Exact Solutions Of Dirac Equation Research Articles

Recent advances in relativistic excitation of atomic hydrogen

Within the framework of the relativistic regime ( E k > 2700 eV) and the first Born approximation, we investigate the inelastic excitation of atomic hydrogen by electron impact from 1 s to 2 p state. The incoming and outgoing electrons are described by free Dirac spinors, while the hydrogen atom target is described by the exact solutions of Dirac equation with a central potential. A detailed study has been devoted to the non-relativistic regime as well as the relativistic regime of the process, and the results obtained show the differences between the two regimes. These differences have revealed that the differential cross section (DCS) is influenced by the parameters of the 1 s and 2 p states, along with significant effects on diffusion during small pulse transfers. The relativistic effects are responsible for the significant changes in the DCS as a function of diffusion angles which depend on the chosen geometry.

Discussion on Exact Solution of Dirac Equation with Generalized Exponential Potential in the Presence of Generalized Uncertainty Principle

Discussion on Exact Solution of Dirac Equation with Generalized Exponential Potential in the Presence of Generalized Uncertainty Principle

Bound states of Dirac equation using the proper quantization rule

Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.

Exact solutions of Dirac equation on a static curved space–time

In this work we study the Dirac equation in a curved static space–time, where the metric depends on two arbitrary functions f(r) and g(r), namely the line element is ds2=e2f(r)dt2−e2g(r)dr2−r2dθ2−r2sin2θdϕ2. We exactly solve the angular equation for arbitrary f(r) and g(r). Then, we uncouple the radial component of the spinor for a metric model defined by f=g and find the exact solutions for three quantum systems: Hydrogen atom, Dirac–Morse oscillator and a linear radial potential by V(r)=ar.

Wave Functions for Time-Dependent Dirac Equation under GUP**Supported by the National Natural Science Foundation of China under Grant No. 11565009

In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials.

Dirac方程式の厳密解の可視化

Dirac方程式の厳密解の可視化

Relativistic Energy Analysis of Five-Dimensionalq-Deformed Radial Rosen-Morse Potential Combined withq-Deformed Trigonometric Scarf Noncentral Potential Using Asymptotic Iteration Method

We study the exact solution of Dirac equation in the hyperspherical coordinate under influence of separableq-deformed quantum potentials. Theq-deformed hyperbolic Rosen-Morse potential is perturbed byq-deformed noncentral trigonometric Scarf potentials, where all of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equationlD-1have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations are also obtained by using this method. The relativistic energy levels are numerically calculated using Matlab, and the increase of radial quantum numberncauses the increase of bound state relativistic energy level in both dimensionsD=5andD=3. The bound state relativistic energy level decreases with increasing of both deformation parameterqand orbital quantum numbernl.

Exact Solution of Dirac Equation for q-Deformed Trigonometric Scarf potential with q-Deformed Trigonometric Tensor Coupling Potential for Spin and Pseudospin Symmetries Using Romanovski Polynomial

The bound state solutions of Dirac equation for q-deformed trigonometric Scarf potential with q-deformed trigonometric cotangent and cosecant tensor coupling potential under spin and pseudospin symmetric limits are investigated using Supersymmetric Quantum Mechanics (SUSY QM) method. The new tensor potentials proposed is inspired by superpotential form in SUSY quantum mechanics. The Dirac equations for Scarf potential coupled by new tensor potential in the pseudospin and spin symmetric cases reduce to Schrodinger type equations for shape invariant potential since the proposed new potentials are equivalent to the superpotential of q-deformed trigonometric Scarf potential. The relativistic wave functions are exactly obtained by using SUSY operator method and the relativistic energy equation are exactly obtained by using SUSY method and the idea of shape invariance in the approximation scheme of centrifugal term. The new tensor potential causes the energy degeneracies is omit both for pseudospin and spin symmetric case.

Exact solution of Dirac equation for Scarf potential with new tensor coupling potential for spin and pseudospin symmetries using Romanovski polynomials

The bound state solutions of Dirac equations for a trigonometric Scarf potential with a new tensor potential under spin and pseudospin symmetry limits are investigated using Romanovski polynomials. The proposed new tensor potential is inspired by superpotential form in supersymmetric (SUSY) quantum mechanics. The Dirac equations with trigonometric Scarf potential coupled by a new tensor potential for the pseudospin and spin symmetries reduce to Schrödinger-type equations with a shape invariant potential since the proposed new tensor potential is similar to the superpotential of trigonometric Scarf potential. The relativistic wave functions are exactly obtained in terms of Romanovski polynomials and the relativistic energy equations are also exactly obtained in the approximation scheme of centrifugal term. The new tensor potential removes the degeneracies both for pseudospin and spin symmetries.

