One-dimensional Dirac Equation Research Articles

Exact solution of the 1D Dirac equation for a pseudoscalar interaction potential with the inverse-square-root variation law

We present the exact solution of the one-dimensional stationary Dirac equation for the pseudoscalar interaction potential, which consists of a constant and a term that varies in accordance with the inverse-square-root law. The general solution of the problem is written in terms of irreducible linear combinations of two Kummer confluent hypergeometric functions and two Hermite functions with non-integer indices. Depending on the value of the indicated constant, the effective potential for the Schrödinger-type equation to which the problem is reduced can form a barrier or well. This well can support an infinite number of bound states. We derive the exact equation for the energy spectrum and construct a rather accurate approximation for the energies of bound states. The Maslov index involved turns out to be non-trivial; it depends on the parameters of the potential.

Solutions of one-dimensional Dirac equation associated with exceptional orthogonal polynomials and the parametric symmetry

We consider one-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Pöschl–Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Pöschl–Teller cases, a new family of Dirac scalar potentials is generated using the idea of parametric symmetry and their solutions are obtained in terms of conventional as well as exceptional orthogonal polynomials.

Energy Power Series Analysis of the Bound States of the One-dimensional Dirac Equation

In this work we consider the one-dimensional Dirac equation including an electrostatic potential with compact support, and focus on the regime of bound states. We obtain exact expressions for both the characteristic function and the eigenfunctions in L 2 (ℝ, ℂ2), given in the form of power series of the energy parameter. This approach is meant for arbitrary bounded potentials, so that a square potential is a special case of the theory here presented. We derive an efficient numerical method for the calculation of approximate eigen-energies of the bound states. Finally, we investigate the physical sense of the eigen-energies that are forbidden in the non-relativistic regime in terms of the Klein tunneling.

Dirac Equation with Morse Potential under the Influence of Position-Dependent Mass and Local Fermi Velocity

We solve the one-dimensional Dirac equation by taking into account the possibility of position-dependence in the mass function. We also take the Fermi velocity to act as a local variable and examine the combined effects of the two on the solvability of the Dirac equation with respect to the Morse potential. Our results for the wave functions and the energy levels corresponding to such an extended scheme are furnished in closed forms.

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers (N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

Exact solution of time-independent one-dimensional Dirac equation with position-dependent mass and vector potential of hyperbolic type by generalized SUSY QM approach

Exact solution of time-independent one-dimensional Dirac equation with position-dependent mass and vector potential of hyperbolic type by generalized SUSY QM approach

Relativistic tunneling time under the generalized uncertainty principle

The one-dimensional Dirac equation is studied for square-well potential in the presence of generalized uncertainty principle (GUP). Based on the exact solution, the general and explicit relation between the phase time and the dwell time is given and quantified. The phase time, dwell time and self-interference delay are calculated as well and it is shown that the GUP has effects on the tunneling time. Moreover, the effect of GUP on the tunneling time showed a slight change with the change of mass, and goes to zero with the increase of kinetic energy.

DFT-NEGF Study of Electron Transfer Property in Graphene Nanoribbon Double Barrier System

Current-voltage (I – V) characteristics in armchair graphene nanoribbon (AGNR) double barrier systems (DBSs) are investigated by the density functional theory (DFT) and the non-equilibrium Green’s function (NEGF) methods. The hexagonal boron nitride (h-BN) which has an analogous structure with a graphene shows wide bandgap and becomes an electrical barrier in DBSs. The heterojunction structure of AGNR and two h-BNs with one six-membered ring width has properties as a resonant tunnelling device. Due to the resonance, sharp peaks are formed in the transmission function around the Fermi energy EF . Our previous studies have explained the origins of some peaks by solving the one-dimensional Dirac equation for massless particles. Here, by using the two-dimensional Dirac equation, we show that the remaining sharp peaks depend on the transverse nodal pattern of the wavefunction as well as the anisotropy of the Fermi velocity. Moreover, in the I – V characteristics, a flat part is formed without the negative differential resistance appearing. By partial density of state (PDOS) analysis, we reveal that the flat part in the current is caused by the fact that the peak positions in the resonance region are stable against the bias voltage.

