The Darboux Transformation for the One-Dimensional Stationary Dirac Equation with a Scalar Potential

This paper deals with the Darboux transformation for the Dirac equation with a scalar-type potential. Formulas are derived for the potential difference and for the solutions of the transformed equations. The relationship between the Darboux transforms for Dirac and Schrödinger equations is analyzed. New transparent potentials and a potential with a Coulomb asymptotics are obtained as examples.     

Nonlinear Schrödinger equation with a Dirac delta potential: finite difference method*

The nonlinear Schrödinger equation with a Dirac delta potential is considered in this paper. It is noted that the equation can be transformed into an equation with a drift-admitting jump. Then following the procedure proposed in Chen and Deng (2018 Phys. Rev. E 98 033302), a new second-order finite difference scheme is developed, which is justified by numerical examples.     

Relativistic theory of a hydrogen atom in a superstrong magnetic field

The Dirac equation for a hydrogen atom in a static, uniform magnetic field is transformed by a modified FWT transformation into a mixed form; relativistic for perpendicular motion and semi-relativistic for parallel motion. In this way the adiabatic approximation can easily be obtained.     

The Darboux Transformation for the One-Dimensional Stationary Dirac Equation with a Pseudoscalar Potential

This paper consider the Darboux transform for the Dirac equation with a pseudoscalar-type potential. Formulas for the potential difference and for the solutions of the transformed equation are derived. The relationship between the Darboux transforms for Dirac and Schrödinger equations is analyzed. New potentials with the spectrum of a relativistic harmonic oscillator are obtained as examples.     

Solutions of the Maxwell equations and photon wave functions

Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.     

On a relation between Maxwell and Dirac theories

It is shown that the Dirac equation can be written in a form similar to Maxwell equations, where the Maxwell tensor is written as a bilinear expression of the Dirac field and the current is a simple function of the external potential and the Dirac field. Similarly, the Maxwell equations can be written as a self-coupled Dirac equation where the potential is a simple function of the Dirac field itself. It is illustrated by examples how the new formalism helps to find solutions of the coupled field equations.     

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.     

高频强场中Kramers-Henneberger变换在狄拉克方程中的应用

在强激光场中,通过Kramers-Henneberger(KH)变换和高频近似可以将狄拉克方程的求解从一个含时问题转换成一个稳态问题来研究。以氢原子为例,详细研究了相对论框架下该稳态问题中的有效库仑势的特点,并和非相对论框架下的结果进行了比较。研究发现,由于相对论有效质量的引入,有效库仑势的分布范围并不像非相对论条件下随着激光场强度的增强单调变大,而是增加到一定值后不再变化。另外,通过比较偶极近似和非偶极近似下有效库仑势的分布,发现对高频强场,偶极近似不再适用。     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

On the Mechanism of Fermion-Boson Transformation

We study a fermion-boson transformation. Our approach is based on the 3 × 3 equations which are subequations of both the Dirac and Duffin-Kemmer-Petiau equations and thus provide a link between these equations. We show that solutions of the free Dirac equation can be converted to solutions of spin- 0 Duffin-Kemmer-Petiau equation and vice versa. Mechanism of this transition assumes existence of a constant spinor.     

A transformation method of generating exact analytic solutions of the Schrödinger equation

A transformation method is presented which consists of a coordinate transformation and a functional transformation that allow generation of normalized exact analytic bound-state solutions of the Schrödinger equation, starting from an analytically solved quantum problem. The coordinate transformation is the basic transformation, which is supplemented by the functional transformation so that one can choose the dimension of the space of the transformed system. By repeated application of the method, it is possible to generate a number of solved quantum problems in the case that the original quantum system has a multiterm potential. It is shown that the eigenfunction of the transformed system can be easily normalized in most cases.     

电磁学与电动力学中的磁单极-Ⅱ

本系列文章一共4篇,在电磁学和电动力学框架内用尽量科普的方式分别介绍磁单极的若干奇特性质.本篇文章主要介绍狄拉克磁单极是如何展示矢量势的规范变换的.我们首先简要介绍规范变换与规范对称性及狄拉克磁单极与狄拉克弦,然后讨论狄拉克磁单极与规范变换的联系.我们显式演示狄拉克弦摆动产生的规范变换,弦摆动区域对场点所张的立体角正比于规范变换的变换函数.磁偶极子则可以由两个无穷靠近的正反狄拉克磁单极构成.相应两条狄拉克弦位置的变化都对应磁偶极子矢量势的规范变换,特别当两条弦重合时弦效应相互抵消,只剩下纯的磁偶极子.传统的由磁偶极子产生的矢量势的规范变换则可以图像化为组成磁偶极子的正反狄拉克磁单极的狄拉克弦的摆动.我们显式地计算了位于坐标原点弦为直线的狄拉克磁单极,并进一步构造了没有奇异的吴大峻-杨振宁磁单极.     

