Radial Dirac Equation Research Articles

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

Spin and pseudospin symmetries in radial Dirac equation and exceptional hermite polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Mudirac: A Dirac equation solver for elemental analysis with muonic X‐rays

We present a new open‐source software for the integration of the radial Dirac equation developed specifically with muonic atoms in mind. The software, called mudirac, is written in C++ and can be used to predict frequencies and probabilities of the transitions between levels of the muonic atom. In this way, it provides an invaluable tool in helping with the interpretation of muonic X‐ray spectra for elemental analysis. We introduce the details of the algorithms used by the software, show the interpretation of a few example spectra and discuss the more complex issues involved with predicting the intensities of the spectral lines. The software is available publicly at https://github.com/muon-spectroscopy-computational-project/mudirac.

On the Impact of Substrate Uniform Mechanical Tension on the Graphene Electronic Structure

Employing density functional theory calculations, we obtain the possibility of fine-tuning the bandgap in graphene deposited on the hexagonal boron nitride and graphitic carbon nitride substrates. We found that the graphene sheet located on these substrates possesses the semiconducting gap, and uniform biaxial mechanical deformation could provide its smooth fitting. Moreover, mechanical tension offers the ability to control the Dirac velocity in deposited graphene. We analyze the resonant scattering of charge carriers in states with zero total angular momentum using the effective two-dimensional radial Dirac equation. In particular, the dependence of the critical impurity charge on the uniform deformation of graphene on the boron nitride substrate is shown. It turned out that, under uniform stretching/compression, the critical charge decreases/increases monotonically. The elastic scattering phases of a hole by a supercritical impurity are calculated. It is found that the model of a uniform charge distribution over the small radius sphere gives sharper resonance when compared to the case of the ball of the same radius. Overall, resonant scattering by the impurity with the nearly critical charge is similar to the scattering by the potential with a low-permeable barrier in nonrelativistic quantum theory.

On solutions of the Dirac equation in a strong disappearing at infinity electrostatic field

It is shown that the one-particle radial Dirac equation with a spherically symmetric narrow rectangular well remains valid also in the supercritical region. Positron resonances in scattering arising at low supercriticality are in no way related to the process of spontaneous production of electron-positron pairs from vacuum.

Discontinuous radial potentials, bound states and Dirac equations

Discontinuous, radial ‘square-well’ potentials of scalar and (time-component) vector types require special quantization conditions in Dirac equations transformed to second-order differential equations. Quantization conditions for positive energies are derived from first-order and second-order differential equations using amplitude-phase methods. Analytic condition for no orbital angular momentum of the large spinor component is obtained. Numerical applications relevant to independent-nucleon shell models illustrate level shifts due to shape (square-well/Woods–Saxon) differences.

A Novel Exact Solution of the 2+1-Dimensional Radial Dirac Equation for the Generalized Dirac Oscillator with the Inverse Potentials

The generalized Dirac oscillator as one of the exact solvable models in quantum mechanics was introduced in 2+1-dimensional world in this paper. What is more, the general expressions of the exact solutions for these models with the inverse cubic, quartic, quintic, and sixth power potentials in radial Dirac equation were further given by means of the Bethe ansatz method. And finally, the corresponding exact solutions in this paper were further discussed.

Coulomb Problem for Z > Zcr in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Zcr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z Zcr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions e0 and widths γ of quasistationary states as a function of supercriticality. The values of e0* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Zcr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.

The Generalized Stationary Scattering Problems

We introduce and investigate the important notion of the generalized stationary scattering function for Schrodinger and Dirac radial equations. In particular, this notion is fruitful in the case of Coulomb-type potentials. Although generalized dynamical scattering functions were introduced and actively studied in the previous century, the notion of the generalized stationary scattering function is new.

Solutions of the three-dimensional radial Dirac equation from the Schrödinger equation with one-dimensional Morse potential

New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.

