Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry

In this Letter the approximately analytical bound state solutions of the Dirac equation with the Manning-Rosen potential for arbitrary spin-orbit coupling quantum number k are carried out by taking a properly approximate expansion for the spin-orbit coupling term. In the case of exact spin symmetry, the associated two-component spinor wave functions of the Dirac equation for arbitrary spin-orbit quantum number k are presented and the corresponding bound state energy equation is derived. We study briefly two special cases; the general s-wave problem and the equal scalar and vector Manning-Rosen potential.     

Exact solution of the Dirac equation with the Mie-type potential under the pseudospin and spin symmetry limit

We investigate the exact solution of the Dirac equation for the Mie-type potentials under the conditions of pseudospin and spin symmetry limits. The bound state energy equations and the corresponding two-component spinor wave functions of the Dirac particles for the Mie-type potentials with pseudospin and spin symmetry are obtained. We use the asymptotic iteration method in the calculations. Closed forms of the energy eigenvalues are obtained for any spin-orbit coupling term κ. We also investigate the energy eigenvalues of the Dirac particles for the well-known Kratzer-Fues and modified Kratzer potentials which are Mie-type potentials.     

On the Dirac oscillator

In the present work we obtain a new representation for the Dirac oscillator based on the Clifford algebra C?₇. The symmetry breaking and the energy eigenvalues for our model of the Dirac oscillator are studied in the non-relativistic limit.     

Quasi-exact solvability of Dirac equation with Lorentz scalar potential

We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and sl (2)-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.     

Quasi-exact solvability of Dirac-Pauli equation and generalized Dirac oscillators

In this paper we demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the sl(2) symmetry are discussed separately in the spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.     

The spin symmetry for deformed generalized Pöschl-Teller potential

In the case of spin symmetry we solve the Dirac equation with scalar and vector deformed generalized Pöschl-Teller (DGPT) potential and obtain exact energy equation and spinor wave functions for s-wave bound states. We find that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition λ<−η for the wave functions. The energy eigenvalue approaches a constant when the potential parameter α goes to zero. Two special cases such as generalized PT potential and standard PT potential are also briefly discussed.     

The scattering of the Manning-Rosen potential with centrifugal term

In this Letter the approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Manning-Rosen potential are carried out by a proper approximation to the centrifugal term. The normalized radial wave functions of l-wave scattering states are presented and the calculation formula of phase shifts is derived. It is well shown that the poles of the S-matrix in the complex energy plane correspond to bound states for real poles and scattering states for complex poles in the lower half of the energy plane. We consider and verify two special cases: the l=0 and the s-wave Hulthén potential.     

Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials

The Dirac equation is exactly solved for a pseudoscalar linear plus Coulomb-like potential in a two-dimensional world. This sort of potential gives rise to an effective quadratic plus inversely quadratic potential in a Sturm-Liouville problem, regardless the sign of the parameter of the linear potential, in sharp contrast with the Schrödinger case. The generalized Dirac oscillator already analyzed in a previous work is obtained as a particular case.     

Tunneling times in complex potentials

Tunneling times in absorptive media, described by complex potentials, are investigated. A general expression relating relevant tunneling times is extended by introducing a new term τa. For both single Dirac δ potential and rectangular potential τa exhibits a maximum for a specific complex potential configuration. The optimal shape of a complex rectangular potential in respect to the maximization of τa differs drastically in the short or long wavelength limits.     

(1+1)-Dirac Particle with Position-Dependent Mass in Complexified Lorentz Scalar Interactions: Effectively $\mathcal{PT}$ -Symmetric

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the “quasi-parity” on the Dirac particles’ spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x−ib) (an inversely linear plus linear, leading to a symmetric oscillator model), and S(x)=S _r (x)+iS _i (x) (a -symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator and an η-weak-pseudo-Hermiticity generator are presented and a complexified -symmetric periodic-type model is used as an illustrative example.     

Hidden supersymmetry in quantum bosonic systems

We show that some simple well-studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov-Bohm, the Dirac delta and the Pöschl-Teller potential problems, in which the unbroken and broken N = 2 supersymmetry of linear and nonlinear (polynomial) forms is revealed.     

Trapping neutral fermions with a PT-symmetric trigonometric potential in the presence of position-dependent mass

The relativistic problem of neutral fermions subject to PT-symmetric trigonometric potential (∼iαtanαx)$(\sim \mathrm{i}\alpha \tan \alpha x)$ in 1+1$1+1$ dimensions is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Dirac equation with the vector coupling scheme and obtain the bound state solutions in closed form. The behavior of the energy spectra is discussed in detail.     

Fermions tunneling of higher-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter

The 1/2 spin fermions tunneling at the horizon of n-dimensional Kerr–Anti-de Sitter black hole with one rotational parameter is researched via semi-classical approximation method, and the Hawking temperature and fermions tunneling rate are obtained in this Letter. Using a new method, the semi-classical Hamilton–Jacobi equation is gotten from the Dirac equation in this Letter, and the work makes several quantum tunneling theories more harmonious.     

Spin-0 and spin-1/2 particles in a constant scalar-curvature background

We study the Klein-Gordon and Dirac equations in the presence of a background metric ds²=−dt²+dx²+e^−2gx(dy²+dz²) in a semi-infinite lab (x>0). This metric has a constant scalar-curvature R=6g² and is produced by a perfect fluid with equation of state p=−ρ/3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E²_min=m²c⁴+g²c²?² for bosons (E_KG>E_min), while the fermions have no specific ground state (E_Dirac>mc²).     

Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass

We present a new method to construct the exactly solvable PT-symmetric potentials within the framework of the position-dependent effective mass Dirac equation with the vector potential coupling scheme in 1 + 1 dimensions. In order to illustrate the procedure, we produce three PT-symmetric potentials as examples, which are PT-symmetric harmonic oscillator-like potential, PT-symmetric potential with the form of a linear potential plus an inversely linear potential, and PT-symmetric kink-like potential, respectively. The real relativistic energy levels and corresponding spinor components for the bound states are obtained by using the basic concepts of the supersymmetric quantum mechanics formalism and function analysis method.     

Exactly Solvable Combinations of Scalar and Vector Potentials for the Dirac Equation Interrelated by Riccati Equations

New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form ofa Riccati equation. The present note generalizes a series of former articles.     

Exact Solutions of the Dirac Equation for an Electron in a Magnetic Field with Shape Invariant Method

Based on the shape invariance property we obtain exact solutions oI the Dirac equation for an electron moving in the presence of a certain varying magnetic field, then we also show its non-relativistic limit.     

Exact L series solution of the Dirac-Coulomb problem for all energies

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.     

The uncertainties in radial position and radial momentum of an electron in the non-relativistic hydrogen-like atom

Similar to the case of a simple harmonic oscillator, an increase in azimuthal quantum number l will result in simultaneous decrease in both the uncertainty in radial position and the uncertainty in radial momentum for the same principal quantum number n in the non-relativistic hydrogen-like atom. Thus, in some cases of hydrogen-like atom and in the case of a simple harmonic oscillator, the more precisely the position is determined, the more precisely the momentum is known in that instant, and vice versa.     

Solutions to the 1d Klein-Gordon equation with cut-off Coulomb potentials

In a recent paper by Barton [G. Barton, J. Phys. A: Math. Gen. 40 (2007) 1011], the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V₁(|x|)=−α/(|x|+a). In the present Letter, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cut-off Coulomb potential V₂(|x|)=−α/|x|, |x|>a, and otherwise V₂(|x|)=−α/a.