Two-component Spinor Research Articles

Approximate analytical solutions of the relativistic equations with the Deng–Fan molecular potential including a Pekeris-type approximation to the (pseudo or) centrifugal term

By employing the Pekeris-type (or a new improved approximation) to deal with the (pseudo or) centrifugal term, we solve the Klein–Gordon and Dirac equations with equally mixed scalar and vector Deng–Fan molecular potentials for all values of l (orbital the angular momentum quantum number) and κ (spin–orbit coupling quantum number), respectively. Using the formalism of the Nikiforov–Uvarov method, the approximate analytical bound state energy equations and the associated two-component spinors corresponding to the two relativistic equations are obtained. Also, special cases including the non-relativistic limits of the relativistic equation are obtained.

Approximate Solutions to the Dirac Equation with Effective Mass for the Manning–Rosen Potential in N Dimensions

The solutions of the effective mass Dirac equation for the Manning–Rosen potential with the centrifugal term are studied approximately in N dimension. The relativistic energy spectrum and two-component spinor eigenfunctions are obtained by the asymptotic iteration method. We have also investigated eigenvalues of the effective mass Dirac–Manning–Rosen problem for α = 0 or α = 1. In this case, the Manning–Rosen potential reduces to the Hulthen potential.

The Classical Two-Component Spinor Formalisms for General Relativity I

The classical two-component spinor formalisms for general relativity as built up by Infeld and van der Waerden afford an elegant approach to spacetime geometry. Deeply involved in the inner structure of these frameworks is the beautiful theory of spin densities of Schouten. In this work Schouten’s theory is presented in detail. The elementary aspects of the spin-affine geometry borne by the formalisms are then described in a systematic way. It is particularly shown that the symmetric parts of any admissible spin connections behave covariantly under the action of the Weyl gauge group.

The Classical Two-Component Spinor Formalisms for General Relativity II

The Classical Two-Component Spinor Formalisms for General Relativity II

Raising and lowering operators for the Dirac-Woods-Saxon potential in the presence of spin and pseudospin symmetry

In the case of spin symmetry and pseudospin symmetry, raising and lowering operators and the bound state solutions of the Dirac equation for the spherically Woods-Saxon potential are presented within the context of Supersymmetric Quantum Mechanics. The energy equation and corresponding two-component spinors of the two Dirac particles are obtained in the closed form for arbitrary spin-orbit quantum number k by using the Pekeris approximation. The Hamiltonian hierarchy method and the shape invariance property are used in the calculations.

On the Integrability Conditions for Infeld-Van der Waerden Spin-Affine Connexions

The integrability condition that must be formally fulfilled by any Infeld-Van der Waerden spin-affine connexion is derived explicitly. Some geometric properties of the action of torsionless covariant-derivative commutators on arbitrary spin tensors and densities are then brought out. The relevant calculations supply a set of new differential expressions which will presumably ensure the consistency of any formulation that involves utilizing normally the traditional two-component spinor methods for classical general relativity.

Fundamental dynamical equations for spinor wavefunctions: I. Lévy-Leblond and Schrödinger equations

A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Lévy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Lévy-Leblond and Schrödinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrödinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrödinger equation are discussed.

Two-component natural spinors from two-step spin-orbit coupled wave functions

We developed an algorithm to obtain the natural orbitals (natural spinors) from the two-step spin-orbit coupled wave functions. These natural spinors are generally complex-valued, mixing two spin components, and they can have similar symmetry properties as the j-j spinors from the one-step spin-orbit coupling calculations, if the reduced density equally averages all the components of a multi-dimensional irreducible representation. Therefore, the natural spinors can serve as an approximation to the j-j spinors and any wave function analysis based on the j-j spinors can also be performed based on them. The comparison between the natural spinors and the j-j spinors of three representative atoms, Tl, At, and Lu, shows their close similarity and demonstrates the ability of the natural spinors to approximate the j-j spinors.

Dirac equation with spin symmetry for the modified Pöschl–Teller potential in D dimensions

We present solutions of the Dirac equation with spin symmetry for vector and scalar modified P\"oschl-Teller potential within framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov-Uvarov method and the two-component spinor wavefunctions are obtain are in terms of the Jacobi polynomials. It is found that there exist only positive-energy states for bound states under spin symmetry, and the energy levels increase with the dimension and the potential range parameter $\alpha$.

Spin symmetry in the relativistic modified Rosen—Morse potential with the approximate centrifugal term

By applying a Pekeris-type approximation to the centrifugal term, we study the spin symmetry of a Dirac nucleon subjected to scalar and vector modified Rosen—Morse potentials. A complicated energy equation and associated two-component spinors with arbitrary spin—orbit coupling quantum number k are presented. The positive-energy bound states are checked numerically in the case of spin symmetry. The relativistic modified Rosen—Morse potential cannot trap a Dirac nucleon in the limiting case α → 0.

TOPOLOGICAL FOUR-DIMENSIONAL GRAVITY AND KODAMA STATE BEYOND INSTANTONS

By using the two-component spinor formalism and the symplectic geometry of the phase space, it is shown that the Kodama state actually exist within a topological phase where the Weyl spinor does not vanish. This topological phase, belonging to the complete topological sector of general relativity (GR), is bigger than the one associated with gravitational instantons. The conditions of projection onto the sector of gravitational instantons are discussed as well.

