Radial Spinor Research Articles

Exact solution of Dirac–Rosen–Morse problem in curved space–time

In this work, we extend the analysis of the relativistic Dirac–Rosen–Morse problem in curved space–time. For that, we consider the Dirac equation in curved space–time with line element d s2 = (1 + α2 U( r))2(d t2 − d r2) − r2dθ2 − r2sin 2θdϕ2, where α is fine structural constant, U( r) is a scalar potential, and in the presence of the electromagnetic field Aμ = ( V( r), cA( r), 0, 0). Because of the spherical symmetry, the angular spinor is given in terms of the spherical harmonics. For the radial spinor, we apply a unitary transformation and define the vector component of the electromagnetic field A( r) written as a function of V( r) and U( r) so as to solve the radial spinor for Dirac–Rosen–Morse problem. Graphical analyses were performed comparing the eigenenergies and the probability densities in curved and flat space–time to visualize the influence of curvature in space–time on the two-component radial spinor, with the upper and lower components representing the particle and antiparticle, respectively.

Supersymmetry of Bianchi attractors in gauged supergravity

Bianchi attractors are near horizon geometries with homogeneous symmetries in the spatial directions. We construct supersymmetric Bianchi attractors in $\mathcal{N}=2, d=4,5$ gauged supergravity. In $d=4$ we consider gauged supergravity coupled to vector and hypermultiplets. In $d=5$ we consider gauged supergravity coupled to vector multiplets with a generic gauging of symmetries of the scalar manifold and the $U(1)_R$ gauging of the $R$-symmetry. Analyzing the gaugino conditions we show that when the fermionic shifts do not vanish there are no supersymmetric Bianchi attractors. This is analogous to the known condition that for maximally supersymmetric solutions, all the fermionic shifts must vanish. When the central charge satisfies an extremization condition, some of the fermionic shifts vanish and supersymmetry requires that the symmetries of the scalar manifold do not be gauged. This allows supersymmetric Bianchi attractors sourced by massless gauge fields and a cosmological constant. In five dimensions in the Bianchi I class we show that the anisotropic $AdS_3\times\mathbb{R}^2$ solution is $1/2$ BPS. We also construct a new class of $1/2$ BPS Bianchi III geometries labeled by the central charge. When the central charge takes a special value the Bianchi III geometry reduces to the known $AdS_3\times\mathbb{H}^2$ solution. For the Bianchi V and VII classes the radial spinor breaks all of supersymmetry. We briefly discuss the conditions for possible massive supersymmetric Bianchi solutions by generalizing the matter content to include tensor, hypermultiplets and a generic gauging on the R symmetry.

Relativistic Energies and Scattering Phase Shifts for the Fermionic Particles Scattered by Hyperbolical Potential with the Pseudo(spin) Symmetry

In this paper, we studied the approximate scattering state solutions of the Dirac equation with the hyperbolical potential with pseudospin and spin symmetries. By applying an improved Greene-Aldrich approximation scheme within the formalism of functional analytical method, we obtained the spin-orbit quantum numbers dependent scattering phase shifts for the spin and pseudospin symmetries. The normalization constants, lower and upper radial spinor for the two symmetries, and the relativistic energy spectra were presented. Our results reveal that both the symmetry constants (Cps and Cs) and the spin-orbit quantum number κ affect scattering phase shifts significantly.

Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant

It is shown that solutions of the second-order decoupled radial Dirac equations satisfy Ermakov-type invariants. These invariants lead to amplitude-phase-type representations of the radial spinor solutions, with exact relations between their amplitudes and phases. Implications leading to a Bohr-Sommerfeld quantization condition for bound states, and a few particular atomic/ionic and nuclear/hadronic bound-state situations are discussed.

ON THE SUPERSYMMETRY OF THE DIRAC–KEPLER PROBLEM PLUS A COULOMB-TYPE SCALAR POTENTIAL IN (D+1) DIMENSIONS AND THE GENERALIZED LIPPMANN–JOHNSON OPERATOR

We will study the Dirac–Kepler problem plus a Coulomb-type scalar potential by generalizing the Lippmann–Johnson operator to D spatial dimensions. From this operator, we construct the supersymmetric generators to obtain the energy spectrum for discrete excited eigenstates and the radial spinor for the SUSY ground state.

Analysis of the asymptotic and short-range behavior of quasilocal Hartree-Fock and Dirac-Fock-Coulomb electron-electron interaction potentials

The analytic origin behavior of Hartree-Fock and Dirac-Fock-Coulomb electron-electron interaction potentials is derived. This yields explicit expressions, which depend on $〈{1/r}^{n}〉$ matrix elements. The direct part of the electron-electron interaction including the self-interaction is equal for all shells. If the self-interaction is subtracted the Coulomb part will become shell dependent. Additional shell dependence of the origin behavior originates from the true exchange-interaction terms. For the Dirac-Fock-Coulomb operator we find singular behavior for non s shells, if a Coulomb-type electron-nucleus potential is used, while it is nonsingular for s shells. For all shells in case of finite nuclear models, the electron-electron interaction potentials are nonsingular at the origin. A comparison of potentials for the small and large component of the relativistic radial spinor shows that they are equal in the long-range limit, while this is not true, in general, at the origin. The analytic results presented are tested by comparing them to numerical results obtained for the zinc atom.