Spin-1/2 relativistic particle in a magnetic field in NC phase space

This work provides an accurate study of the spin-1/2 relativistic particle in a magnetic field in NC phase space. By detailed calculation we find that the Dirac equation of the relativistic particle in a magnetic field in noncommutative space has similar behaviour to what happens in the Landau problem in commutative space even if an exact map does not exist. By solving the Dirac equation in NC phase space, we not only obtain the energy level of the spin-1/2 relativistic particle in a magnetic field in NC phase space but also explicitly offer some additional terms related to the momentum-momentum non-commutativity.     

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.     

Scattering and bound states of a relativistic neutral spin-1/2 particle in a magnetic field

By using the Dirac equation we have investigated the scattering and bound states of a relativistic spin-1/2 particle in the system ofN magnetic barriers (or magnetic wells) andN magnetic delta functions. Using theM-matrix method we have studied tunnelling with spin and band structure.     

Dirac Equation at Finite Temperature

In this paper, we propose finite temperature Dirac equation, which can describe the quantum systems in an arbitrary temperature for a relativistic particle of spin-1/2. When the temperature T=0, it become Dirac equation. With the equation, we can study the relativistic quantum systems in an arbitrary temperature.     

Quantum mechanics of ‘free’ spin-1/2 particles in an expanding universe

This paper deals with the gravi-quantum mechanical interaction on the level of the first quantisation and in the framework of a metric theory of gravitation (no field quantisation). The interaction is introduced by embedding the quantum mechanics of the otherwise unaffected (i.e. free) spin-1/2 particle in the given curved space-time of the 3-flat expanding Robertson-Walker universe. The metric acts thereby as an external field. The corresponding Hilbert space formalism is established in interpreting the generally covariant theory of the Dirac field in the Riemann space in question as the Dirac representation of the spin-1/2 particle in the Schrödinger picture. The evolution operator is then extracted out of the general relativistic Dirac equation, while contractions of the symmetric energy momentum tensor with the tetrad vectors of the reference system lead to the operators of energy, linear momentum and total angular momentum. The temporal behaviour of the corresponding expectation values is calculated.     

Aharonov–Casher effect for spin-1 particles in a non-commutative space

In this work, the Aharonov–Casher (AC) phase is calculated for spin-1 particles in a non-commutative space. The AC phase has previously been calculated from the Dirac equation in a non-commutative space using a gauge-like technique. In the spin-1 case, we use the Kemmer equation to calculate the phase in a similar manner. It is shown that the holonomy receives non-trivial kinematical corrections. By comparing the new result with the already known spin-1/2 case, one may conjecture a generalized formula for the corrections to holonomy for higher spins. PACS 02.40.Gh; 03.65.Pm     

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.     

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 Aharonov–Casher effect for spin-1 particles in non-commutative quantum mechanics

By using a generalized Bopp’s shift formulation, instead of the star product method, we investigate the Aharonov–Casher (AC) effect for a spin-1 neutral particle in non-commutative (NC) quantum mechanics. After solving the Kemmer equations both on a non-commutative space and a non-commutative phase space, we obtain the corrections to the topological phase of the AC effect for a spin-1 neutral particle both on a NC space and a NC phase space. PACS 02.40.Gh, 11.10.Nx, 03.65.-w     

Mesoscopic superposition states in relativistic Landau levels

We show that a linear superposition of mesoscopic states in relativistic Landau levels can be built when an external magnetic field couples to a relativistic spin 1/2 charged particle. Under suitable initial conditions, the associated Dirac equation produces unitarily superpositions of coherent states involving the particle orbital quanta in a well-defined mesoscopic regime. We demonstrate that these mesoscopic superpositions have a purely relativistic origin and disappear in the nonrelativistic limit.     

