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.     

一类相对论性非球谐振子系统的束缚态

给出了具有形式为12r²+A2r²的非球谐振子型标量势和矢量势 的相对论系 统在两种势相等的条件下三维Klein-Gordon方程，二维和三维Dirac方程的s波束缚态解. 关键词： 三维非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

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.     

Spinning-particle model for the Dirac equation and the relativistic Zitterbewegung

We construct the relativistic particle model without Grassmann variables which meets the following requirements. A) Canonical quantization of the model implies the Dirac equation. B) The variable which experiences Zitterbewegung, represents a gauge non-invariant variable in our model. Hence our particle does not experience the undesirable Zitterbewegung. C) In the non-relativistic limit spin is described by three-vector, as it could be expected.     

The equation-transform model for Dirac–Morse problem including Coulomb tensor interaction

The approximate solutions of Dirac equation with Morse potential in the presence of Coulomb-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 obtained. By using convolution integral, the corresponding radial wave functions are presented in terms of confluent hypergeometric functions.     

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.     

Makarov势的Dirac方程的束缚态解

给出了Makarov型标量势与矢量势相等条件下的Dirac方程的束缚态解. Dirac方程的角向方程用因子分解方法求解，在得出角向波函数的过程中，自然地得到了属于同一本征值的不同角向波函数间的递推操作. 径向束缚态波函数用合流超几何函数表示，束缚态的能量方程可由径向波函数满足的边界条件得到. 关键词： Makarov势 Dirac方程 束缚态 因子分解方法     

Derivation of Dirac's Equation from the Evans Wave Equation

The Evans wave equation [1] of general relativity is expressed in spinor form, thus producing the Dirac equation in general relativity. The Dirac equation in special relativity is recovered in the limit of Euclidean or flat spacetime. By deriving the Dirac equation from the Evans equation it is demonstrated that the former originates in a novel metric compatibility condition, a geometrical constraint on the metric vector qused to define the Einstein metric tensor. Contrary to some claims by Ryder, it is shown that the Dirac equation cannot be deduced unequivocally from a Lorentz boost in special relativity. It is shown that the usually accepted method in Clifford algebra and special relativity of equating the outer product of two Pauli spinors to a three-vector in the Pauli basis leads to the paradoxical result X = Y = Z = 0. The method devised in this paper for deriving the Dirac equation from the Evans equation does not use this paradoxical result.     

On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

In this work we study in detail the connection between the solutions to the Dirac and Weyl equations and the associated electromagnetic four-potentials.First,it is proven that all solutions to the Weyl equation are degenerate,in the sense that they correspond to an infinite number of electromagnetic four-potentials.As far as the solutions to the Dirac equation are concerned,it is shown that they can be classified into two classes.The elements of the first class correspond to one and only one four-potential,and are called non-degenerate Dirac solutions.On the other hand,the elements of the second class correspond to an infinite number of four-potentials,and are called degenerate Dirac solutions.Further,it is proven that at least two of these fourpotentials are gauge-inequivalent,corresponding to different electromagnetic fields.In order to illustrate this particularly important result we have studied the degenerate solutions to the forcefree Dirac equation and shown that they correspond to massless particles.We have also provided explicit examples regarding solutions to the force-free Weyl equation and the Weyl equation for a constant magnetic field.In all cases we have calculated the infinite number of different electromagnetic fields corresponding to these solutions.Finally,we have discussed potential applications of our results in cosmology,materials science and nanoelectronics.     

Solution of Dirac equation for Eckart potential and trigonometric Manning Rosen potential using asymptotic iteration method

The Dirac equation for Eckart potential and trigonometric Manning Rosen potential with exact spin symmetry is obtained using an asymptotic iteration method. The combination of the two potentials is substituted into the Dirac equation,then the variables are separated into radial and angular parts. The Dirac equation is solved by using an asymptotic iteration method that can reduce the second order differential equation into a differential equation with substitution variables of hypergeometry type. The relativistic energy is calculated using Matlab 2011. This study is limited to the case of spin symmetry. With the asymptotic iteration method, the energy spectra of the relativistic equations and equations of orbital quantum number l can be obtained, where both are interrelated between quantum numbers. The energy spectrum is also numerically solved using the Matlab software, where the increase in the radial quantum number nr causes the energy to decrease. The radial part and the angular part of the wave function are defined as hypergeometry functions and visualized with Matlab 2011. The results show that the disturbance of a combination of the Eckart potential and trigonometric Manning Rosen potential can change the radial part and the angular part of the wave function.     

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     

The Dirac Equation in the Bertotti-Robinson Space-Time

The Dirac equation is considered in the uniform electromagnetic field space of Bertotti-Robinson with charge coupling. The methods of separation of variables and decoupling are easily achieved. The separated axial equation is reduced to a rare Riccati type of differential equation. The behaviour of potentials, their asymptotic solutions and the conserved currents of the Dirac equation are found.     

Spin and Pseudospin Symmetries of Hellmann Potential with Three Tensor Interactions Using Nikiforov-Uvarov Method

The Dirac equation with Hellmann potential is presented in the presence of Coulomb-like tensor (CLT), Yukawa-like tensor (YLT), and Hulthen-type tensor (HLT) interactions by using Nikiforov-Uvarov method. The bound state energy spectra and the radial wave functions are obtained approximately within the framework of spin and pseu- dospin symmetries limit. We have also reported some numerical results and figures to show the effects of the tensor interactions. Special cases of the potential are also discussed.     

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     

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.     

Dirac Equation from the Hamiltonian and the Case With a Gravitational Field

Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation—in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-spacetime Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the static-gravitational case is not equivalent to the standard (Fock-Weyl) gravitational extension of the Dirac equation.     

Exact solutions of Dirac equation with Pöschl–Teller double-ring-shaped Coulomb potential via Nikiforov–Uvarov method

Exact analytical solutions of the Dirac equation are reported for the Pöschl-Teller double-ring-shaped Coulomb potential. The radial, polar, and azimuthal parts of the Dirac equation are solved by using the Nikiforov-Uvarov method, and exact bound state energy eigenvalues and the corresponding two-component spinor wavefunctions are reported.     

Spin Symmetry in Dirac-Attractive Radial Problem and Tensor Potential

In this paper,we solve the Dirac equation under spin symmetry limit for attractive radial potential including a Coulomb-like tensor interaction.By using the parametric generalization of the Nikiforov-Uvarov method,the energy eigenvalues equation and the corresponding wave functions have been obtained in closed forms.Some numerical results are given too.