Time-independent Dirac Equation Research Articles

Spatial quasi-bound states of Dirac electrons in graphene monolayer

Our study investigated the emergence of spatial quasi-bound states (QBSs) in graphene monolayers induced by rectangular potential barriers. By solving the time-independent Dirac equation and using the transfer matrix formalism, we calculated the resonance energies and identify the QBSs based on probability density functions (PDF). We analyzed two types of structures: single and double barriers, and we find that the QBSs are located within the barrier region, at energies higher than the single barrier. Additionally, we observe QBSs in the double barrier and their position depends on the distance and width of the well between the two barriers. The width and height of the barrier significantly impact the QBSs while the well width influences the resonance energy levels of the QBSs in the double barrier. Interestingly, the QBSs can be manipulated in the graphene system, offering potential for optoelectronic devices. Finally, our results demonstrated that the spatial localization of these states is counter-intuitive and holds great promise for future research in optolectronic devices.

The Λ-core model for studying the binding energies of Λ-hypernuclei under pseudo-spin symmetry using the Hellmann potential

We have studied the binding energies of a group of single Λ-hypernuclei in a relativistic approach and modeled the single Λ-hypernuclei as a Λ-core binary system. Since the Hellmann potential is ideal for defining nucleon-core interaction and the single Λ-hypernuclei can be assumed as a single nucleon coupled to the whole nuclei but does not suffer from Pauli blocking, we have selected this potential for interaction between the Λ particle and the core. The time-independent Dirac equation is a reasonable option for defining the relativistic bound states that correspond to a spin-1/2 Λ hyperon in the hypernuclei. We solved this equation by using the Hellmann potential under the presence of pseudo-spin symmetry in terms of the generalized parametric Nikiforov-Uvarov method, a way to make the application of the Nikiforov-Uvarov method as plain as possible. Our results were in good agreement with experimental values and other theoretical works. Hence, this model is applicable for the Λ-hypernuclei.

Approximate Treatment of the Dirac Equation with Hyperbolic Potential Function

The time independent Dirac equation is solved analytically for equal scalar and vector hyperbolic potential function in the presence of Greene and Aldrich approximation scheme. The bound state energy equation and spinor wave functions expressed by the hypergeometric function have been obtained in detail with asymptotic iteration approach. In order to indicate the accuracy of this different approach proposed to solve second order linear differential equations, we present that in the non-relativistic limit, analytical solutions of the Dirac equation converge to those of the Schrodinger one. We introduce numerical results of the theoretical analysis for hyperbolic potential function. Bound states corresponding to arbitrary values of n and l are reported for potential parameters covering a wide range of interaction. Also, we investigate relativistic vibrational energy spectra of alkali metal diatomic molecules in the different electronic states. It is observed that theoretical vibrational energy values are consistent with experimental Rydberg–Klein–Rees (RKR) results and vibrational energies of NaK, $$K_2$$ and KRb diatomic molecules interacting with hyperbolic potential smoothly converge to the experimental dissociation limit $$D_e=2508cm^{-1}$$ , $$254cm^{-1}$$ and $$4221cm^{-1}$$ , respectively.

Relativistic ground state of diatomic molecules from the numerical solution of the Dirac equation on parallel computers

The parallel implementation of a variational method for the numerical solution of the time-independent Dirac equation is presented. This is utilized in the calculation of diatomic molecule relativistic wave functions and ground state energy, in the Born-Oppenheimer approximation. Following [1], a functional equation predicting the same spectrum as the Dirac equation is discretized over a set of basis functions. We find that the combination of B-spline basis functions and prolate spheroidal coordinate systems facilitate the vectorization of the technique using domain decomposition. This procedure results in a non-linear eigenvalue problem which is solved by iteration using Brent's method. We validate our methodology by comparing with existing results and show that it yields very accurate ones. We also analyze the parallelization and demonstrate that our implementation has a good scaling performance. Finally, the utilization of this method for laser-matter interaction and other physical applications will be discussed.

Numerical solution of the time-independent Dirac equation for diatomic molecules:Bsplines without spurious states

Two numerical methods are used to evaluate the relativistic spectrum of the two-centre Coulomb problem (for the $H_{2}^{+}$ and $Th_{2}^{179+}$ diatomic molecules) in the fixed nuclei approximation by solving the single particle time-independent Dirac equation. The first one is based on a min-max principle and uses a two-spinor formulation as a starting point. The second one is the Rayleigh-Ritz variational method combined with kinematically balanced basis functions. Both methods use a B-spline basis function expansion. We show that accurate results can be obtained with both methods and that no spurious states appear in the discretization process.

