Wigner's pseudo-particle relativistic dynamics in external potential field

The Wigner's pseudo-particle formalism has been generalized to describe quantum dynamics of relativistic particle in external potential field. As a simplest application of the developed formalism the time evolution of the 1D relativistic quantum harmonic oscillator been considered. Due to the complex structure of the evolution equation for Wigner function, the only numerical treatment is possible by combining Monte Carlo and molecular dynamics methods. Relativistic dynamics results in appearance of the new physical effects as opposed to non-relativistic case. Interesting is the complete changing of the shape of the momentum and coordinate distribution functions as well as formation of ‘unexpected’ protuberances. To analyze the influence of relativistic effects on average values of quantum operators, the dependencies on time of average momentum, position, their dispersions and energy have been compared for the non-relativistic and relativistic dynamics.     

The uncertainties in radial position and radial momentum of an electron in the non-relativistic hydrogen-like atom

Similar to the case of a simple harmonic oscillator, an increase in azimuthal quantum number l will result in simultaneous decrease in both the uncertainty in radial position and the uncertainty in radial momentum for the same principal quantum number n in the non-relativistic hydrogen-like atom. Thus, in some cases of hydrogen-like atom and in the case of a simple harmonic oscillator, the more precisely the position is determined, the more precisely the momentum is known in that instant, and vice versa.     

High momentum component in the deuteron

The high momentum component in the deuteron, which stems from the short range part of the nucleon-nucleon interaction, is studied in they-scaling function and the structure functionF ₂ of the deuteron. We use not only some non-relativistic wave functions but also relativistic ones. It is shown that the relativistic mechanism or a six-quark state in the nucleon-nucleon interaction yields a large high momentum component.     

Fisher-Shannon plane and statistical complexity of atoms

Using the Hartree-Fock non-relativistic wave functions in the position and momentum spaces, the statistical measure of complexity C, due to López-Ruiz, Mancini, and Calbet for the neutral atoms as well as their monopositive and mononegative ions with atomic number Z=1-54 are reported. In C, given by the product of exponential power Shannon entropy and the average density, the latter is then replaced by the Fisher measure to obtain the Fisher-Shannon plane. Our numerical results suggest that in overall the Fisher-Shannon plane reproduces the trends given by C, with significantly enhanced sensitivity in the position, momentum and the product spaces in all neutral atoms and ions considered.     

Theory of complete orthonormal sets of relativistic tensor wave functions and Slater tensor orbitals of particles with arbitrary spin in position, momentum and four-dimensional spaces

Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the 2(2s+1)-component relativistic tensor wave functions and Slater tensor orbitals in position, momentum and four-dimensional spaces are derived, where s=1/2,1,3/2,2,… . The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic Slater tensor orbitals in position space are also derived.     

The ideal relativistic spinning gas: Polarization and spectra

We study the physics of the ideal relativistic rotating gas at thermodynamical equilibrium and provide analytical expressions of the momentum spectra and polarization vector for the case of massive particles with spin 1/2 and 1. We show that the finite angular momentum J entails an anisotropy in momentum spectra, with particles emitted orthogonally to J having, on average, a larger momentum than along its direction. Unlike in the non-relativistic case, the proper polarization vector turns out not to be aligned with the total angular momentum with a non-trivial momentum dependence.     

Relativistic quantum particle in a homogeneous external field

The model of the relativistic quantum particle in a homogeneous external field is proposed. This model is realized in the one-dimensional relativistic configurational x-space and is described by the finite-difference equation. The momentum p-space in our case is the one-dimensional Lobachevsky space. We have found the wave functions and propagator for the model under study in both x- and p-representations.     

Matter Density and Relativistic Models of Wave Function Collapse

Mathematical models for the stochastic evolution of wave functions that combine the unitary evolution according to the Schrödinger equation and the collapse postulate of quantum theory are well understood for non-relativistic quantum mechanics. Recently, there has been progress in making these models relativistic. But even with a fully relativistic law for the wave function evolution, a problem with relativity remains: Different Lorentz frames may yield conflicting values for the matter density at a space-time point. We propose here a relativistic law for the matter density function. According to our proposal, the matter density function at a space-time point x is obtained from the wave function ψ on the past light cone of x by setting the i-th particle position in |ψ|² equal to x, integrating over the other particle positions, and averaging over i. We show that the predictions that follow from this proposal agree with all known experimental facts.     

