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.     

包含非中心电耦极矩的环状非谐振子势场赝自旋对称性的三对角化表示

提出了一个包含非中心电耦极矩分量的环状非谐振子势模型,在能够负载Dirac波动算子三对角化表示的完全平方可积L~2空间讨论了这一势场的赝自旋对称性.利用三对角化矩阵方案,使得求解Dirac方程转换为寻求波函数展开系数满足的三项递推关系式.角向波函数和径向波函数分别以Jacobi多项式和Laguerre多项式表示.由径向分量展开系数递推关系式的对角化条件得到束缚态的能量谱,显示出这一势模型具有严格的赝自旋对称性     

Solution of the Dirac equation in the tridiagonal representation with pseudospin symmetry for an anharmonic oscillator and electric dipole ring-shaped potential

An anharmonic oscillatory potential is proposed in which a noncentral electric dipole is included. The pseudospin symmetry for this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three-term recursion relation for the expansion coefficients of the wavefunctions (both angular and radial) are presented. The angular/radial wavefunction is written in terms of Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by the diagonalization of the radial recursion relation. The algebraic properties of the energy equation are also discussed, showing the exact pseudospin symmetry.     

Solutions to the Dirac equation for symmetric and asymmetric trigonometric Rosen-Morse potential using SUSYQM

In this paper, we calculate the energy spectra and the corresponding wavefunction for the symmetric and asymmetric trigonometric Rosen-Morse potential of the Dirac equation within the framework of spin and pseudospin symmetry limits including the tensor interaction using the supersymmetric quantum mechanics (SUSYQM) formalism. We have also reported some numerical results and figures to show the effect of tensor interaction.     

用Laplace变换求Hartmann势的精确解及波函数的递推关系

本文求解了在球坐标下Hartmann势的Schrdinger方程,得到了能量方程和归一化的波函数.用Laplace变换使径向的二阶微分方程退化为一阶微分方程,直接积分后用级数展开,应用Laplace逆变换得出本征函数.讨论了径向本征函数的像函数的递推关系,从而得出径向波函数的递推关系.     

库仑势加新环形势的相对论束缚态

在标量势等于矢量势的条件下，获得了库仑势加新环形势的Klein-Gordon方程和Dirac方程的束缚态的精确解. 对于Klein-Gordon方程，获得了精确的能谱方程和归一化的波函数. 对于Dirac方程，给出了精确的能谱方程和归一化的旋量波函数. 关键词： 库仑势加新环形势 束缚态 精确解     

Dirac方程的数值解法及其在原子总能量计算中的应用

中心势近似下径向Dirac方程的求解是相对论性原子(离子)结构计算的基础.本文通过相对论性方程中径向波函数大分量与非相对论方程径向波函数的类比,提出了径向Dirac方程的一种数值解法.为了验证数值解法的精度和可靠性,首先将数值结果与类氢势作用下的解析解进行比较.然后,将这种算法扩展到基于解析势的相对论性原子结构计算中,并将计算出的总能量与实验结果和其他方法得到的结果进行对比.     

环形非球谐振子标量势与矢量势的Klein-Gordon和Dirac方程的束缚态

通过求解环形非球谐振子势的 Klein-Gordon 和 Dirac方程,给出了束缚态波函数与能量方程的精确解,并指出非相对论Schr?dinger方程与具有相等标量势和矢量势的Klein-Gordon 方程以及Dirac方程之间具有数学结构上的等价性。     

Solution of the Dirac equation by separation of variables in spherical coordinates for a large class of non-central electromagnetic potentials

A systematic and intuitive approach for the separation of variables of the three-dimensional Dirac equation in spherical coordinates is presented. Using this approach, we consider coupling of the Dirac spinor to electromagnetic four-vector potential that satisfies the Lorentz gauge. The space components of the potential have angular (non-central) dependence such that the Dirac equation becomes separable in all coordinates. We obtain exact solutions for a class of three-parameter static electromagnetic potential whose time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wave functions are obtained. The Aharonov–Bohm and magnetic monopole potentials are included in these solutions.     