Nucleon–anti-nucleon intruder state of Dirac equation for nucleon in deep scalar potential well

We solve the Dirac radial equation for a nucleon in a scalar Woods–Saxon potential well of depth [Formula: see text] and radius [Formula: see text]. A sequence of values for the depth and radius are considered. For shallow potentials with [Formula: see text] the wave functions for the positive-energy states [Formula: see text] are dominated by their nucleon component [Formula: see text]. But for deeper potentials with [Formula: see text] the [Formula: see text]s begin to have dominant anti-nucleon component [Formula: see text]. In particular, a special intruder state enters with wave function [Formula: see text] and energy [Formula: see text]. We have considered several [Formula: see text] values between 2 and 8[Formula: see text]fm. For [Formula: see text] and the above [Formula: see text] values, [Formula: see text] is the only bound positive-energy state and has its [Formula: see text] closely equal to [Formula: see text], both having a narrow wave packet shape centered around [Formula: see text]. The [Formula: see text] of this state is practically independent of [Formula: see text] for the above [Formula: see text] range and obeys closely the relation [Formula: see text].

Induced vacuum charge of massless fermions in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions

We study the vacuum polarization of zero-mass charged fermions in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions. For this, we construct the Green’s function of the two-dimensional Dirac equation in the considered field configuration and use it to find the density of the induced vacuum charge in so-called subcritical and supercritical regions. The Green’s function is represented in regular and singular (in the source) solutions of the Dirac radial equation for a charged fermion in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions and satisfies self-adjoint boundary conditions at the source. In the supercritical region, the Green’s function has a discontinuity related to the presence of singularities on the nonphysical sheet of the complex plane of “energy,” which are caused by the appearance of an infinite number of quasistationary states with negative energies. Ultimately, this situation represents the neutral vacuum instability. On the boundary of the supercritical region, the induced vacuum charge is independent of the self-adjoint extension. We hope that the obtained results will contribute to a better understanding of important problems in quantum electrodynamics and will also be applicable to the problem of screening the Coulomb impurity due to vacuum polarization in graphene with the effects associated with taking the electron spin into account.

The role of shape invariance potentials in the relativistic quantum mechanics

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.

Vacuum polarization of planar charged fermions with Coulomb and Aharonov–Bohm potentials

Vacuum polarization of charged massless fermions is investigated in the superposition of Coulomb and Aharonov–Bohm (AB) potentials in 2 + 1 dimensions. For this purpose, we construct the Green function of the two-dimensional Dirac equation with Coulomb and AB potentials (via the regular and irregular solutions of the radial Dirac equation) and then calculate the vacuum polarization charge density in the so-called subcritical and supercritical regimes. In the supercritical regime, the Green function has a discontinuity in the complex plane of “energy” due to the singularities on the negative energy axis; these singularities are situated on the unphysical sheet and related to the creation of infinitely many quasistationary fermionic states with negative energies. We expect that our results will be helpful in gaining deeper understanding of the fundamental problem of quantum electrodynamics which can be applied to the problems of charged impurity screening in graphene taking into consideration the electron spin.

Spherical Dirac equation on the lattice and the problem of the spurious states

With the development of radioactive ion beam facilities, the study of exotic nuclei with unusual N/Z ratio has attracted much attention. Compared with the stable nuclei, the exotic nuclei have many novel features, such as the halo phenomenon. In order to describe the halo phenomenon with the diffused density distribution, the correct asymptotic behaviors of wave functions should be treated properly. The relativistic continuum Hartree-Bogoliubov (RCHB) theory which provides a unified and self-consistent description of mean field, pair correlation and continuum has achieved great success in describing the spherical exotic nuclei. In order to study the halo phenomenon in deformed nuclei, it is necessary to extend RCHB theory to the deformed case. However, solving the relativistic Hartree-Bogoliubov equation in space is extremely difficult and time consuming. Imaginary time step method is an efficient method to solve differential equations in coordinate space. It has been used extensively in the nonrelativistic case. For Dirac equation, it is very challenging to use the imaginary time step method due to the Dirac sea. This problem can be solved by the inverse Hamiltonian method. However, the problem of spurious states comes out. In this paper, we solve the radial Dirac equation by the imaginary time step method in coordinate space and study the problem of spurious states. It can be proved that for any potential, when using the three-point differential formula to discretize the first-order derivative operator, the energies of the single-particle states respectively with quantum numbers and - are identical. One of them is a physical state and the other is a spurious state. Although they have the same energies, their wave functions have different behaviors. The wave function of physical state is smooth in space while that of spurious state fluctuates dramatically. Following the method in lattice quantum chromodynamics calculation, the spurious state in radial Dirac equation can be removed by introducing the Wilson term. Taking Woods-Saxon potential for example, the imaginary time step method with the Wilson term is implanted successfully and provides the same results as those from the shooting method, which demonstrates its future application to solving the Dirac equation in coordinate space.

