Time-evolution of highly localized positive-energy states of the free Dirac electron

Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.

The velocity of probability transport in quantum mechanics

In his study of the 'time of arrival' problem in the nonrelativistic quantum mechanics of a single particle, Allcock [1] noted that the direction of the probability flux vector is not necessarily the same as that of the mean momentum of a wave packet, even when the packet is composed entirely of plane waves with a common direction of momentum. Packets can be constructed, for example for a particle moving under a constant force, in which probability flows for a finite time in the opposite direction to the momentum. A similar phenomenon occurs for the Dirac electron. The maximum amount of probabilitiy backflow which can occur over a given time interval can be calculated in each case.

STRUCTURE OF THE ELECTROMAGNETIC FIELD ALLOWING EXACT SOLUTION OF THE SCHRODINGER EQUATION IN SUPERPOSITION WITH AN AHARONOV-BOHM FIELD

A charged particle is considered in a complex external electromagnetic field. The field is a superposition of an Aharonov-Bohm field and some additional field. Here we describe all additional fields known up to the present time that allow exact solution of the Schrodinger equation in a complex field.

Colossal anisotropy in diluted magnetic topological insulators

We consider dilute magnetic doping in the surface of a three dimensional topological insulator where a two dimensional Dirac electron gas resides. We find that exchange coupling between magnetic atoms and the Dirac electrons has a strong and peculiar effect on both. First, the exchange-induced single ion magnetic anisotropy is very large and favors off-plane orientation. In the case of a ferromagnetically ordered phase, we find a colossal magnetic anisotropy energy, of the order of the critical temperature. Second, a persistent electronic current circulates around the magnetic atom and, in the case of a ferromagnetic phase, around the edges of the surface.

Solution of the time dependent Dirac-Fock-Slater equation for many-electron collisions systems using a time window method

We present a new scheme to solve the time dependent Dirac-Fock-Slater equation (TDDFS) for heavy many electron ion-atom collision systems. Up to now time independent self consistent molecular orbitals have been used to expand the time dependent wavefunction and rather complicated potential coupling matrix elements have been neglected. Our idea is to minimize the potential coupling by using the time dependent electronic density to generate molecular basis functions. We present the first results for 16 MeV S{^16+} on Ar.

All electron fully relativistic Dirac-Fock calculation for darmstadtium carbide using prolapse free basis set

For the first time, ab inito all electron fully relativistic and correlated Dirac-Fock calculations with prolapse free basis set are reported for a Super Heavy Element. We investigated the relativistic effects on bonding and on some spectroscopic constants for the darmstadtium carbide and our results at DF/CCSD(T) with a prolapse free basis set suggest for R-e, omega(e) and D-e the values of 174 pm, 1114 cm(-1) and 7.29 eV, respectively. These values are very similar to the values for PtC found on literature. It was also found that prolapse free basis set may be important to estimate the dissociation energy using Relativistic 4-components correlated methods. (c) 2007 ELsevier B.V. All rights reserved.

Ab Initio correlated all electron dirac-fock calculations for Eka-Francium Fluoride (E119F)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Ab Initio correlated all electron Dirac-Fock calculations for eka-francium fluoride (E119F)

Resultados obtidos por cálculos Dirac-Fock correlacionados de 4 componentes para o fluoreto do elemento E119 (Eka-Frâncio) com base estável e precisa, livre de prolapso variacional, são reportados neste trabalho. No nível CCSD(T), a distância de equilíbrio R_e, frequência harmônica ω_e e energia de dissociação D_e são 2,432 Å, 354,97 cm_-1 e 116,92 kcal mol_-1, respectivamente. Também são reportados base livre de prolapso variacional de 4 componentes para o elemento 119, uma curva analítica de energia potencial precisa e o espectro vibracional a partir dos dados obtidos no nível CCSD(T). Nossos resultados sugerem que a molécula E119F deva ser menos iônica que seus fluoretos alcalinos homólogos mais leves, em contraste com o senso químico comum baseado nas propriedades periódicas - era de se esperar nesta molécula a ligação química mais iônica possível. Também encontramos que a correção do tipo modelo de carga para negligenciar as integrais do tipo SS resulta em erros insignificantes e acelera os cálculos cerca de 3 vezes no nível CCSD(T) e cerca de 4 vezes no nível DFT/B3LYP.

Photoionization of Co+ and electron-impact excitation of Co2 + using the Dirac R-matrix method

Modelling of massive stars and supernovae (SNe) plays a crucial role in understanding galaxies. From this modelling we can derive fundamental constraints on stellar evolution, mass-loss processes, mixing, and the products of nucleosynthesis. Proper account must be taken of all important processes that populate and depopulate the levels (collisional excitation, de-excitation, ionization, recombination, photoionization, bound–bound processes). For the analysis of Type Ia SNe and core collapse SNe (Types Ib, Ic and II) Fe group elements are particularly important. Unfortunately little data is currently available and most noticeably absent are the photoionization cross-sections for the Fe-peaks which have high abundances in SNe. Important interactions for both photoionization and electron-impact excitation are calculated using the relativistic Dirac atomic R-matrix codes (DARC) for low-ionization stages of Cobalt. All results are calculated up to photon energies of 45 eV and electron energies up to 20 eV. The wavefunction representation of Co III has been generated using GRASP0 by including the dominant 3d⁷, 3d⁶[4s, 4p], 3p⁴3d⁹ and 3p⁶3d⁹ configurations, resulting in 292 fine structure levels. Electron-impact collision strengths and Maxwellian averaged effective collision strengths across a wide range of astrophysically relevant temperatures are computed for Co III. In addition, statistically weighted level-resolved ground and metastable photoionization cross-sections are presented for Co II and compared directly with existing work.