Exact Solution of Dirac Equation with Charged Harmonic Oscillator in Electric Field: Bound States

In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case equivalent of the Stark effect for a charged harmonic oscillator (HO) in a uniform electric field of specific strength (HO in an external dipole field). We obtain the exact s-wave solutions of the Dirac equation for some potential models which are linear combination of single exactly solvable potentials (ESPs). In the framework of the spin and pseudospin symmetric concept, we calculate analytical expressions for the energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by using the Nikiforov-Uvarov (NU) method, in closed form. The nonrelativistic limit of the solution is also studied and compared with the other works.

DIRAC EQUATION FOR A SPHERICALLY PSEUDOHARMONIC OSCILLATORY RING-SHAPED POTENTIAL

In this paper, we investigate exact solutions of Dirac equation with spherically pseudoharmonic oscillatory ring-Shaped potential. The generalized parametric Nikiforov-Uvarov method is used obtaining analytical solutions of radial and polar angle parts of Dirac Equation. Energy eigenvalues and corresponding eigenfunctions are obtained in closed forms.

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.

Dirac equation for the harmonic scalar and vector potentials and linear plus coulomb-like tensor potential; the SUSY approach

The problem of analytical solutions of the 3-dimensional Dirac equation is usually studied via techniques such as The Nikiforov–Uvarov (NU) method. Here, we see that one of the most attractive potentials can be brought into a well-known form of Schrödinger-like problem possessing known solutions via the methodology of supersymmetry (SUSY). Next, using the idea of shape invariance, we calculate exact solutions of Dirac equation for quadratic scalar and vector potentials in the presence of a tensor potential that depends on the radial component either linearly or inversely. The tensor potential itself, besides its applications, removes degeneracy, too.

Exact Solutions of Dirac Equation on (1+1)-Dimensional Spacetime Coupled to a Static Scalar Field

We use a generalized scheme of supersymmetric quantum mechanics to obtain the energy spectrum and wave function for Dirac equation in (1+1)-dimensional spacetime coupled to a static scalar field.

Exact Solutions of Dirac Equation and Particle Creation in (1+3)-Dimensional Robertson-Walker Spacetime

The exact solutions of the Dirac equation are discussed for a Robertson-Walker spacetime with asymptotically Minkowskian in and out regions. We obtain the mode solutions which reduce to positive and negative Minkowskian spinors in asymptotically regions. Using the obtained solutions we compute the density of created particles.

EXACT SOLUTION OF THE CONTINUOUS STATES FOR GENERALIZED ASYMMETRICAL HARTMANN POTENTIALS UNDER THE CONDITION OF PSEUDOSPIN SYMMETRY

Under the condition of pseudospin symmetry, the exact solution of Dirac equation is studied and that no bound solutions are observed for generalized asymmetrical Hartmann potential, which is in agreement with that for Coulomb potential. With the analytic continuation method, the unbound solutions are presented by mapping the wave functions of bound states in the complex momentum plane. Furthermore, the scattering phase shifts are obtained from the radial wave function by analyzing the asymptotic behavior of the confluent hypergeometric functions.

Exact solution of Dirac equation in 2+1 dimensional gravity

We find exact solutions of the Dirac equation in the 2+1 dimensional curved background by separation of variables. These solutions are given in terms of hypergeometric functions. We also perform the Gordon decomposition for the Dirac current to discuss the time dependence of the polarization densities and the magnetization density, and to show that the polarization densities are more effective than the magnetization density in the pair production in finite time intervals.

The Kepler problem in Dirac theory for a particle with position-dependent mass

Exact solution of Dirac equation for a particle whose potential energy and mass are inversely proportional to the distance from the force centre has been found. The bound states exist provided the length scale $a$ which appears in the expression for the mass is smaller than the classical electron radius $e^2/mc^2$. Furthermore, bound states also exist for negative values of $a$ even in the absence of the Coulomb interaction. Quasirelativistic expansion of the energy has been carried out, and a modified expression for the fine structure of energy levels has been obtained. The problem of kinetic energy operator in the Schr\"odinger equation is discussed for the case of position-dependent mass. In particular, we have found that for highly excited states the mutual ordering of the inverse mass and momentum operator in the non-relativistic theory is not important.

Exact solutions of Dirac equation on a 2D gravitational background

We obtain classes of two-dimensional static Lorentzian manifolds, which through the supersymmetric formalism of quantum mechanics admit the exact solvability of Dirac equation on these curved backgrounds. Specially in the case of a modified supersymmetric harmonic oscillator the wave function and energy spectrum of Dirac equation is given explicitly.

EXACT SOLUTIONS OF DIRAC AND SCHRÖDINGER EQUATIONS FOR A LARGE CLASS OF POWER-LAW POTENTIALS AT ZERO ENERGY

We obtain exact solutions of Dirac equation for radial power-law relativistic potentials at rest mass energies. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrödinger equation at zero energy carrying representations of SO(2,1) Lie algebra. The latter are obtained by point canonical transformations of the exactly solvable problem of the three dimensional oscillator. The wave function solutions are written in terms of the confluent hypergeometric functions and almost always square integrable. For most cases these solutions support bound states at zero energy. Some exceptional unbounded states are normalizable for nonzero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.