Klein paradox for the klein gordon equation winter, rolf g. We develop the scattering theory for the charged klein gordon equation on, when the electrostatic potential has different asymptotics as. Accordingly, the kleingordon kg equation and dirac equation are restudied and compared with the relativistic stationary schr. Our work has implications to the analog quantum simulations of many exotic phenomena in quan. If the well is wide enough, a seemingly constant current is emitted. Klein gordon equation relativistic quantum mechanics duration. We will come back to the question what it actually describes later on. Isnt that for the schrodinger equation, rather than kleingordon. If you like you can view the klein gordon equation as a \toy model for the maxwell equations. Later, the equation was revived and is now known as kleingordon equation. The general solutions found, could be used for future explorations on the study for other speci. Jul 20, 2005 the results are compared with those obtained from similar computations employing the dirac equation. The klein paradox is reassessed by considering the properties of a finite square well or barrier in the dirac equation. Having shown elsewhere 1 that the klein paradox for the kleingordon kg equation of spinzero particle manifests exactly the same kind of wave propagation and negative refraction phenomena, which also exist in the scattering of tm transversemagnetic polarized.
Both the klein gordon and the dirac equation are no 1particle wave equations, but relativistic. Comparison among kleingordon equation, dirac equation and. Chiral tunnelling and the klein paradox in graphene. Covariant notation for the kleingordon equation if an equation is consistent with the principles of special relativity, it should be possible to write it in covariant form, that is, in a form that is the same in all lorentz frames. Pdf whenever we consider any relativistic quantum wave equation we are confronted with the klein paradox, which asserts that incident particles will. Jul 20, 2005 the behavior of a beam of spinzero particles incident on a region of large potential increase is examined. Therefore, as long as energy and momentum follows the einsteins relation eq. The signs in the formula for w1are taken arbitrarily. Boundary value problems for the kleingordon equation.
Virtual beams and the klein paradox for the kleingordon equation. In other words, the kleingordon equation, if useful at all, does not describe the probability wave, which the schrodinger equation does, but describes something else. Klein tunneling in singlelayer graphene transmission probabilities through a 100nmwide barrier as a function of the angle of incidence for single layer grahene. The klein paradox has not been observed yet, but a simulation was performed by using trapped ions 7,8. The klein paradox of klein gordon kg equation is discussed to show that kg equation is selfconsistent even at oneparticle level and the wave function for. We consider the scattering of a klein gordon particle of energy and momentum from a electrostatic stepfunction potential as shown in figure 4. Pdf virtual beams and the klein paradox for the klein. It shows interesting quantum relativistic phenomena like klein paradox and zitterbewegung, a rapid vibrating movement of quantum relativistic. The results are compared with those obtained from similar computations employing the dirac equation. The same effect was also obtained from the solutions of the kleingordon equation. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. While initially discarded this equation of many fathers can be used in understanding spinless particles that consequently led to the discovery of pions and other subatomic particles. After this, the equation was rediscovered by pauli, klein, gordon and fock, among others pauli called it the equation with many fathers, see 4, and today.
The plane waves 10 are basic solutions and the field 9 is constructed by. It is secondorder in space and time and manifestly lorentzcovariant. However, klein s result showed that if the potential is of the order of the electron mass. 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. So the klein gordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study.
In the corresponding relativistic situation, we employ the timeindependent kleingordon equation. This behavior of the relativistic particles is named as the klein paradox in the literature 2,3. Pdf on the kleins paradox in the presence of a scalar potential. When an electron is bound in a hydrogenlikeatom with pointlikenucleus havingcharge number z, thecritical value of z, zc, equals to 7 in dirac equation whereas. Virtual beams and the klein paradox for the kleingordon. The method of multiple scales for nonlinear kleingordon. Both the kleingordon and the dirac equation are no 1particle waveequations, but relativistic. It is a quantized version of the relativistic energymomentum relation. In both equations the laplacian appears on the right hand side.
The behavior of a beam of spinzero particles incident on a region of large potential increase is examined. But there is a similar schwingers proper time method that works for kleingordon. Download pdf 110 kb abstract the klein paradox of kleingordon kg equation is discussed to show that kg equation is selfconsistent even at oneparticle level and the wave function for antiparticle is uniquely determined by the reasonable explanation of klein paradox. In fact it corresponds to a situation that cannot be formulated consistently within the framework of a field theory with given potentials. The klein paradox of klein gordon kg equation is discussed to show that kg equation is selfconsistent even at oneparticle level and the wave function for antiparticle is uniquely determined by the reasonable explanation of klein paradox. On the other hand, the klein paradox, as of this writing, is a hot topic in the area of graphene physics. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation.
Relativistic quantum mechanics janos polonyi university of strasbourg dated. So, none of the objections that led to discarding the klein equation held up with time. Negative refraction gives rise to the klein paradox durdu o. Following recent results on the dirac equation, we propose a solution to this paradox for the kleingordon case by introducing virtual beams in a natural wellposed. In the nonrelativistic speeds, such a behavior of the quantum particles does not exist.
Optoelectronic simulation of the klein paradox based on. Negative refraction gives rise to the klein paradox. If you like you can view the kleingordon equation as a \toy model for the maxwell equations. The relativisticallyinvariant quantum equation describing spinless scalar or pseudoscalar particles, for example, and mesons. Klein gordon eld is characterized by \particles with rest mass mand no other structure e. Klein tunnelling and the klein paradox international.