Relativistic effects on complexity indexes in atoms in position and momentum spaces

Three different statistical measures of complexity are explored for the atoms He to Ra. The measures are analysed in both position and momentum spaces. Relativistic effects on the complexity indexes are systematically studied. These effects are discussed in terms of the information content factor and the disorder terms of the complexity indexes. Relativistic and non-relativistic complexity indexes are calculated from Optimized Effective Potential densities.     

Estimation of shift parameters of a quantum state

Estimation of shift parameters such as arrival time, phase, angle of rotation, position of a non-relativistic or relativistic particle is considered. An approach from the point of view of quantum estimation theory enables to give a proper definition of the time observable and the position observables of a massless relativistic particle, i.e. observables to which there do not correspond self-adjoint operators. Some new inequalities for estimates of shift parameters are obtained; in particular a rigorous uncertainty relation for coordinates of the photon is established.     

Relativistic quantum particle in a time-dependent homogeneous field

A model of the relativistic quantum particle under the action of a time-dependent force is considered. This exactly solvable model is realized in the one-dimensional relativistic configurational x-space and is described by the finite-difference equation. The momentum p-space is one-dimensional Lobachevsky space. We have explicitly constructed the wave functions and propagators for this model in both x- and p-representations. We have also found a solution of a definite class of partial differential and finite-difference equations, which can be interpreted as the operator identities.     

探测二茂铁外价轨道(e,2e)反应中的扭曲波效应

利用第三代高效率电子动量谱仪,分别在600和1500 eV两种不同入射电子能量下获得了二茂铁(ferrocene)分子外价轨道的电离能谱和电子动量谱的相关实验结果.并利用非相对论与标量相对论密度泛函方法计算出了二茂铁的重叠型和交错型两种不同构象的理论动量谱.两种构象的外价轨道一一对应,理论电子动量谱基本一样.对二茂铁的外价轨道,在低动量区观测到了强烈的扭曲波效应,这与这些轨道主要由铁原子的3d轨道构成有关.通过相对论和非相对论计算结果的比较,表明相对论效应对于二茂铁的外轨道动量分布几乎没有影响. 关键词： 二茂铁分子 电子动量谱 相对论效应 扭曲波效应     

A Numerical Investigation of the Pinning Phenomenon in Quasi-Periodic Frenkel Kontrova Model Under an External Force

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in Benedikter et al. (Arch Rat Mech Anal 221(1): 273–334, 2016). The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation.     

Defining statistical relative complexity measure: Application to diversity in atoms

A statistical relative complexity measure, based on the Kullback-Leibler distance measure defining the relative information and the Carbó quantum similarity index defining the relative disequilibrium is proposed. It is shown that with the specific choice of prior density corresponding to the atom at the beginning of the subshell, this measure reveals the diversity of atoms as the subshells are filled across the periodic table. Numerical tests are reported using the non-relativistic Hartree-Fock as well as the relativistic Dirac-Fock density for all atoms in the periodic table.     

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Jüttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.     

Oscillator strengths in the boron isoelectronic sequence

The quantum defect orbital formalism in both its relativistic and non-relativistic versions has been applied to the calculation of oscillator strengths in the sharp spectral series of some boron-like systems. The results are presented in both tabular and graphical forms.     

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

We present a systematic procedure for constructing mass operators with continuous spectra for a system of particles in a manner consistent with Galilean relativity. These mass operators can be used to construct what may be called point-form Galilean dynamics. As in the relativistic case introduced by Dirac, the point-form dynamics for the Galilean case is characterized by both the Hamiltonian and momenta being altered by interactions. An interesting property of such perturbative terms to the Hamiltonian and momentum operators is that, while having well-defined transformation properties under the Galilei group, they also satisfy Maxwell’s equations. This result is an alternative to the well-known Feynman-Dyson derivation of Maxwell’s equations from non-relativistic quantum physics.     

Gravitational geometric phase in the presence of torsion

We investigate the relativistic and non-relativistic quantum dynamics of a neutral spin-1/2 particle subject to an external electromagnetic field in the presence of a cosmic dislocation. We analyze the explicit contribution of the torsion in the geometric phase acquired in the dynamics of this neutral spinorial particle. We discuss the influence of the torsion in the relativistic geometric phase. Using the Foldy–Wouthuysen approximation, the non-relativistic quantum dynamics is studied and the influence of the torsion on the Aharonov–Casher and He–McKellar–Wilkens effects are discussed. An erratum to this article can be found at