新环状非球谐振子势的Dirac方程束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下, 给出了Dirac 方程的束缚态解.通过分离变量得到Dirac方程相应的角向方程和径向方程,得出了用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数;获得了精确的束缚态能谱方程并对结果作适当讨论与结论。     

Approximate Relativistic Bound State Solutions of the Tietz–Hua Rotating Oscillator for Any κ-State

Approximate analytical solutions of the Dirac equation with Tietz–Hua (TH) potential are obtained for arbitrary spin–orbit quantum number κ using the Pekeris approximation scheme to deal with the spin–orbit coupling terms κ(κ ± 1)r ^?2. In the presence of exact spin and pseudo-spin symmetric limitation, the bound state energy eigenvalues and associated two-component wave functions of the Dirac particle moving in the field of attractive and repulsive TH potential are obtained using the parametric generalization of the Nikiforov–Uvarov method. The cases of the Morse potential, the generalized Morse potential and non-relativistic limits are studied.     

An extended class of L-series solutions of the wave equation

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.     

新环状非球谐振子势的Dirac方程束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下, 给出了Dirac 方程的束缚态解.通过分离变量得到Dirac方程相应的角向方程和径向方程,得出了用广义连带勒让德多项式表示的归一化角向波函数和用合流超几何函数表示的归一化径向波函数;获得了精确的束缚态能谱方程并对结果作适当讨论与结论.     

Simple Proof for Global Existence of Bohmian Trajectories

We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schrödinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide an alternative proof of the known global existence result for spinless Schrödinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result is conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.     

Pseudospin Symmetry in Position-Dependent Mass Dirac-Coulomb Problem by Using Laplace Transform and Convolution Integral

The exact pseudospin symmetry solutions of Dirac equation with position-dependent mass (PDM) Coulomb potential in the presence of Colulomb-like tensor potential are obtained by using Laplace transform (LT) approach. The energy eigenvalue equation of the Dirac particles is found and some numerical results are given. By using Laplace convolution integral, the corresponding radial wave functions are presented in terms of confluent hypergeometric functions.     

一类环状非球谐振子势场中相对论粒子的束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下，给出了其Klein-Gordon方程和Dirac方程的束缚态解. Klein-Gordon方程的θ角向波函数以超几何函数表示，径向波函数可用合流超几何函数或广义拉盖尔多项式表示，能谱方程由径向波函数满足的束缚态边界条件得到. Dirac方程的旋量波函数可用Klein-Gordon方程的解构造. 关键词： 环状非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

Triality and Quantization of Singularities in Massive Fermion

It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has closk relation with the. triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefunction round any closed curve with the possibility of there being singularities in our vector potential will lead to the law of quantization of physical constants including the mass. The triality properties of Dirac's spinors are studied and it leads to a double covering vector representation of Dirac spinor field. It is proved that massive Dirac equation in the bosonic representation is self-dual.     

Wave Functions for Time-Dependent Dirac Equation under GUP

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.     

Exact Pseudospin Symmetric Solution of the Dirac Equation for Pseudoharmonic Potential in the Presence of Tensor Potential

Under the pseudospin symmetry, we obtain exact solution of the Dirac equation for the pseudoharmonic potential in the presence of the tensor potential with arbitrary spin–orbit coupling quantum number κ. The energy eigenvalue equation of the Dirac particles is found and the corresponding radial wave functions are presented in terms of confluent hypergeometric functions. We investigate the tensor potential dependence of the energy of the each state in the pseudospin doublet. It is shown that degeneracy between members of the pseudospin doublet is removed by tensor interaction. Furthermore, the radial node structure of the Dirac spinor is discussed.     

具有自旋轨道耦合项的Manning-Rosen势的赝自旋对称解

近似研究了具有任意自旋轨道耦合项的Manning-Rosen势的Dirac方程。在赝自旋对称的条件下获得了关联的二分量的旋量及相应的能量方程。最后,对 和Hulthén 势两类特殊情况进行了简单 讨论。