One-dimensional Variable-mass Dirac Equation and Spinor slow Llght

One-dimensional Variable-mass Dirac Equation and Spinor slow Llght

Position-dependent mass Dirac equation and local Fermi velocity

We present a new approach to study the one-dimensional Dirac equation in the background of a position-dependent mass m. Taking the Fermi velocity v f to be a local variable, we explore the resulting structure of the coupled equations and arrive at an interesting constraint of m turning out to be the inverse square of v f. We address several solvable systems that include the free particle, shifted harmonic oscillator, Coulomb and nonpolynomial potentials. In particular, in the supersymmetric quantum mechanics context, the upper partner of the effective potential yields a new form for an inverse quadratic functional choice of the Fermi velocity.

Density and temperature dependence of the cross sections after excitation of Ar XVIII by electron impact

We propose a novel distorted wave approach within the fundamental framework of relativity theory to calculate the dynamics of magnetic sublevels excitations of highly charged ions by electron impact within a quantum plasma. These cross sections can be used to predict the alignment parameter and subsequent angular pattern of the photoemission. Two kinds of modified Shukla and Eliasson [Phys. Rev. Lett. 108, 165007 (2012)] models, based on the quantum hydrodynamic theory, are adopted to determine the effects of screened Coulomb interaction among charged particles. The bound and continuous state (radial) wave functions are addressed through the one-dimensional Dirac equations, where the numerical convergence is checked. As an application, the 1s1∕2 → 2p3∕2 transition in Ar XVIII is used as a probe for the assessment of the quantum plasma shielding effects and the exchange-correlation effects on the energies, excitation cross sections, alignment parameters, and angular patterns of x-ray emission vs electron densities, from 1019 to 1024 cm−3, and compared to existing data. Our results highlight that the inclusion of the effect of plasma density and temperature not only leads to a significant reduction in the magnetic sublevels cross section, but also produces a change in the alignment parameter and angular pattern of the photoemission. The exchange-correlation effect is found to be crucial to the atomic structure, yet it weakens in the scattering process. In addition to insights into collision dynamics, this study opens the door to a further understanding of the polarization and angular distribution properties of x-ray radiation under extreme conditions. The present study is of interest for the warm density matter physics and high-energy density physics applications. The obtained atomic structure, sublevel cross section, and angular pattern of the photoemission are essential fundamental properties for advanced diagnostics and equation of state calculations in planet interior, inertial confinement fusion, and Z pulsed power facility.

Topological guided-mode resonances at non-Hermitian nanophotonic interfaces

Abstract The topological properties of photonic microstructures are of great interest because of their experimental feasibility for fundamental study and potential applications. Here, we show that robust guided-mode-resonance states exist in photonic domain-wall structures whenever the complex photonic band structures involve certain topological correlations in general. Using the non-Hermitian photonic analogy of the one-dimensional Dirac equation, we derive essential conditions for photonic Jackiw-Rebbi-state resonances taking advantage of unique spatial confinement and spot-like spectral features which are remarkably robust against random parametric errors. Therefore, the proposed resonance configuration potentially provides a powerful method to create compact and stable photonic resonators for various applications in practice.

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

The thermal properties of the one-dimensional boson particles in Rindler spacetime

In this paper we study the one-dimension Klein-Gordon (KG) equation in the Rindler spacetime. The solutions of the wave equation in an accelerated reference frame are obtained. As a result, (i) we derive a compact expression for the energy spectrum associated in an accelerated reference, and (ii) we show that the non-inertial effect of the accelerated reference frame mimics an external potential in the Klein-Gordon equation and, moreover, allows the formation of bound states. In addition, the thermal properties of the Klein-Gordon from the partition function, have been investigated, and the effect of the accelerated reference frame parameter a on these properties has been tested. This study is extended to the case of the one-dimensional Dirac equation where the spectrum of energy is well determined and has an exact form. As a result, we will see that the behavior of the thermal quantities of the fermion particles is similar to the case of boson particles.

