Dirac Equation Research Articles

Nonrelativistic limits of the Klein-Gordon and Dirac equations in the Amelino-Camelia DSR

Nonrelativistic limits of the Klein-Gordon and Dirac equations in the Amelino-Camelia DSR

Controllable quantum scars induced by spin–orbit couplings in quantum dots

Spin–orbit couplings (SOCs), originating from the relativistic corrections in the Dirac equation, offer nonlinearity in the classical limit and are capable of driving chaotic dynamics. In a nanoscale quantum dot confined by a two-dimensional parabolic potential with SOCs, various quantum scar states emerge quasi-periodically in the eigenstates of the system, when the ratio of confinement energies in the two directions is nearly commensurable. The scars, displaying both quantum interference and classical trajectory features on the electron density, due to relativistic effects, serve as a bridge between the classical and quantum behaviors of the system. When the strengths of Rashba and Dresselhaus SOCs are identical, the chaos in the classical limit is eliminated as the classical Hamilton’s equations become linear, leading to the disappearance of all quantum scar states. Importantly, the quantum scars induced by SOCs are robust against small perturbations of system parameters. With precise control achievable through external gating, the quantum scar induced by Rashba SOC is fully controllable and detectable.

Investigation of bound state energy spectra for fermionic particles in the presence of ultra generalized exponential hyperbolic potential model

By considering the ultra generalized exponential hyperbolic potential, which covers many potential types, the solutions of the Dirac equation with spin/pseudo-spin symmetric limits are achieved. In both approaches, the relation giving the bound state energy eigenvalues is obtained in a closed form. By using these relations, the energy values are calculated numerically for both symmetry cases via the software program. In addition, it has been identified how the bound state energy eigenvalues depend on the parameters. Besides, the energy equations for the Schrödinger and Klein–Gordon particles in the limit states are derived.

VAMPyR-A high-level Python library for mathematical operations in a multiwavelet representation.

Wavelets and multiwavelets have lately been adopted in quantum chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few), they provide a framework that narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for multiwavelet calculations (algebra and integral and differential operator application) and exposes the required functionality to write a simple Python code to solve, among others, the Hartree-Fock equations, the generalized Poisson equation, the Dirac equation, and the time-dependent Schrödinger equation up to any predefined precision. In this study, we will outline the main features of multiresolution analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms.

Approximate solutions of the spin and pseudospin symmetries under coshine Yukawa tensor interaction

The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model.

Massive Dirac particles based on gapped graphene with Rosen-Morse potential in a uniform magnetic field

Abstract In this paper, we explore the gapped graphene structure in the two-dimensional plane in the presence of the Rosen-Morse potential and the external uniform magnetic field. In order to describe the corresponding structure, we consider the propagation of electrons in graphene as relativistic fermion quasi-particles, and analyze it by the wave functions of two-component spinors with pseudo-spin symmetry using the Dirac equation. Next, to solve and analyze the Dirac equation, we obtain the eigenvalues and eigenvectors using the Legendre differential equation. After that, we obtain the bounded states of energy depending on the coefficients of Rosen-Morse and magnetic potentials in terms of quantum numbers of principal n and spin-orbit k. Then, the values of the energy spectrum for the ground state and the first excited state are calculated, and the wave functions and the corresponding probabilities are plotted in terms of coordinates r. In what follows, we explore the band structure of gapped graphene by the modified dispersion relation and write it in terms of the two-dimensional wave vectors K x and K y . Finally, the energy bands are plotted in terms of the wave vectors K x and K y with and without the magnetic term.

Boosting energy levels in graphene magnetic quantum dots through magnetic flux and inhomogeneous gap

We study the effects of a magnetic flux and an inhomogeneous gap on the energy spectrum of graphene magnetic quantum dots (GMQDs). By considering the Dirac equation in the infinite mass framework, we can analytically obtain eigenspinor expressions. By applying boundary conditions, we obtain an energy spectrum equation in terms of system parameters such as radius, magnetic field, energy, flux, and gap. In the infinite limit, we recover Landau levels for graphene in a magnetic field. We show that the energy spectrum increases significantly in the presence of flux and a gap inside the GMQDs, which prolongs the lifetime of the trapped electron states. We show that higher flux also produces new Landau levels of negative angular momentum. Meanwhile, we find that the gap increases the separation between the electron and hole energy bands. As shown in the radial probability analysis, flux and gap emerge as influential factors in controlling electron mobility, affecting confinement, and prolonging the presence of quasi-bound states.

Non-inertial interpretation of the Dirac oscillator

Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.

