Wigner function for the Dirac oscillator in spinor space

The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space. Secondly, the matrix form of the Wigner function is proven to support the Dirac equation. Thirdly, by solving the Dirac equation, energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained.     

The Three-Dimensional Dirac Oscillator in the Presence of Aharonov-Bohm and Magnetic Monopole Potentials

No Heading We study the Dirac equation in 3+1 dimensions with non-minimal coupling to an isotropic radial three-vector potential and in the presence of a static electromagnetic potential. The space component of the electromagnetic potential has angular (non-central) dependence such that the Dirac equation separates completely in spherical coordinates. We obtain solutions for the case where the three-vector potential is linear in the radial coordinate (Dirac oscillator) and the time component of the electro-magnetic potential vanishes. The relativistic energy spectrum and spinor eigenfunctions are obtained.     

非对易相空间中的狄拉克振子的能级和Wigner函数

非对易几何、弦论和圈量子引力理论的发展,使非对易空间受到越来越多的关注.非对易量子理论不同于平常的量子理论,它是弦尺度下的特殊的物理效应,处理非对易量子力学问题需要特殊方法.本文首先介绍了Moyal方程与Wigner函数,利用Moyal-Weyl乘法与Bopp变换将H(x,p)变换成^H(^x,^p),考虑坐标—坐标、动量—动量的非对易性,实现对非对易相空间中星乘本征方程的求解.并利用非对易相空间量子力学的代数关系,讨论了非对易相空间中狄拉克振子的Wigner函数和能级,研究结果发现非对易相空间中狄拉克振子的能级明显依赖于非对易参数.     

Two-Dimensional Linear Dependencies on the Coordinate Time-Dependent Interaction in Relativistic Non-Commutative Phase Space

In this paper, the Non-Commutative phase space and Dirac equation, time-dependent Dirac oscillator are introduced. After presenting the desire general form of a two-dimensional linear dependency on the coordinate time-dependent potential, the Dirac equation is written in terms of Non-Commutative phase space parameters and solved in a general form by using Lewis-Riesenfield invariant method and the time-dependent invariant of Dirac equation with two-dimensional linear dependency on the coordinate time-dependent potential in Non-Commutative phase space has been constructed, then such latter operations are done for time-dependent Dirac oscillator. In order to solve the differential equation of wave function time evolution for Dirac equation and time-dependent Dirac oscillator which are partial differential equation some appropriate ordinary physical problems have been studied and at the end the interesting result has been achieved.     

Pseudospin Symmetry for a Ring-Shaped Non-spherical Harmonic Oscillator Potential

The pseudospin symmetry for a ring-shaped non-spherical harmonic oscillator potential is investigated by solving the Dirac equation with equal mixture of scalar and vector potentials with opposite signs. The normalized spinor wave function and energy equation are obtained, the algebraic property of the energy equation and some particular cases are also discussed.     

The Wigner Functions for a Spin-1/2 Relativistic Particle in the Presence of Magnetic Field

This paper provides a study of Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field. Since the Dirac equation is described as a matrix equation, it is necessary to describe the Wigner function as a matrix function in phase space. What’s more, this function is then proved to satisfy the Dirac equation with ⋆-product. Finally, by solving the ⋆-product Dirac equation, the energy levels as well as the Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field are obtained.     

Pseudospin symmetry and a double ring-shaped spherical harmonic oscillator potential

A new double ring-shaped spherical harmonic oscillator potential is presented. The pseudospin symmetry in this system is investigated by solving the Dirac equation with equal mixture of scalar and vector potentials with opposite signs. The normalized spinor wave function and energy equation are obtained and some particular cases are discussed.      

