Dirac Wave Function Research Articles

Gaussian-Transform for the Dirac Wave Function and its Application to the Multicenter Molecular Integral Over Dirac Wave Functions for Solving the Molecular Matrix Dirac Equation

Gaussian-transform formula is derived for the Dirac wave function. Using it, one can derive the multicenter molecular integral over Dirac wave functions for any physical quantity. As the first application of it, multicenter molecular integrals over Dirac wave functions are derived for the homogeneous charge density distribution model and the Gauss-type charge density distribution model. Such integrals are necessary for solving the gauge-invariant molecular matrix Dirac equation with using the restricted magnetic balance.

Ordinary Muon Capture on 136Ba: Comparative Study Using the Shell Model and pnQRPA

In this work, we present a study of ordinary muon capture (OMC) on 136Ba, the daughter nucleus of 136Xe double beta decay (DBD). The OMC rates at low-lying nuclear states (below 1 MeV of excitation energy) in 136Cs are assessed by using both the interacting shell model (ISM) and proton–neutron quasiparticle random-phase approximation (pnQRPA). We also add chiral two-body (2BC) meson-exchange currents and use an exact Dirac wave function for the captured s-orbital muon. OMC can be viewed as a complementary probe of the wave functions in 136Cs, the intermediate nucleus of the 136Xe DBD. At the same time, OMC can be considered a powerful probe of the effective values of weak axial-type couplings in a 100 MeV momentum exchange region, which is relevant for neutrinoless DBD. The present work represents the first attempt to compare the ISM and pnQRPA results for OMC on a heavy nucleus while also including the exact muon wave function and the 2BC. The sensitivity estimates of the current and future neutrinoless DBD experiments will clearly benefit from future OMC measurements taken using OMC calculations similar to the one presented here.

The noncommutative quantum Hall effect with anomalous magnetic moment in three different relativistic scenarios

In the present paper, we investigate the bound-state solutions of the noncommutative quantum Hall effect with anomalous magnetic moment in three different relativistic scenarios, namely: the Minkowski spacetime (inertial flat case), the spinning cosmic string (CS) spacetime (inertial curved case), and the spinning CS spacetime with noninertial effects (noninertial curved case). In particular, in the first two scenarios, we have an inertial frame, while in the third, we have a rotating frame. With respect to bound-state solutions, we focus primarily on eigenfunctions (Dirac spinor and wave function) and on energy eigenvalues (Landau levels), where we use the flat and curved Dirac equation in polar coordinates to reach such solutions. However, unlike the literature, here we consider a CS with a non-null angular momentum and also the NC of the positions, and therefore, we seek a more general description for the QHE. Once the solutions are obtained, we discuss the influence of all parameters and physical quantities on relativistic energy levels. Finally, we analyze the nonrelativistic limit, and we also compared our problem with other works, where we verified that our results generalize some particular cases of the literature.

Solutions of the Dirac equation with position-dependent mass for q-deformed Pöschl–Teller and Coulomb-like tensor potentials

We consider the Dirac equation with position-dependent mass for a q-deformed Pöschl–Teller potential plus a Coulomb-like tensor interaction in the limits of spin and pseudospin symmetries. Under the condition of spin symmetry, the bound state energy eigenvalue equation and the two radial components of the Dirac wave function in terms of the Jacobi polynomials are obtained approximately for an arbitrary shifted spin-orbit quantum number λκ = κ + H and for any value of the deformation parameter q ≥ 1. In the case of the pseudospin symmetry limit, there are no bound states. The Dirac equation describes a free physical system and the two components of the wave function are expressed in terms of the spherical Bessel functions.

Heuristic Solution to the Conundrum of the <i>Zitterbewegung</i>

Assuming the Dirac wavefunction describes the state of a single particle. We propose that the relation derived by Schrödinger, which contains the Zitterbewegung term, is a position equation for an amplitude modulated wave. Namely, the elementary constituents are amplitude modulated waves. Indeed, we surmise that a second wave is associated with the particle, which corresponds to a signal. At the same time, we interpret that Broglie’s wave corresponds to a carrier. Furthermore, the quantum object is a recording medium and, like in a hologram, information encoded on its surface. We suggest a description and the cause of the Zitterbewegung heretofore never considered regarding the previous assertions. Hereunder, we shall also apply the quantum amplitude modulation interpretation to the single-photon wave function by Bialynicki-Birula. The predictions are testable, thence providing evidence for the proposed hypothesis.