Green's function method in covariant density functional theory

This article provides a brief review on the Green's function covariant density functional theory (GF-CDFT), and its appli-cation for the nuclear structure. By reviewing the research status of exotic nuclei and the problems of traditional theory in treating continuum, the Green's function method in nuclear physics is introduced. The GF-CDFT theory is then presented in detail, including a short introduction to the Green's function method, Green's function relativistic mean field (GF-RMF) theory, and Green's function relativistic continuum Hartree-Bogoliubov (GF-RCHB) theory, in which the detailed equa-tions in solving the radial Dirac equation and RCHB equation by Green's function method are given. As to applications, GF-RMF theory is applied to study the single-particle resonant states, in which the energies and widths for the resonant states of 120 Sn are provided; GF-RCHB theory is applied to study the giant halos in the neutron-rich Zr isotopes, in which the extended density distribution as well as the energies and widths of the resonant states playing important roles for giant halos are well described.

Single particle states in the Fermi sea and Dirac sea in spherical nuclei

The Dirac equation sits in the center of nuclear covariant density functional theories. In this review, we will introduce briefly the numerical methods of solving the Dirac equation for spherical nuclei and discuss several relevant topics. We introduce the shooting method which is used in solving the radial Dirac equation in the coordinate space and the Woods-Saxon basis expansion approach including the Schr\odinger-Woods-Saxon basis and the Dirac-Woods-Saxon basis. For the Dirac-Woods-Saxon basis, the completeness of the basis requires that not only the positive energy states in the Fermi sea but also the negative energy states in the Dirac sea should be included in the basis. The pseudospin symmetry of those negative energy states, or equivalently, the spin symmetry in the anti-nucleon spectra are discussed.

Influence of dense quantum plasmas on fine-structure splitting of Lyman doublets of hydrogenic systems

Relativistic calculations are performed to study the effects of oscillatory quantum plasma screening on the fine-structure splitting between the components of Lyman-α and β line doublets of atomic hydrogen and hydrgen-like argon ion within dense quantum plasmas, where the effective two-body (electron–nucleus) interaction is modeled by the Shukla–Eliasson oscillatory exponential cosine screened-Coulomb potential. The numerical solutions of the radial Dirac equation for the quantum plasma-embedded atomic systems reveal that the oscillatory quantum screening effect suppresses the doublet (energy) splitting substantially and the suppression becomes more prominent at large quantum wave number kq. In the absence of the oscillatory cosine screening term, much larger amount of suppression is noticed at larger values of kq, and the corresponding results represent the screening effect of an exponential screened-Coulomb two-body interaction. The Z4 scaling of the Lyman doublet splitting in low-Z hydrogen isoelectron...

Does Dirac equation for a generalized Coulomb-like potential in D+1 dimensional flat space–time admit any solution for [formula omitted

The relativistic hydrogen atom in a Euclidean space–time of arbitrary number of space dimensions (D) plus one time dimension is revisited. In particular, numerical solutions of the radial Dirac equation for a generalized Coulombian potential proportional to 1/r(D−2) are investigated. It is argued that one could not find any physical solution for D≥4.