Electromagnetic negative refraction in metamaterials has attracted increasingly great interest, since its first experimental verification in 2001. It potentially leads to the applications superior to conventional devices including compact antennas for mobile stations, imaging beyond the diffraction limit, and high-resolution radars, not to mention the anamolous wave propagation in fundamental optics. Here, we report how metamaterials could be used to simulate the "negative refraction of spin-zero particles interacting with a strong potential barrier", which gives rise to the Klein paradox--a counterintuitive relativistic process. We address the underlying physics of analogous wave propagation behaviours in those two entirely different domains of quantum and classical.
The Klein Gordon equation was the first attempt at unifying special relativity and quantum mechanics. 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. The equation leads to the development of Dirac equation and hence quantum field theory. It shows interesting quantum relativistic phenomena like Klein Paradox and "Zitterbewegung", a rapid vibrating movement of quantum relativistic particles. The simulation of such quantum equations initially motivated Feynman t…
The Dirac equation has been applied to fermions scattering from the downward potential step. The results show some particles do not fall off the edge of the step and reflect. Also, based on de Broglie-Bohm interpretation of quantum mechanics (Bohmian mechanics) and Bohmian trajectories we have resolved the problem. Lastly, a phenomenological study of the Bohmian trajectory of the Klein paradox has been discussed.
We analyse a little known aspect of the Klein paradox. A Klein-Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein-Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier's height is enhanced yet remains fin…
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. 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 well-posed generalization of the method of images in the theory of partial differential equations. Thus, our solution considers a global reflection coefficient obtained from the two contributions, the reflected particles plus…
We figure out the famous Klein's paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle's energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein-Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials …
The "particle in a box" problem is investigated for a relativistic particle obeying the Klein-Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for "shallow" wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wavefunction outside the well vanish…
We investigate the Schrödinger (non-relativistic) and the Dirac (``relativistic") billiards in the universal regime. The study is based on a non-ideal quantum resonant scattering numerical simulation. We show universal results that reveal anomalous behavior on the conductance, on the shot-noise power and on the respective eigenvalues distributions. In particular, we demonstrate the Klein's paradox in the graphene and tunable suppression/amplification transitions on the typical observables of the quantum dots.
Reflection and transmission of electrons scattered by a rectangular potential step in the presence of an external magnetic field parallel to the electron beam is described with the use of the Dirac equation. It is shown that in addition to the known effects present in the so called Klein paradox, the presence of magnetic field gives rise to electron components with reversed spin in the reflected and transmitted beams. The spin-flip scattering processes are caused by the spin-orbit interaction activated by electric field of the potential step and transverse momentum components of electron motio…
We investigate the band structure of ABCA-tetralayer graphene (ABCA-TTLG) subjected to an external potential $V$ applied between top and bottom layers. Using the tight-binding model, including the nearest $t$ and next-nearest-neighbor $t'$ hopping, low-energy model and two-band approximation model we study the band structure variation along the lines $Γ-M-K-Γ$ in the first Brillouin zone, electronic band gap near Dirac point $K$ and transmission properties, respectively. Our results reveal that ABCA-TTLG exhibits markedly different properties as functions of $t'$ and $V$. We show that the hopp…
Quantization Mapping on Dirac Dynamics via Voltage-Driven Charge Density in Monolayer Graphene: A Klein Paradox and Entropy-Ruled Wavevector Mechanics Study
KPR Institute of Engineering and Technology（KPR工程技术学院）
A comparison of the new exact solutions of the relativistic wave equations of a charged particle propagating in a strong laser field in an underdense plasma