Confinement limit of the Dirac particle in one dimension

For a particle of mass $m$ that obeys the time-independent Dirac equation in one dimension with a symmetric potential, Unanyan et al. [Phys. Rev. A 79, 044101 (2009)] recently pointed out that the inequality $\ensuremath{\Delta}x=\sqrt{\ensuremath{\langle}{x}^{2}}\ensuremath{\rangle}g\ensuremath{\lambda}/2$ can be derived simply from the Dirac equation. Here $\ensuremath{\lambda}=\ensuremath{\hbar}/(\mathit{mc})$ is the Compton wavelength, $x$ is the particle coordinate, and $\ensuremath{\langle}{x}^{2}\ensuremath{\rangle}$ is the expectation value of ${x}^{2}$. We conjecture that a new, more stringent limit $\ensuremath{\Delta}x\ensuremath{\geqslant}\ensuremath{\lambda}/\sqrt{2}$ holds for any symmetric potential. We present a model analysis on which the conjecture is based.

Real spectra for the non-Hermitian Dirac equation in [formula omitted] dimensions with the most general coupling

The most general combination of couplings of fermions with external potentials in 1 + 1 dimensions, must include vector, scalar and pseudoscalar potentials. We consider such a mixing of potentials in a PT-symmetric time-independent Dirac equation. The Dirac equation is mapped into an effective PT-symmetric Schrödinger equation. Despite the non-hermiticity of the effective potential, we find real energies for the fermion.

The gravitational analogue to the hydrogen atom**A summer study at the borders of quantum mechanics and general relativity.

This paper reports on a student summer project performed in 2006 at the University of Frankfurt. It is addressed to undergraduate students familiar with the basic principles of relativistic quantum mechanics and general relativity. The aim of the project was to study the Dirac equation in curved spacetime. To obtain the general relativistic Dirac equation we use the formulation of gravity as a gauge theory in the first part. After these general considerations we restrict the further discussion to the special case of the Schwarzschild metric. This setting corresponds to the hydrogen atom, with the electromagnetic field replaced by gravity. Although there is a singularity at the event horizon it turns out that a regular solution of the time-independent Dirac equation exists. Finally the Dirac equation is solved numerically using suitable boundary conditions.

Magnetic order at a single-crystal surface in the diffuse-scattering theory

A theoretical description of incoherent spin-dependent multiple scattering of electrons at a magnetically disordered single-crystal surface is reported. A formalism in which the spin operators specify the magnetic state of a surface atom is used for the description of magnetic order at the surface. The theory is based upon the concepts used in multiple scattering spin-dependent diffuse LEED theory (DSPLEED) theory. In the present considerations, this theory is extended to the case of magnetic materials by using the time-independent Dirac equation with an effective magnetic field. Thus, an expression for incoherent spin-dependent intensity for magnetic material is obtained. It depends on the Fourier transform on the surface lattice of the spin-pair correlation function and, as a consequence, on the magnetic properties of the surface. The equations for the description of magnetization and various correlation functions in the frame of effective field theory are derived and the results of the numerical calculations are presented for the particular case of Ni(1 0 0) surface. The spin–orbit induced and exchange asymmetries are calculated. It is found that the magnetic DSPLEED is sensitive to the properties of the surface characterized by the spin-pair correlation functions. Thus, it is demonstrated that the magnetic DSPLEED can be an effective method in the investigation of critical behaviour of magnetic surfaces.

Theory of incoherent spin dependent scattering of electrons at adisordered surface

A theoretical description of incoherent spin dependent multiple scatteringof electrons at a single-crystal surface partially covered with adisordered overlayer is reported. A formalism in which occupation operatorsspecify the occupancy of a chemisorption site is used for the statisticaldescription of a disordered adlayer. The theory is based upon the conceptsused in multiple-scattering DLEED theory. In the present considerationsspin effects are introduced to this theory by using the time independentDirac equation. Thus, a simple expression for incoherent spin dependentintensity is obtained. The spin-orbit induced `up/down' asymmetry is thencalculated. As an example the particular case of CO/Pt(111) is examined. Itis found that diffuse SPLEED is sensitive to the geometrical parameterscharacterizing the substrate/disordered adsorbate system as well as to theparameters related to the statistical distribution of the occupiedchemisorption sites. It is demonstrated that the method is applicable toany disordered surface suitable for DLEED investigation and in this wayenlarges the number of experimental data convenient for disordered surfacestructure analysis.

Minimax variational approach to the relativistic two-electron problem.

A minimax formulation for the time-independent Dirac equation is extended to the two-particle problem. The question of solutions of the minimax formulation is discussed and related to the continuum dissolution problem. The approach is applied to the ground state and the n=2 complex of states of He-like ions. Variational calculations are made for bases with a small number of basis functions, varying the nonlinear parameters, as well as for large-dimensional configuration-interaction-type bases. \textcopyright{} 1996 The American Physical Society.