Global bifurcation in nonlinear Dirac problems with spectral parameter in the boundary condition

In this paper we consider nonlinear eigenvalue problems for a one-dimensional Dirac equation with spectral parameter in the boundary condition. We investigate local and global bifurcations of nontrivial solutions to these problems. The existence of unbounded continua of nontrivial solutions bifurcating from points and intervals of the line of trivial solutions is shown.

I-V Characteristics of In-Plate Graphene Nanoribbon/h-BN Heterojunctions and Resonant Tunneling

We present here the first principle calculations of the electrical properties of in-plate heterojunctions of armchair graphene nanoribbon/h-BN(AGNR/h-BN)s. The calculations are carried out using SIESTA package, which is comprised of numerical codes of the density functional theory(DFT) and the non-equilibrium Green’s function(NEGF). Especially, adopting the conductive (3n-1)-family of AGNR((3n-1)-AGNR) makes the lead parts on both side of the model metallic. Two transverse arrays of h-BN, which is a wide-gap semiconductor, are embedded in the middle of (3n-1)-AGNR and act as a double barrier system.The quantum double barrier tunneling is found in the transmission functions(TF) and I-V characteristics of 8, 11, 14-AGNR/h-BN. The TF shows significantly spiky peaks in the neighborhood of the Fermi energy, and consequently, it results in step-sise I-V characteristics. Simple one-dimensional Dirac equation model for the double barrier system is also proposed to analyze numerical results. Our model reproduces most of the peaks of the transmission functions nearby the Fermi energy, as a result of quantum tunneling.

Darboux transformations of the one-dimensional stationary Dirac equation with linear potential and its new solutions

We study the Darboux transformation method for one-dimensional stationary Dirac equation with linear potential. We obtain new exactly solvable potentials associated to new solutions of the first Hamiltonian and a series of new Hamiltonians are obtained. Finally, as a particular case, we introduce another form of the Darboux operator that is applicable to generating new potentials.

Relativistic properties and invariants of the Du Fort–Frankel scheme for the one-dimensional Schrödinger equation

The Du Fort–Frankel scheme for the one-dimensional Schrödinger equation is shown to be equivalent, under a time-dependent unitary transformation, to the Ablowitz–Kruskal–Ladik scheme for the Klein–Gordon equation. The Schrödinger equation describes a non-relativistic quantum particle, while the Klein–Gordon equation describes a relativistic particle. The conditional convergence of the Du Fort–Frankel scheme to solutions of the Schrödinger equation arises because solutions of the Klein–Gordon equation only approximate solutions of the Schrödinger equation in the non-relativistic limit. The time-dependent unitary transformation is the discrete analog of the transformation that arises from seeking a non-relativistic limit using the interaction picture of quantum mechanics to decompose the Klein–Gordon Hamiltonian into the relativistic rest energy and a remainder. The Ablowitz–Kruskal–Ladik scheme is in turn decomposed into a quantum lattice gas automaton for the one-dimensional Dirac equation, which is also the one-dimensional discrete time quantum walk. This relativistic interpretation clarifies the origin of the known discrete invariant of the Du Fort–Frankel scheme as expressing conservation of probability for the 2-component wavefunction in the one-dimensional Dirac equation under discrete unitary evolution. It also leads to a second invariant, the matrix element of the evolution operator, whose imaginary part gives a discrete approximation to the expectation of the non-relativistic Schrödinger Hamiltonian.

Optical simulation of the free Dirac equation

We present a classical optics simulation of the one-dimensional Dirac equation for a free particle. Positive and negative energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle. Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator - known as zitterbewegung (ZB). We are also able to measure the so called mean-position operator which is a single-particle observable and presents no oscillations. Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.

Superdiffusive transport in one-dimensional disordered Dirac model

We have considered discrete one-dimensional Dirac equation in the framework of quantum dimer model with a Bernoulli distribution. It has been shown, that superdiffusive transport regime is realized at several combinations of particle mass and amplitude of potential. For the case of non-zero mass it has been shown, that the only one point exists, where Lyapunov exponent is equal to zero. In the framework of dimer model with “random masses” it has been established, that localization is absent.