Can we make a Bohmian electron reach the speed of light, at least for one instant?

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light-a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current (psi) over bar gamma(mu)psi is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function psi : R(4) -> C(4) that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of C(4) along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions psi : R(4) -> C(4), Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation. (C) 2010 American Institute of Physics. [doi:10.1063/1.3520529]

Localizing the relativistic electron

A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac's position operator x. It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the four-vector density (rho(x, t), j(x, t)/c) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive energy states. A suitable spin operator can be diagonalized at the same time.

Theoretical study of elastic electron scattering off stable and exotic nuclei

Results for elastic electron scattering by nuclei, calculated with charge densities of Skyrme forces and covariant effective Lagrangians that accurately describe nuclear ground states, are compared against experiment in stable isotopes. Dirac partial-wave calculations are performed with an adapted version of the ELSEPA package. Motivated by the fact that studies of electron scattering off exotic nuclei are intended in future facilities in the commissioned GSI and RIKEN upgrades, we survey the theoretical predictions from neutron-deficient to neutron-rich isotopes in the tin and calcium isotopic chains. The charge densities of a covariant interaction that describes the low-energy electromagnetic structure of the nucleon within the Lagrangian of the theory are used to this end. The study is restricted to medium- and heavy-mass nuclei because the charge densities are computed in mean-field approach. Because the experimental analysis of scattering data commonly involves parameterized charge densities, as a surrogate exercise for the yet unexplored exotic nuclei, we fit our calculated mean-field densities with Helm model distributions. This procedure turns out to be helpful to study the neutron-number variation of the scattering observables and allows us to identify correlations of potential interest among some of these observables within the isotopic chains.

Distorted-wave calculation of cross sections for inner-shell ionization by electron and positron impact

The relativistic distorted-wave Born approximation is used to calculate differential and total cross sections for inner shell ionization of neutral atoms by electron and positron impact. The target atom is described within the independent-electron approximation using the self-consistent Dirac-Fock-Slater potential. The distorting potential for the projectile is also set equal to the Dirac-Fock-Slater potential. For electrons, this guarantees orthogonality of all the orbitals involved and simplifies the calculation of exchange T-matrix elements. The interaction between the projectile and the target electrons is assumed to reduce to the instantaneous Coulomb interaction. The adopted numerical algorithm allows the calculation of differential and total cross sections for projectiles with kinetic energies ranging from the ionization threshold up to about ten times this value. Algorithm accuracy and stability are demonstrated by comparing differential cross sections calculated by our code with the distorting potential set to zero with equivalent results generated by a more robust code that uses the conventional plane-wave Born approximation. Sample calculation results are presented for ionization of K- and L-shells of various elements and compared with the available experimental data.

Optical-model potential for electron and positron elastic scattering by atoms

An optical-model potential for systematic calculations of elastic scattering of electrons and positrons by atoms and positive ions is proposed. The electrostatic interaction is determined from the Dirac-Hartree-Fock self-consistent atomic electron density. In the case of electron projectiles, the exchange interaction is described by means of the local-approximation of Furness and McCarthy. The correlation-polarization potential is obtained by combining the correlation potential derived from the local density approximation with a long-range polarization interaction, which is represented by means of a Buckingham potential with an empirical energy-dependent cutoff parameter. The absorption potential is obtained from the local-density approximation, using the Born-Ochkur approximation and the Lindhard dielectric function to describe the binary collisions with a free-electron gas. The strength of the absorption potential is adjusted by means of an empirical parameter, which has been determined by fitting available absolute elastic differential cross-section data for noble gases and mercury. The Dirac partial-wave analysis with this optical-model potential provides a realistic description of elastic scattering of electrons and positrons with energies in the range from ~100 eV up to ~5 keV. At higher energies, correlation-polarization and absorption corrections are small and the usual static-exchange approximation is sufficiently accurate for most practical purposes.

Electron scattering in isotonic chains as a probe of the proton shell structure of unstable nuclei

Electron scattering on unstable nuclei is planned in future facilities of the GSI and RIKEN upgrades. Motivated by this fact, we study theoretical predictions for elastic electron scattering in the N=82, N=50, and N=14 isotonic chains from very proton-deficient to very proton-rich isotones. We compute the scattering observables by performing Dirac partial-wave calculations. The charge density of the nucleus is obtained with a covariant nuclear mean-field model that accounts for the low-energy electromagnetic structure of the nucleon. For the discussion of the dependence of scattering observables at low-momentum transfer on the gross properties of the charge density, we fit Helm model distributions to the self-consistent mean-field densities. We find that the changes shown by the electric charge form factor along each isotonic chain are strongly correlated with the underlying proton shell structure of the isotones. We conclude that elastic electron scattering experiments on isotones can provide valuable information about the filling order and occupation of the single-particle levels of protons.