Electron scattering in the monolayer graphene with a band-asymmetric annular potential well

Electron scattering in the monolayer graphene has been considered within the framework of our model of short-range defects proposed earlier. Electronic properties are determined by the Dirac equation for the two-component spinor wave function. Perturbation is modeled by the annular well with a band-asymmetric potential. Band-asymmetry of the potential stems from the local structure defect and leads to the mass (gap) perturbation in the Dirac equation. Transitions between the K and K’ critical points in the Brillouin zone are neglected, which is valid provided that the short-range perturbation has a finite radius. Exact explicit formulas for the scattering matrix have been derived. Results are presented in terms of the scattering phases and in the geometrical form of a relation between some 2-vectors. The characteristic equation for the bound and resonance states has been obtained in the form of an orthogonality condition. An approximate calculation of observables in terms of the scattering theory results is outlined.

Spin dependent 2D electron scattering by nanomagnets

The 2D scattering problem of an electron by a magnetized nanoparticle is solved in the Born approximation with account of the dipole–dipole interaction of the magnetic moments of electron and nanomagnet. The scattering amplitudes in this problem are the two-component spinors. They are obtained as functions of the electron spin orientation, the electron energy and show anisotropy in scattering angle. The initially polarized beam of electrons scattered by the nanomagnet consists of electrons with no spin flipped and spin flipped. The majority of electrons with no spin flipped are scattered by small angles. The majority electrons with spin flipped are scattered in the vicinity of the scattering angles π / 2 and 3 π / 2 . This can be used as one more method of controlling the spin currents.

Solution of Dirac equation with spin and pseudospin symmetry for an anharmonic oscillator

We present the exact solutions of Dirac equation with anharmonic oscillator potential using the Nikiforov–Uvarov method. Taking into account potentials of vector field V(r) and scalar field S(r) in Dirac Hamiltonian, the bound state energy eigenvalues and the corresponding upper and lower two-component spinors of fermion are obtained. These solutions are considered in the framework of the spin and pseudospin symmetry concept.

EXACT SOLUTIONS OF DIRAC EQUATION WITH HARTMANN POTENTIAL BY NIKIFOROV–UVAROV METHOD

We investigate the exact solution of the Dirac equation for the Hartmann potential. The radial and polar parts of the Dirac equation are solved by Nikiforov–Uvarov method. The bound state energy eigenvalues and the corresponding two-component spinor wave functions of the Dirac particles are obtained.

A Direct Road to Majorana Fields

A concise discussion of spin-1/2 field equations with a special focus on Majorana spinors is presented. The Majorana formalism which describes massive neutral fermions by the help of two-component or four-component spinors is of fundamental importance for the understanding of mathematical aspects of supersymmetric and other extensions of the Standard Model of particle physics, which may play an increasingly important role at the beginning of the LHC era. The interplay between the two-component and the four-component formalism is highlighted in an introductory way. Majorana particles are predicted both by grand unified theories, in which these particles are neutrinos, and by supersymmetric theories, in which they are photinos, gluinos and other states.

Projector augmented-wave method: Application to relativistic spin-density functional theory

Applying the projector augmented-wave (PAW) method to relativistic spin-density functional theory (RSDFT) we derive PAW Dirac-Kohn-Sham equations for four-component spinor pseudo-wave-functions. The PAW freedom to add a vanishing operator inside the PAW spheres allows us to transform these PAW Dirac-type equations into PAW Pauli-type equations for two-component spinor pseudo-wave-functions. With these wave functions, we get the frozen-core energy as well as the charge and magnetization densities of RSDFT, with errors comparable to the largest between $1/{c}^{2}$ and the transferability error of the PAW data sets. Presently, the latter limits the accuracy of the calculations, not the use of the Pauli-type equations. The theory is validated by applications to isolated atoms of Fe, Pt, and Au, and to the band structure of fcc-Pt, fcc-Au, and ferromagnetic bcc-Fe.

Bound-state solutions of the Dirac-Rosen-Morse potential with spin and pseudospin symmetry⋆

The energy spectra and the corresponding two- component spinor wavefunctions of the Dirac equation for the Rosen-Morse potential with spin and pseudospin symmetry are obtained. The $s-$wave ($\kappa = 0$ state) solutions for this problem are obtained by using the basic concept of the supersymmetric quantum mechanics approach and function analysis (standard approach) in the calculations. Under the spin symmetry and pseudospin symmetry, the energy equation and the corresponding two-component spinor wavefunctions for this potential and other special types of this potential are obtained. Extension of this result to $\kappa \neq 0$ state is suggested.

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the ( 3 + 1 ) -dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov–Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases κ = ± 1 ( l = l ˜ = 0 ,i.e . , s -wave ) , the constant mass and the non-relativistic limits are briefly investigated.

Polarization suppression and nonmonotonic local two-body correlations in the two-component Bose gas in one dimension

We study the interplay of quantum statistics, strong interactions, and finite temperatures in the two-component (spinor) Bose gas with repulsive delta-function interactions in one dimension. Using the Thermodynamic Bethe Ansatz, we obtain the equation of state, population densities, and local density correlation numerically as a function of all physical parameters (interaction, temperature, and chemical potentials), quantifying the full crossover between low-temperature ferromagnetic and high-temperature unpolarized regimes. In contrast to the single component, Lieb-Liniger gas, nonmonotonic behavior of the local density correlation as a function of temperature is observed.