Klein paradox for the charged kleingordon equation. The klein paradox role of chirality klein tunneling in singlelayer graphene klein tunneling and conductivity klein tunneling in bilayer graphene. The solution just like in the case with the wave function in the schrodinger equation just comes out to be some scalar function of x, y, z, and t. Solving the klein gordon equation is quite similar to solving schrodingers equation. As we can see now, the apparent problems were manifestations of the new physics.
How to derive general solution to the klein gordon equation. In the nonrelativistic speeds, such a behavior of the quantum. So the kleingordon eld is physically and mathematically, too the simplest of the relativistic elds that one can study. This comparison yields an instructive illustration of the difference between particles and antiparticles in spin zero and spin onehalf singleparticle theory. Kleingordon equation in schrodinger form 3 which is the usual relativistic formula for the energy of a particle. Fok as a wave equation under the conditions of cyclicity in the fifth coordinate and was shortly afterwards deduced by several authors for example, w. Pdf the klein paradox is reassessed by considering the properties of a finite square. Heuristic derivation of the klein gordon equation 11 b. Introduction to the klein paradoxes seventy years ago klein 1 published a paper where he calculated the re. The equation leads to the development of dirac equation and hence quantum field theory.
Holstein department of physics and astronomy, university of massachusetts, amherst, massachusetts 01003 received 24 september 1997. Its explanation in terms of electronpositron production is reassessed. Whenever we consider any relativistic quantum wave equation we are confronted with the klein paradox, which asserts that incident particles will suffer a surplus. Because of this reason, the kleingordon equation was abandoned for a while. Klein paradox pdf solid state engineering quantum mechanics. The klein paradox of kleingordon kg equation is discussed to show that kg equation is selfconsistent even at oneparticle level and the wave function for. This problem is an archetype problem in nonrelativistic quantum mechanics. We assume that the incoming boson has a positive charge and its initial quantum. Its solutions include a quantum scalar or pseudoscalar field, a field whose. The barrier would be becoming more transparent if the potential approached to infinity.
This comparison yields an instructive illustration of the. In order to numerically verify this approximation we want to present numerical methods in this chapter which enable us to obtain a good numerical solution of both the grosspitaevskii equation 3. 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. The kleingordon equation is a linear homogeneous secondorder partial differential equation with constant coefficients. The kleingordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field. Ruijsenaars instituut lorentz, university of leiden, leiden, netherlands received december 19, 1975 in this paper we give a mathematically rigorous analysis of the klein paradox in the manyparticle formulation of. 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. 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. Klein tunnelling is a property of relativistic wave equations, not necessarily connected with particle emission. Gordon equation like the dirac equation has negative energy solutions. The klein gordon equation was the first attempt at unifying special relativity and quantum mechanics. Klein paradox in the phase space without decoherence duration. We explain the underlying physics of the similar wave propagation behaviors in the kleingordon and the maxwellian pictures.
Hence, they give rise to the klein paradoxa counterintuitive phenomenon in relativistic quantum mechanics. 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. The klein paradox as a many particle problem pdf free. Kleins paradox is shown to be resolved by a careful consideration of feynmans picture of antiparticles as negative energy solutions traveling backward in. The kleingordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly nonvanishing mass m m on some possibly curved spacetime lorentzian manifold. Quantum simulation of klein gordon equation and observation of. Klein paradox for the kleingordon equation nasaads.
The vacuum charge and lifetime of the well are estimated. The same effect was also obtained from the solutions of the klein gordon equation. Covariant notation for the klein gordon equation if an equation is consistent with the principles of special relativity, it should be possible to write it in covariant form, that is, in a form that is the same in all lorentz frames. The absence of a klein paradox in a field theoretical treatment is demonstrated. Jun 19, 2019 shortly afterwards, klein solved a simple potential step problem for the dirac equation and encountered an apparent paradox. Kleingordon eld is characterized by \particles with rest mass mand no other structure e. Kleingordon equation a relativistic that is satisfying the requirements of the theory of relativity quantum. The vacuum charge and lifetime of the wellbarrier are calculated. Kleingordon equation relativistic quantum mechanics. The kleingordon equation is applied in the description of mesons and corresponding fields. Apr 12, 2016 how to derive general solution to the klein gordon equation. Underlying physics the same for kleingordon equation, but paradox not so visible. Relativistic quantum mechanics kleingordon equation dirac.
The klein paradox is more than an example demonstrating the problems of negative energy states of the single particle dirac equation, as it was meant to be by klein. These equations appear in the study of relativistic and quantum physics. It is a quantum relativistic wave equation, used in the description of particles with spin 0. In 1928, dirac proposed a wave equation to describe relativistic electrons1. Pdf in this paper, we have studied the kleins paradox in the presence of. Whenever we consider any relativistic quantum wave equation we are confronted with the klein paradox, which asserts that incident particles will suffer a surplus of reflection when dispersed by a discontinuous potential. That is where cryptohermitian approach can be applied. Both the kleingordon and the dirac equation are no. The method of multiple scales for nonlinear kleingordon and. These phenomena are transient whereas the tunnelling first calculated by klein is timeindependent. In this case, the conserved energy is not positive definite klein paradox. 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. Shortly afterwards, klein solved a simple potential step problem for the dirac equation and encountered an.
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