in the theory of partial differential equations. Solutions of the one dimensional dirac equation with piecewise constant potentials are presented using standard methods. Quantum simulation of the klein paradox quantum optics.
However, some extra physical detail can be inferred, and this suggests that the most extreme case involves pair production within the potential barrier. In 1929, physicist oskar klein obtained a surprising result by applying the dirac equation to the familiar problem of electron scattering from a potential barrier. Now, suppose a solution to the klein gordon equation is a free particle with energy eand momentum p ip ne x 11 1. Relativistic quantum mechanics janos polonyi university of strasbourg dated. Its explanation in terms of electronpositron production is reassessed. Applications to graphene systems are also discussed. Using the complex wkb method, the timeindependent scattering theory in terms of incoming and outgoing plane wave solutions is established.
This is interpreted as the creation of particleantiparticle pairs, where the negative transmission coefficient just shows the flux of. Shortly afterwards, klein solved a simple potential step problem for the dirac equation and encountered an. Transmission probabilities through a 100nmwide barrier as a function of the angle of incidence for single layer grahene. The generalization to quaternionic potentials 20 could be useful to understand many hidden aspects of the dirac solutions. This is a numerical simulation of the dirac equation in two dimensions that illustrates the klein paradox.
Klein paradox and scattering theory for the semiclassical. Kleingordon equation the quest for a mathematical theory of quantum mechanics began with great am bition. Ii we turn to the underlying physics of the klein paradox and show that particle production and klein tunnelling arise naturally in the dirac equation. We will try to find a relativistic quantum mechanical description of the electron. With the linear confining potential, we show that the dirac equation presents no bound state. Pdf solution of dirac equation for a step potential and. These solutions show that the klein paradox is nonexistent and represents a failure to correctly match solutions across a step potential. Klein paradox and scattering theory for the semiclassical dirac equation khochman, abdallah.
However, klein s result showed that if the potential is of the order of the electron mass. Although klein s gedanken experiment is now well understood, the notion of paradox is still widely used27, perhaps because the e. A wave packet is directed towards to a steep potential wall that is high enough to. The explanation of this effect in terms of electronpositron production is reassessed.
The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31 making the standard substitution, p. The klein gordon equation also shows the paradox 12 where it is an artifact of its not being a proper quantum mechanical equation with a positive definite probability density. Klein paradox and scattering theory for the semiclassical dirac equation klein paradox and scattering theory for the semiclassical dirac equation khochman, abdallah 20090101 00. Neither of the above papers are completely correct in my opinion.
Using the complex wkb method, the timeindependent scattering. In its free form, or including electromagnetic interactions, it describes all spin1 2 massive particles such as electrons and quarks for which parity is a symmetry. Both the klein gordon and the dirac equation are no 1particle wave equations, but relativistic. What we see is that the negative energy solutions of the dirac equation become accessible on the barrier side thus allowing both reflection and transmissi. The rst point to be elucidated is that the potential in the extended form of the dirac equation is. The effect that is known as the klein paradox 1, 2 1 is one of the cornerstones in the development of relativistic quantum mechanics. The dirac equation has mainly been used for the description of the highenergy relativistic physics of electrons and photons. Virtual beams and the klein paradox for the kleingordon. The explanation of this e ect in terms of electronpositron production is reassessed.
Both are often mentioned in the current literature on this equation and both give rise to controversy among researchers. Both these results can be identified as fine examples of the klein paradox. Chiral tunnelling and the klein paradox in graphene. It is shown that a potential well or barrier in the dirac equation can produce positron or electron emission spontaneously if the potential is strong enough.
We study the klein paradox for the semiclassical dirac operator on r with potentials having constant limits, possibly different at infinity. It resulted from the analysis of dirac equation for a particle that is subject to a onedimensional impulsive repulsive force, and the epithet paradox was given because the solution was in discord with anything that the intuition would have expected. Relativistic quantum mechanics kleingordon equation dirac. The conclusion is that one needs to take the correct solution with positive probability current in the klein paradox regime. The early papers by klein, sauter and hund which investigate scattering off a high step potential in the context of the dirac equation are discussed to derive the paradox first obtained by klein. Klein paradox and scattering theory for the semiclassical dirac equation article pdf available in asymptotic analysis 6534 december 2007 with 31 reads how we measure reads. The klein paradox role of chirality klein tunneling in singlelayer graphene klein tunneling and conductivity klein tunneling in bilayer graphene. The second of these aspects is the question of how closely the graphene spectrum resembles the dirac spectral properties and states. The elecron concentration n outside the barrier is chosen as 0. Pdf klein paradox and scattering theory for the semi. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic.
We find that in the case of potential well, the bound states disappear from the spectrum for large enough potential depth. We also discuss the socalled klein paradox that can actually be seen, doing away with its paradoxical status forever. Understanding the dirac equation and the electronvacuum system william c. In 1928, dirac proposed a wave equation to describe relativistic electrons1. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. At the very least it should be consistent with the theory of spe cial relativity that had revolutionized classical physics. However, here we shall be dealing with electron transport at much lower energy and hence temperature. Perfect andreev reflection due to the klein paradox in a. Second, the connection between the number of space dimensions and the number of spinor components.
A 4space formulation of dirac s equation gives results formally identical to those of the usual klein paradox. Pio neers in quantum mechanics, edited by ranabir dull and asim k ray, wiley eastern limited, 1993. The massless dirac equation in the refrigerator springerlink. Also we would like to have a consistent description of the spin of the electron that in the nonrelativistic theory has to be added by hand. It is shown that a potential well or barrier in the dirac equation can become supercritical and. The dirac equation also predicted a number of counterintuitive effects, of which zitterbewegung a quivering motion of a free dirac particle and the klein paradox are the best known. Note that there is still a klein paradox which is just transmission through a huge potential and that unitarity is preserved.
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