Real and damped modes for an interacting fermion–antifermion pair: exciton in monolayer medium

In this manuscript, we investigate the interaction between a fermion–antifermion pair in (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensions, governed by a Coulomb-type inter-particle potential. We explore analytical solutions for a fully covariant two-body Dirac equation derived from quantum electrodynamics, focusing on a spinless composite system. We present a non-perturbative second-order wave equation and derive a solution using the generators of the sℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\ell _{2}$$\\end{document} algebra. Subsequently, we determine the relativistic frequency modes of the system and extend our findings to electron–hole pairs in a monolayer medium in the vicinity of substrate. We analyze the impact of the effective dielectric constant (ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon$$\\end{document}) of the surrounding medium on the real and damped modes of an exciton. Our results indicate that the ground state binding energy of the system can vary from the order of eV to a few meV, depending on changes in ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon$$\\end{document}. Additionally, we observe that the decay time for the ground state of such an exciton is on the order of ∼10-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 10^{-12}$$\\end{document} s. These findings suggest that the evolution of such a pair can be controlled by adjusting the effective dielectric constant of the surrounding medium.

On n-Dimensional Maxwell and Dirac Equations in Curved Space-Time and Its Applications in SO(P,Q) Group Theoretic Image Processing

Maxwell equations in p time and q spatial dimensions are formulated. Properties of the Green’s function for the associated (p,q)-wave operator are derived based on contour integration. The notions of electric and magnetic fields in (p,q)-dimensions is explored by analogy with four dimensional physics and the problem of far field computation of the radiation field generated by charges and currents in \((1,n-1)\) space-time is analyzed. SO(p,q) invariance properties of the Maxwell equations are deduced and used to formulate SO(p,q)-group theoretic image processing problems in (p,q)-dimensions. Dirac’s equation in (p,q)-dimensional space-time is derived using the Clifford algebra of the Dirac gamma matrices. SO(p,q)-invariance of the Dirac equation based on the spinor representation of \(SO(p,q)\) is mentioned. The Maxwell’s equations in (p,q)-dimensional curved space-time is analyzed using perturbation theory in the context of maximally symmetric spaces which play a fundamental role in \((p,q)\)-dimensional cosmological models for homogeneous and isotropic spaces. The Einstein-Maxwell equations for (p,q)-dimensional gravity and electromagnetism is studied and used to derive the equations of motion of point charges carrying mass moving under mutual gravitational and electromagnetic interactions in general relativity. Finally, Dirac’s equation in (p,q)-dimensional curved space-time interacting with the (p,q)-dimensional electromagnetic field is looked at. \(U(1)\)-Gauge, local SO(p,q) Lorentz and diffeomorphism invariance of this equation is analyzed. Local \(SO(p,q)\) invariance of the curved space-time Dirac equation is deduced based on transformation properties of the Dirac matrices under the spinor representation. The appendix presents some applications of group representation theoretic statistical image processing on a manifold to the situation when the image field is described by the six component electromagnetic field tensor on which the Lorentz group acts.

Neutrino oscillations in the Non-Kerr black hole with quantum phenomenon

Abstract In this paper, we have investigated the mathematical components of the Dirac equation in curved space-time and how it can be applied to the analysis of neutrino oscillations. More specifically, we have developed a method for calculating the phase shift in flavor neutrino oscillations by utilizing a Taylor series expansion of the action, taking into account $\Delta m^4$ orders. In addition, we have used this method to assess how the phase difference in neutrino mass eigenstates changes due to the gravitational field described by the Johannsen spacetime.

Effect of Extreme Variations of Fundamental Constants on the Structure of Atoms and Molecules.

The fundamental constants (FCs) of physics are promoted to dynamic quantities in modern theories. Heretofore, astrophysical observations, atomic clock experiments, and the Oklo natural nuclear reactor phenomenon all have pointed to small variations of FCs happening on a cosmologically long time scale. In this paper, we investigate the novel regime of extreme but transient variations of FCs. We focus on the speed of light (c) and show that its variation can dramatically change the electronic structure and chemistry of atoms and molecules. These changes are induced by increased relativistic effects when c is reduced from its nominal value. To model these changes, we solve the fully relativistic Dirac equation at different values of c. We show that at extreme variations of c, the periodic table is truncated, the nominal ground states of atoms can change, water fails to serve as a universal solvent, and the ammonia molecule becomes planar.

Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equations

Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equations

Boundary Value Problems for the Perturbed Dirac Equation

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ.