Dirac equation and quantum relativistic effects in a single trapped ion

We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space.We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field.By solving the Klein-Gordon equation in NC phase space,we obtain the energy levels of the Klein-Gordon oscillators,where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space. We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field. By solving the Klein-Gordon equation in NC phase space, we obtain the energy levels of the Klein-Gordon oscillators, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

Relativistic quantum mechanics of fermions

We present a modification of the Dirac equation that allows us to formulate a relativistic quantum mechanics for spin-1/2 fermions in an external electromagnetic field, with a probabilistic interpretation similar to that in nonrelativistic quantum mechanics and based on an indefinite charge density. We find that stationary states cannot be interpreted in this manner, and we replace them by quasistationary states. We also include a general discussion of the difficulties and possible generalizations of this approach.     

Behavior of a spin-1/2 massive charged particle in Schwarzschild immersed in an electromagnetic universe

The Dirac equation is considered in a spacetime that represents a Schwarzschild metric coupled to a uniform external electromagnetic field. Due to the presence of electromagnetic field from the surroundings, the interaction with the spin-1/2 massive charged particle is considered. The equations of the spin-1/2 massive charged particle are separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular equations obtained are similar to the Schwarzschild geometry. For the radial equations we manage to obtain the one dimensional Schrödinger-type wave equations with effective potentials. Finally, we study the behavior of the potentials by plotting them as a function of radial distance and expose the effect of the external parameter, charge and the frequency of the particle on them.     

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pschl-Teller(tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ± 1)r-2.In view of spin and pseudo-spin(p-spin) symmetries,the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method(AIM).We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ.The non-relativistic limit is also obtained.     

无穷深势阱中相对论粒子的矩阵元及其经典近似

对于无穷深势阱中自旋为０(满足Klein-Gordon方程)和自旋为1/2(满足Dirac方程)的相对论粒子, 分别计算了坐标、动量以及速度算符的矩阵元. 在大量子数极限下, 这些矩阵元给出相应的经典物理量(这里是狭义相对论中的有关量), 并且满足正确的经典关系. 从而表明, Heisenberg对应原理对这样的相对论体系也适用. 关键词： 无穷深势阱 Klein-Gordon方程 Dirac方程 Heisenberg对应原理     

Berry联络与吴－杨单极势

通过研究一半整数自旋值的中性粒子在球对称磁场中的Ｂｏｒｎ－Ｏｐｐｅｎｈｅｉｍｅｒ方程，发现在自旋空间Ｂｅｒｒｙ联络呈非阿贝尔形式．在绝热近似下，Ｂｅｒｒｙ联络相当于阿贝尔形式的吴－杨单极场．由于拓扑的非平庸性，利用纤维丛中截面的概念研究了位形空间的动力学，发现中心势场中粒子的角动量和能量均取量子化值，但数值发生移动，这是纯几何起源的现象．     

Nonlinear quantum mechanics as weyl geometry of a classical statistical ensemble

We derive nonlinear relativistic and non-relativistic wave equations for spin-0 and 1/2 particles. For a suitable choice of coupling constants, the equations become linear and Weyl gauge invariant in the spin-0 case. The Dirac particle is much more subtle. When a suitable gauge is chosen and, when the Compton wavelength of the particle is much larger than Planck's length, we recover the standard Dirac equation. Nonlinear corrections to the Schrödinger equation are obtained and these appear as the first-order relativistic corrections to the non-relativistic Hamilton-Jacobi equation. Consequently, we construct nonbilinear homogeneous realizations of anapproximate Galilean symmetry. We put forth the idea that not only a modification of quantum mechanics might be necessary in order to accommodate gravity, but quantum mechanics itself might have a geometrical origin with Planck's constant as the coupling between matter and curvature.1. We thank L. Boya for this remark.2. If we wish to have nodes for stationary states then we must require that has an inflection point at the node, i.e., ² is zero at such node.3. I. Bialynicki-Biruli and J. Mycielski,Ann. Phys. (N. Y.) 100, 62–93 (1976).