Quantum equations from Brownian motions

Classical Schr?dinger and Dirac equations have been derived from Brownian motions of a particle, it has been shown that the classical Schr?dinger equation can be transformed to usual Schr?dinger Quantum equation on applying Heisenberg uncertainty principle between position and momentum while Dirac Quantum equation follows from it’s classical counter part on applying Heisenberg uncertainty principle between energy and time without applying any analytical continuation.     

Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any <Emphasis Type="Italic">κ</Emphasis>-value

The Dirac equation, with position-dependent mass, is solved approximately for the generalized Hulthén potential with any spin-orbit quantum number κ. Solutions are obtained by using an appropriate coordinate transformation, reducing the effective mass Dirac equation to a Schrödinger-like differential equation. The Nikiforov-Uvarov method is used in the calculations to obtain energy eigenvalues and the corresponding wave functions. Numerical results are compared with those given in the literature. Analytical results are also obtained for the case of constant mass and the results are in good agreement with the literature.     

Hawking thermal radiation of the dirac particle in spherically symmetric nonstatic space-time

The dynamical properties of Dirac spinor particles in a spherically symmetric nonstatic space-time are studied. The explicit representative of the four-component wave function of Dirac particles is obtained. The Dirac equation can be reduced to the standard form of the wave equation near the event horizon by the proper coordinate transformation. The event horizon location and Hawking radiation temperature are obtained.     

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.     

Connections on Clifford bundles and the Dirac operator

It is shown, how, in the setting of Clifford bundles, the spin connection (or Dirac operator) may be obtained by averaging the Levi-Civita connection (or Kähler-Dirac operator) over the finite group generated by an orthonormal frame of the base manifold.The familiar covariance of the Dirac equation under a simultaneous transformation of spinors and matrix representations emerges very naturally in this scheme, which can also be applied when the manifold does not possess a spin structure.     

Relativistic Confinement of Neutral Fermions with Partially Exactly Solvable and Exactly Solvable PT-Symmetric Potentials in the Presence of Position-Dependent Mass

The relativistic problems of neutral fermions subject to a new partially exactly solvable PT-symmetric potential and an exactly solvable PT-symmetric hyperbolic cosecant potential in 1+1 dimensions are investigated. The Dirac equation with the double-well-like mass distribution in the background of the PT-symmetric vector potential coupling can be mapped into the Schrödinger-like equation with the partially exactly solvable double-well potential. The position-dependent effective mass Dirac equation with the PT-symmetric hyperbolic cosecant potential can be mapped into the Schrödinger-like equation with the exactly solvable modified Pöschl-Teller potential. The real relativistic energy levels and corresponding spinor wavefunctions for the bound states have been given in a closed form.     

The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction

This paper applies the isotopic field-charge spin theory (Darvas in Int. J. Theor. Phys. 50(10):2961–2991, 2011) to the electromagnetic interaction. First, a modified Dirac equation in the presence of a velocity dependent gauge field and isotopic field charges (namely Coulomb and Lorentz type electric charges, as well as gravitational and inertial masses) is derived. This equation is compared with the classical Dirac equation. It is shown that, since the presence of isotopic field-charges would distort the Lorentz invariance of the equation, there is a transformation, which together with the Lorenz transformation restores the invariance of the equation, in accordance with the conservation of the isotopic field-charge spin (Darvas in Concepts Phys. VI 1:3–16, 2009). The paper discusses conclusions derived from the extensions of the Dirac equation. It is shown that in semi-classical approximation the model returns the original Dirac equation, and at significantly relativistic velocities it approaches the Schrödinger equation. Among other conclusions, the clue gives physical meaning to the electric moment. The closing section summarises a few further conclusions and shows a few developments to be discussed in detail in a subsequent paper (Darvas in Int. J. Theor. Phys., 2013).     

Quantum structure emerging from the transformation design of the Dirac equation

It is shown that a quantum structure can be created by a set of chosen constraint conditions that emerge from the transformation design of the Dirac equation in general relativity. As an explanation, the constraints that cause novel bound states with the quantization rule of a 2D Coulomb system are presented. The discussion in this paper provides a systematic way to look for constraints that generate a required quantization rule.