Convergence for Imaginary Time Step evolution in the Fermi and Dirac seas

The convergence for the Imaginary Time Step (ITS) evolution with time step is investigated by performing the ITS evolution for the Schrdinger-like equation and the charge-conjugate Schrdinger-like equation deduced from Dirac equation for the single proton levels of 12C in both the Fermi and Dirac seas. For the guaranteed convergence of the ITS evolution to the exact results,the time step should be smaller than a critical time step Δtc for a given single-particle level. The critical time step Δtc is more s...     

Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential

A new ring-shaped non-harmonic oscillator potential is proposed. The precise bound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 （2005） 94. The positive energy states of system are discussed and the conclusions are made properly.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Solution of Dirac Equation with Generalized Hylleraas Potential

We employ the parametric generalization of the Nikiforov-Uvarov method to solve the Alhaidari formalism of the Dirac equation with a generalized Hylleraas potential of the form V(r) = V0(a+exp(λr))/(b+ exp(λr)) + V1(d+exp(λr))/(b+exp(λr)).We obtain the bound state energy eigenvalue and the corresponding eigenfunction expressed in terms of the Jacobi polynomials.By choosing appropriate parameter in the potential model,the generalized Hylleraas potential reduces to the well known potentials as special cases.     

Approximate Solutions of Dirac Equation with Hyperbolic-Type Potential

The energy eigenvalues of a Dirac particle for the hyperbolic-type potential field have been computed approximately. It is obtained a transcendental function of energy, F(E), by writing in terms of confluent Heun functions.The numerical values of energy are then obtained by fixing the zeros on "E-axis" for both complex functions Re[F(E)]and Im[F(E)].     

Approximate Solutions of Dirac Equation with Hyperbolic-Type Potential

The energy eigenvalues of a Dirac particle for the hyperbolic-type potential field have been computed approximately. It is obtained a transcendental function of energy, F(E), by writing in terms of confluent Heun functions. The numerical values of energy are then obtained by fixing the zeros on "E-axis" for both complex functions Re[F(E)] and Im[F(E)].     

Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.     

The Exact Solution for the Dirac Equation with the Cornell Potential

An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by another function to be determined. The energy levels are obtained and we notice that they obey a band structure.     

Solution of Dirac Equation with Killingbeck Potential by Using Wave Function Ansatz Method under Spin Symmetry Limit

The Dirac equation is solved for Killingbeck potential. Under spin symmetry limit, the energy eigenvalues and the corresponding wave functions are obtained by using wave function ansatz method.     

Dirac Equation in an Axial Coulomb Potential

We consider solutions to the Dirac equation in the presence of an external axial vector potential coupled to the spinor field psi through the interaction term . There turn out to be no bound-state energies in this system consistent with a normalizable wave function.     

Solutions of the Dirac Equation for the Davidson Potential

In the present paper we solve the Dirac equation with Davidson potential by Nikiforov-Uvarov method. The Dirac Hamiltonian contains a scalar S and a vector V Davidson potentials. With equal scalar and vector potential, analytical solutions for bound states of the corresponding Dirac equations are found.     

Rosen-MorseⅡ势函数的Klein-Gordon方程和Dirac方程束缚态的精确解

在标量势和矢量势相等的情形, 研究了Rosen-MorseⅡ势的相对论效应, 应用超对称和形状不变势方法通过求解Klein-Gordon方程和Dirac方程得到了束缚态能量本征值, 最后, 讨论了Rosen-MorseⅡ势的一种特殊情况.     

Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass

We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable for the case with time-independence mass.     

Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass

We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable forthe case with time-independence mass.     

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Dirac equation. The analytical solution for the Makarov potential with pseudospin symmetry is obtained by Nikiforov-Uvarov （N-U） method. The eigenfunctions and eigenenergies are presented with equal mixture of vector and scalar potentials in opposite signs, for which is exact.     

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Dirac equation. The analytical solution forthe Makarov potential with pseudospin symmetry is obtained byNikiforov-Uvarov (N-U) method. The eigenfunctions and eigenenergies arepresented with equal mixture of vector and scalar potentials in oppositesigns, for which is exact.     

Approximate Pseudospin Solutions of the Dirac Equation with the Eckart Potential Including a Coulomb-Like Tensor Potential

We solve the Dirac equation with the Eckart potential including a Coulomb-like tensor potential under pseudospin symmetry limit with arbitrary spin-orbit coupling quantum number κ by using the Nikiforov-Uvarov method. We have obtained closed forms of eigenfunctions, energy eigenvalues and compared our results with other present data.     

Generalized Pseudospectral Method for Solving the Time-Dependent Schrodinger Equation Involving the Coulomb Potential

We present an accurate and efficient generalized pseudospectral method for solving the time-dependent Schrodinger equation for atomic systems interacting with intense laser fields. In this method, the time propagation of the wave function is calculated using the well-known second-order split-operator method implemented by the numerically exact, fast transform between the grid and spectral representations. In the grid representation, the radial coordinate is discretized using the Coulomb wave discrete variable representation （CWDVR）, and the angular dependence of the wave function is expanded in the Gauss-Legendre-Fourier grid. In the spectral representation, the wave function is expanded in terms of the eigenfunctions of the field-free zero-order Hamiltonian. Calculations on the high order harmonic generation and ionization dynamics of hydrogen atom in strong laser pulses are presented to demonstrate the accuracy and efficiency of the present method. This new algorithm will be found more computationally attractive than the close-coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.     

Solution of the Dirac Equation for Ring-Shaped Modified Kratzer Potential

We propose a new exactly solvable potential which is formed by modified Kratzer potential plus a new ring-shaped potential η cot2θ/r2. The solutions of the Dirac equation with equal scalar and vector ring-shaped modified Kratzer potential are found by using the Nikiforov-Uvarov method. The nonrelativistic limit of the energy spectrum has been discussed.