Foldy-Wouthuysen Transformation of Dirac Equation in the Context of Magueijo-Smolin Model of the Doubly Special Relativity

Foldy-Wouthuysen Transformation of Dirac Equation in the Context of Magueijo-Smolin Model of the Doubly Special Relativity

Bound states of relativistic spinless particles in a mix of circularly symmetric vector and scalar harmonic oscillators

We study the dynamics of relativistic spinless particles moving in a plane when there is circular symmetry. The general formalism for solving the Klein–Gordon equation in cylindrical coordinates for such systems is presented, as well as the conserved observables and the corresponding quantum numbers. We look for bound solutions of the corresponding Klein–Gordon equation when one has vector and scalar circularly symmetric harmonic oscillator potentials. Both positive and negative bound solutions are considered when there is either equal vector and scalar potentials or symmetric vector and scalar potentials, and it is shown how both cases are related through charge conjugation. We compute the non-relativistic limit for those cases, and show that for symmetric scalar and vector potentials the limit does not exist in the first order of an harmonic oscillator frequency, recovering a known result from the Dirac equation with the same kind of potentials.

Quantum transport of massless Dirac fermions through wormhole-shaped curved graphene in presence of constant axial magnetic flux

In this work, we have studied the spin-dependent quantum transport of charged fermion on (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional spacetime, whose spatial part is described by a wormhole-type geometry in the presence of constant axial magnetic flux. Choosing the solutions of the Dirac equation associated with real energy and momentum, we explored the spin-dependent transmission probabilities and giant magnetoresistance (GMR) through a single layer of wormhole graphene with an external magnetic field, using the transition matrix (T-Matrix) approach. The spin-up and spin-down components within the A and B sublattices of graphene in the matrix of 4×1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 1$$\\end{document} wave function are coupled to each other due to the wormhole structure and the magnetic field. We have found that transport properties strongly depend on the magnetic field, incident energy, and geometric parameters of the system. We observed that the transmission probability increases as the radius of the wormhole increases, and the length of the wormhole decreases. The higher energies lead to a decrease in the transmission probabilities of particles. Furthermore, we observed that the probability of the spin-flip effect is almost larger than that of the non-spin-flip effect, illustrating that electrons lose their spins during transmission. These findings highlight the complex and interesting behavior of wormhole graphene in the presence of external magnetic fields and suggest that these nano structures can have potential applications in electronic and spintronic devices.

Persistent currents of ultrarelativistic plasma-encased endofullerene molecules entrapping H atom

Abstract In this work, for the first time in the relevant literature, the persistent currents (pc) and induced magnetic fields (imf) of an endofullerene molecule guested a hydrogen atom, under a spherical confinement, are investigated. The endofullerene molecule is enclosed within a spherical region and embedded into a plasma environment. The plasma environment is depicted with the more general exponential cosine screened Coulomb (MGECSC) potential, and its relevant effects are analyzed by considering plasma screening parameters. The relevant model for endohedral confinement is the Woods-Saxon confinement potential, which is compatible with experimental data. The effects of various forms of Cn are thoroughly elucidated through the analysis of confinement depth, spherical shell thickness, the inner radius, and the smoothing parameters. To find the bound states in the spherically confined endofullerene, the decoupling of the second-order Dirac equation for the large and small components of the radial atomic wave functions is considered. The Dirac equation with the interaction potential is solved numerically by using the Runge-Kutta-Fehlberg method through the decoupling formalism. The influence of spin orientations on the pc and imf is also elucidated. The effects of spherical confinement, plasma shielding, and the structural properties of the fullerene on the pc and imf are thoroughly viewed. Moreover, under given physical conditions, the optimal ranges of these effects are determined.

Dirac materials in parallel non-uniform electromagnetic fields generated by SUSY: a chiral Planar Hall Effect

Within a Supersymmetric Quantum Mechanics (SUSY-QM) framework, the (3+1) Dirac equation describing a Dirac material in the presence of external parallel electric and magnetic fields is solved. Considering static but non-uniform electric and magnetic profiles with translational symmetry along the y-direction, the Dirac equation is transformed into two decoupled pairs of Schrödinger equations, one for each chirality of the fermion fields. Taking trigonometric and hyperbolic profiles for the vector and scalar potentials, respectively, we arrive at SUSY partner Pöschl-Teller-like quantum potentials. Restricting to the conditions of the potentials that support an analytic zero-mode solution, we obtain a nontrivial current density perpendicular to the electric and magnetic fields, thus, defining a plane where these three vectors become coplanar, indicating the possibility of realizing the Planar Hall Effect. Furthermore, this non-vanishing current density is the sum of current densities for the left- and right-chiralities, suggesting that the net current is a consequence of chiral symmetry. Possible application in current steering of solitonic nature through a Type-I Weyl semimetal is discussed.

Orthonormal Strichartz estimate for dispersive equations with potentials

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations.