Guang-jiong Ni, Weimin Zhou, Jun Yan
The Klein paradox of Klein-Gordon (KG) equation is discussed to show that KG equation is self-consistent even at one-particle level and the wave function for antiparticle is uniquely determined by the reasonable explanation of Klein paradox. No concept of ``hole'' is needed.
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The original version of Einstein-Podolsky-Rosen (EPR) paradox and the Klein paradox of Klein-Gordon (KG) equation are discussed to show the necessity of existence of antiparticle with its wavefunction being fixed unambiguously. No concept of "hole" is needed.
The original version of Einstein-Podolsky-Rosen (EPR) paradox is discussed to show the completeness of Quantum Mechanics (QM). The unique solution leads to the wave function of antiparticle unambiguously, which implies the essential conformity between QM and Special Relativity (SR).
We study particle interaction with a Dirac step potential. In the standard Klein energy zone, the hypothesis of Klein pair production predicts the existence of free/oscillatory antiparticles. In this paper, we discuss the tunneling energy zone characterized by evanescent wave functions in the classically forbidden region. We ask the question of the nature, particle or antiparticle, of the densities within the classically forbidden region. The answer to this question is relevant to the correct form of the reflection coefficient.
The time-dependent Dirac equation is solved using the three-dimensional Finite Difference-Time Domain (FDTD) method. The dynamics of the electron wave packet in a scalar potential is studied in the arrangements associated with the Klein paradox: potential step barriers and linear potentials. No Klein paradox is observed.
We show analytically that the {\it Zitterbewegung} and Klein Paradox, such well known aspects of the Dirac Equation are not found in the case of Bosons. We use the Kemmer-Duffin-Harish Chandra formalism with $β$ matrices to arrive at our results
While Klein paradox is often encountered in the context of scattering of relativistic particles at a potential barrier, we presently discuss a puzzling situation that arises with the Klein-Gordon equation for bound states. With the usual minimal coupling procedure of introducing the interaction potential, a paradoxical situation arises when the 'hill' becomes a 'well', simulating a bound-state like situation. The phenomenal phenomenon for bound states is contrary to the conventional wisdom of quantum mechanics and is analogous to the well-known Klein paradox, a generic property of relativistic…
We present a resolution of the Klein paradox within the framework of one-particle relativistic quantum mechanics. Not only reflection becomes total but the vacuum remains neutral as well. This is accomplished by replacing the pair production process with virtual negative energy "incidence" within the barrier in a similar manner to what is done with image charges in electrostatic and virtual sources in optics.
It is demonstrated that both transmission and reflection coefficients associated to the Klein paradox at a step barrier are positive and less than unity, so that the particle-antiparticle pair creation mechanism commonly linked to this phenomenon is not necessary. Because graphene is a solid-state testing ground for quantum electrodynamics phenomena involving massless Dirac fermions we suggest that the transport characteristic through a p-n graphene junction can decide between the results obtained in this paper and the common Klein paradox theory, which imply negative transmission and higher-t…
The well known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible ``negative probabilities'' induced by certain potentials in some regimes of energy. The paradox may be resolved employing the notion of electron-positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in Maxwell's equations viewed as the wave equation for photons. In a medium containing ``inverted energy populations'' of excited atoms, e.g. in a LASER medium, one may again compute possible ``negative probabilities''. Th…
The Klein paradox is reassessed by considering the properties of a finite square well or barrier in the Dirac equation. It is shown that spontaneous positron emission occurs for a well if the potential is strong enough. The vacuum charge and lifetime of the well are estimated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Klein tunnelling is a property of relativistic wave equations, not necessarily connected to particle emission. The Coulomb potential is investigated in t…
Solutions of the one dimensional Dirac equation with piece-wise constant potentials are presented using standard methods. These solutions show that the Klein Paradox is non-existent and represents a failure to correctly match solutions across a step potential. Consequences of this exact solution are studied for the step potential and a square barrier. Characteristics of massless Dirac states and the momentum linear band energies for Graphene are shown to have quite different current and momentum properties.
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 well known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible ``negative probabilities'' induced by certain potentials in some regimes of energy. The paradox may be resolved employing the notion of electron-positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in Maxwell's equations viewed as the wave equation for photons. In a medium containing ``inverted energy populations'' of excited atoms, e.g. in a LASER medium, one may again compute possible ``negative probabilities''. Th…
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…
Quantum walks are not only algorithmic tools for quantum computation but also not trivial models which describe various physical processes. The paper compares one-dimensional version of the free particle Dirac equation with discrete time quantum walk (DTQW). We show that the discretized Dirac equation when compared with DTQW results in interesting relations. It is also shown that two relativistic effects associated with the Dirac equation, namely Zitterbewegung (quivering motion) and Klein's paradox, are present in DTQW, which can be implemented within non-relativistic quantum mechanics.
The de Broglie-Bohm pilot wave interpretation of quantum mechanics is shown to provide a consistent explanation for a single relativistic particle (more accurately, a single particle process since pair production is addressed). This is accomplished by incorporating the Feynman-Stueckelberg interpretation of anti-matter. The lack of conserved probability and existence of negative energy solutions is studied through this interpretation. We discuss the resolution of Klein's original paradox through the pilot wave interpretation, and similarly analyze the potential barrier problem. Simulations are…
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 …
We solve the one dimensional Feshbach-Villars equation for spin-1/2 particle subjected to a scalar smooth potential. The eight component wave function is given in terms of the hypergeometric functions and via a limiting procedure, the wave functions of the step potential are deduced. These wave functions are used to test the validity of the boundary conditions deduced from the Feshbach-Villars transformation. The creation of pairs is predicted from the boundary condition of the charge density.
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. The explanation of this effect in terms of electron-positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant current is emitted. This phenomenon is transient whereas the tunnelling first calc…
The purpose of this comment is to clarify two points related to the Dirac equation. First, the Lorentz structure of the potential and its connection with the Klein paradox. Second, the connection between the number of space dimensions and the number of spinor components.
We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external pot…
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…
A few properties of the nonminimal vector interactions in the Duffin-Kemmer-Petiau theory are revised. In particular, it is shown that the space component of the nonminimal vector interaction plays a peremptory role for confining bosons whereas its time component contributes to the leakage. Scattering in a square step potential with proper boundary conditions is used to show that Klein's paradox does not manifest in the case of a nonminimal vector coupling.
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 compute the Bohmian trajectories of the incoming scattering plane waves for Klein's potential step in explicit form. For finite norm incoming scattering solutions we derive their asymptotic space-time localization and we compute some Bohmian trajectories numerically. The paradox, which appears in the traditional treatments of the problem based on the outgoing scattering asymptotics, is absent.
A particle is always not pure. It always contains hiding antiparticle ingredient which is the essence of special relativity. Accordingly, the Klein-Gordon (KG) equation and Dirac equation are restudied and compared with the Relativistic Stationary Schrödinger Equation (RSSE). When an electron is bound in a Hydrogenlike atom with pointlike nucleus having charge number $Z$, the critical value of $Z, Z_c$, equals to 137 in Dirac equation whereas $Z_c=\sqrt{M/μ} (137)$ in RSSE with $M$ and $μ$ being the total mass of atom and the reduced mass of the electron.
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 c…
The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}π=π(n_{m}^{+}-n_{m}^{-})= [δ_{m}(M)+β_{1}]-[δ_{m}(-M)+β_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $δ_{m}$ is named phase shifts. The constants $β_{1}$ and $β_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.
We solve the one-dimensional time-independent Klein-Gordon equation in presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker $M_{κ,μ}(x)$ function, and the antiparticle bound state is discussed in terms of potential parameters.
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 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.
The essence of special relativity is hiding in the equal existence of particle and antiparticle, which can be expressed by two discrete symmetries within one inertial frame --- the invariance under the (newly defined) space-time inversion (${\bf x}\to -{\bf x},t\to -t$), or equivalently, the invariance under a mass inversion ($m\to -m$). The problems discussed are: the evolution of the $CPT$ invariance into a basic postulate, an unique solution to the original puzzle in Einstein-Podolsky-Rosen paradox, the reduced Dirac equation for hydrogenlike atoms, and the negative mass paradox leading to …
Positive Conserved Quantities in the Klein-Gordon Equation
通过将Klein-Gordon方程嵌入一对时间一阶耦合方程，发现两个正守恒积分，解决了其概率解释的历史难题，并给出了Schrödinger方程的相对论形式，为暗物质提供简单解释。
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…
We follow and modify the Feshbach-Villars formalism by separating the Klein-Gordon equation into two coupled time-dependent Schroedinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we fi…
Quantization Mapping on Dirac Dynamics via Voltage-Driven Charge Density in Monolayer Graphene: A Klein Paradox and Entropy-Ruled Wavevector Mechanics Study
本研究通过电压驱动电荷密度的经验关系，利用微分熵控波矢力学对狄拉克材料进行能量量子化映射，揭示了克莱因悖论下的电子动力学，并发现量子态跃迁遵循N(k)=N(U)^3关系。
Perfect transmission of a Dirac particle in one-dimension double square barrier
研究揭示了一维双方形屏障中狄拉克粒子在克莱因区和高于势垒区的连续完美传输现象，表明完美克莱因隧穿与共振传输存在联系，拓展了对克莱因悖论的理解。
Hydrodynamic Analog of the Klein Paradox: Vacuum Instability and Pair Production in a Linear Elastic Medium
通过类比引力和凝聚态物理，研究高势垒下相对论粒子的异常散射现象，揭示真空不稳定性与对称产生机制，复现Hansen和Ravndal的透射系数并恢复Swinger极限。
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitude, Born rule, probability density current, commutation relations, momentum operator, uncertainty relations, rules for including the scalar and vector potentials and existence of antiparticles can be derived from the definition of the mean values of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schroedinger equation and…
Goos-H{ä}nchen Shift for Relativistic Particles Based on Dirac's Equation
本文基于狄拉克方程研究相对论粒子在三维无限势垒上的戈斯-汉肯舍移，发现其可为负值，与非相对论情况不同。
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, probability density current, momentum operator, rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrodinger equation …
We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein parad…
The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because …
The Klein-Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, $V_{v}=V_{s} + \mathrm{const.}$ These isospectral problems are solved in a case of squared trigonometric potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfuntions are discussed in some detail. It is revealed that a spin-0 particle is better localized than a spin-1/2 particle when they have the same mass and are subject to the same potentials.
The Klein-Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, $V_{v}+V_{s}= \mathrm{constant}$. These intrinsically relativistic and isospectral problems are solved in a case of squared hyperbolic potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfuntions are discussed in some detail and the effective Compton wavelength is revealed to be an important physical quantity. It is revealed that a boson is better localized than a fermion when they have the same mass and are subje…
Klein-Gordon oscillators and Bergman spaces
本文研究Minkowski空间中相对论性Klein-Gordon振子的经典与量子动力学，证明其协变相空间为Kähler-Einstein流形$Z_6$，并利用加权Bergman空间给出量子模型的完整解，表明该模型具有Lorentz协变性、幺正性且无非物理态。
Non-Hermitian $\mathcal{PT}$ symmetric Klein-Gordon fields in Lorentz-violating wormholes spacetime background
研究在(3+1)维洛伦兹破坏可穿越虫洞背景下，自旋0的$\mathcal{PT}$对称克莱因-戈登振子场，通过非最小耦合非厄米标量场得到全局正则有效的$\mathcal{PT}$对称振子，径向方程化为合流Heun系统，在多项式截断条件下得到实离散能谱。
Supersymmetric Klein-Gordon and Dirac oscillators
该研究证明超对称相对论振子的协变相空间为$Z_6$的奇切丛，量子化后得到的Dirac振子方程解由两类旋量空间直和构成，且该模型严格可解、洛伦兹协变并具有幺正性。
We derive an explicit manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. This expression involves a one-parameter family of conserved current densities J_a^μ, with a\in(-1,1), that are analogous to the chiral current density for spin half fields. The conservation of J_a^μis related to a global gauge symmetry of the Klein-Gordon fields whose gauge group is U(1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symm…
Klein-Gordon Oscillator with Linear-Fractional Deformed Mass-Shell Invariants in Doubly Special Relativity
研究Klein-Gordon振子在双特殊相对论中的变形特性，揭示不同Casimir变形对能量谱和波函数的影响，提出PT对称和伪厄米特形式。
Schrödinger and Klein-Gordon theories of black holes from the quantization of the Oppenheimer and Snyder gravitational collapse
通过将Oppenheimer-Snyder坍缩量子化，推导出黑洞的薛定谔方程，类比氢原子揭示黑洞为无视界无奇点的量子壳层，并给出克莱因-戈登修正。
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