相对论带电费密子Liouville方程

作为密度矩阵一种形式的Wigner函数是量子相空间里的分布。用它描述相对论费密子时,它的通常表达形式为4×4矩阵函数。本文得到相对论带电费密子的2×2矩阵形式的Wigner函数以及它所满足的Liouville方程。这一方程与量子电动力学里带电费密子满足的Dirac方程完全等价。在描述中能核碰撞的Walecka模型里,当只有矢量介子(或标量介于取平均场近似)时,核子满足一定形式的Dirac方程。本文的方程也与之等价。还证明了(2×2)Wigner函数与相对论费密子的波函数在描述量子体系上起着同样的作用。量子体系的可观察量的全部知识都可以通过这里的Wigner函数得到。 关键词：     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics ofa particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

非对易相空间Dirac oscillator的研究

在极坐标系中研究了非对易相空间中的Dirac oscillator问题.研究显示:系统的波函数可以表示为合流超几何函数,而非对易相空间Dirac oscillator的量子行为类似于朗道问题.最后,对η=0和对易极限两种特殊情况进行了简单讨论.     

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.     

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.     

三维谐振子Wigner函数的星乘解

Wigner函数作为相空间中的一个准概率分布函数,也是密度矩阵的特殊表示形式,具有十分重要的物理意义。首先介绍了Wigner函数的性质及其计算方法,然后利用星本征方程(Moyal方程)计算了三维谐振子的Wigner函数。最后讨论了在相空间中描述声子与电子(或光子)相互作用的方法,并得到了跃迁几率在相空间中所满足的方程。     

相对论性非球谐振子势场中的赝自旋对称性

求解了非球谐振子势场中1/2自旋粒子满足的Dirac方程,Dirac哈密顿量包含有标量非球谐振子势S(r)和矢量非球谐振子势V(r).在Σ(r)=S(r)+V(r)=0和Δ(r)=V(r)-S(r)=0的条件下,解析地得到了Dirac旋量波函数的束缚态解和能谱方程,结果表明非球谐振子势 关键词： 非球谐振子势 Dirac方程 赝自旋对称性 束缚态     

Bounded Solutions of the Dirac Equation with a PT-symmetric Kink-Like Vector Potential in Two-Dimensional Space-Time

The (1+1)-dimensional Dirac equation with a PT-symmetric kink-like vector potential is investigated. By using the basic concepts of the supersymmetric WKB formalism and the function analysis method, we solve exactly the Dirac equation and obtain the bound-state energy levels and two-component spinor components. The PT-symmetric kink-like potential is not Hermitian and absent of bound states in the context of non-relativistic Schrödinger equation, but it possesses two sets of real discrete relativistic energy spectra in the context of the Dirac theory. When the PT symmetry is spontaneously broken, two sets of real energy spectra come into complex conjugate.     

A Spinor from Semiderivatives

Fractional derivatives have been known since the time of Leibniz and have been used in various branches of physics. The present paper shows how they can be used to generate a spinor field, much as the gradient operator generates a vector field. These spinor fields are zero kinetic energy solutions to the Dirac equation.     

Analytical Expressions of Matrix Elements of Physical Quantities for Dirac Oscillator

The analytical expressions of the matrix elements for physical quantities are obtained for the Dirac oscillator in two and three spatial dimensions. Their behaviour for the case of operator's square is discussed in details. The two-dimensional Dirac oscillator has similar behavior to that for three-dimensional one.     

A General Method for Obtaining Degenerate Solutions to the Dirac and Weyl Equations and a Discussion on the Experimental Detection of Degenerate States

In this work, a general method is described for obtaining degenerate solutions of the Dirac equation, corresponding to an infinite number of electromagnetic 4-potentials and fields, which are explicitly calculated. More specifically, using four arbitrary real functions, one can automatically construct a spinor that satisfies the Dirac equation for an infinite number of electromagnetic 4-potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with nonzero mass, the degenerate spinors should be localized, both in space and time. The method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, two experimental methods are proposed for detecting the presence of degenerate states.     

Investigation of Fermions in Non-commutative Space by Considering Kratzer Potential

The two-dimensional Dirac equation for a fermion moving under Kratzer potential in the presence of an external magnetic field is analytically being solved for the energy eigenvalues and eigenfunctions. Subsequently, we have obtained the Wigner function corresponding to the eigenfunctions.