Quantum kinetic theory for unmagnetized and magnetized plasmas

Three different approaches to the treatment of quantum effects in plasmas are reviewed: quantum fluid theory (QFT), phase-space kinetic theory (PKT) and quantum plasmadynamics (QPD). The simplest form of QFT is analogous to a nonrelativistic fluid model for an unmagnetized plasma with a potential electric field, $$\phi $$ . The wave nature of the electron is included through the so-called Bohm term, as in the Madelung equations. The simplest form of PKT is based on the Wigner function and the kinetic equation that it satisfies in a quasi-classical $$\mathbf{p}$$ – $$\mathbf{x}$$ phase space. Further development of PKT involves including additional effects piecemeal. The electron spin is included through a classical model for a spin vector, $$\mathbf{s}$$ , with the phase space extended to include $$\mathbf{s}$$ . The inclusion of degeneracy is straightforward. Electromagnetic effects are described by including the vector potential, $$\mathbf{A}$$ , in Schrodinger’s equation. It is argued that further extensions, to a magnetized plasma and to include relativistic effects, raise conceptual difficulties concerning the phase-space approach. In a magnetic field, the electron (Landau) states are discrete, whereas the kinetic equation in PKT involves a derivative with respect to $$\mathbf{p}$$ . The relativistic case is based on Dirac’s equation, and to derive a Vlasov-like equation for electrons requires excluding the positron and virtual-pair contributions to the Dirac wavefunction, which cannot be achieved exactly. Moreover, specific spin states require a specific choice of spin operator in Dirac’s theory, whereas the Pauli matrices define the only (vectorial) spin operator in PKT. The Dirac wavefunctions for two spin operators are derived and shown to approximate the eigenstates of the Pauli theory only in the nonrelativistic limit. QPD is an exact theory, based on quantum electrodynamics, in which kinetic processes are described using Feynman diagrams. The presence of the plasma is taken into account through a statistical average of the electron propagator, analogous to the use of thermal Green functions. QPD is introduced for the unmagnetized case, and the generalization to include a background magnetic field is presented. It is shown how QPD is used to derive the linear response tensor for an electron gas in the relativistic quantum case for both unmagnetized and magnetized plasmas. A magnetized vacuum is shown to have response tensors analogous to a plasma, allowing processes (such as one-photon pair creation and photon splitting) that are forbidden in an unmagnetized vacuum. The various quantum effects that may be relevant to a plasma are summarized, and their possible application to laboratory and astrophysical plasmas are discussed briefly.

On the Boundary Conditions for the 1D Weyl–Majorana Particle in a Box

In (1+1) space-time dimensions, we can have two particles that are Weyl and Majorana particles at the same time---1D Weyl-Majorana particles. That is, the right-chiral and left-chiral parts of the two-component Dirac wave function that satisfies the Majorana condition, in the Weyl representation, describe these particles, and each satisfies their own Majorana condition. Naturally, the nonzero component of each of these two two-component wave functions satisfies a Weyl equation. We investigate and discuss this issue and demonstrate that for a 1D Weyl-Majorana particle in a box, the nonzero components, and therefore the chiral wave functions, only admit the periodic and antiperiodic boundary conditions. From the latter two boundary conditions, we can only construct four boundary conditions for the entire Dirac wave function. Then, we demonstrate that these four boundary conditions are also included within the most general set of self-adjoint boundary conditions for a 1D Majorana particle in a box.

Unified discrete mechanics III: the hyperincursive discrete Klein-Gordon equation bifurcates to the 4 incursive discrete Majorana and Dirac equations and to the 16 Proca equations

This paper presents the second order hyperincursive discrete Klein-Gordon equation in three spatial dimensions. This discrete Klein-Gordon equation bifurcates to 4 incursive discrete equations. We present the 4 incursive discrete Dubois-Ord-Mann real equations and the corresponding first order partial differential equations. In three spatial dimensions, the two oscillators given by these equations are now entangled with one spatial dimension, so the 4 equations form a whole. Then, we deduce the 4 incursive discrete Dubois-Majorana real equations and the corresponding first order partial differential equations. Next, these Dubois-Majorana real equations are presented in the generic form of the Dirac 4-spinors equation. In making a change in the indexes of the 4 functions in the Dubois-Majorana equations, the 4 first order partial differential equations become identical to the Majorana real 4-spinors equations. Then, we demonstrate that the Majorana equations bifurcate to the 8 real Dirac first order partial differential equations that are transformed to the original Dirac 4-spinors equations. Then, we give the 4 incursive discrete Dirac 4-spinors equations. Finally, we show that there are 16 discrete functions associated to the space and time symmetric discrete Klein-Gordon equation. This is in agreement with the Proca thesis on the 16 components of the Dirac wave function in 4 groups of 4 equations. In this paper, we restricted our derivation of the Majorana and Dirac equations to the first group of 4 equations depending on 4 functions.

Investigating how students relate inner products and quantum probabilities

After instruction, many students lack a functional understanding of quantum states and inner products that would allow them to translate between Dirac notation and wave function representation.

DiracSolver: A tool for solving the Dirac equation

Advantageous numerical methods for solving the Dirac equations are derived. They are based on different stochastic optimization techniques, namely the Genetic algorithms, the Particle Swarm Optimization and the Simulated Annealing method, their use of which is favored fromintuitive, practical, and theoretical arguments. Towards this end, we optimize appropriate parametric expressions representing the radial Dirac wave functions by employing methods that minimize multi parametric expressions in several physical applications. As a concrete application, we calculate the small (bottom) and large (top) components of the Dirac wave function for a bound muon orbiting around a very heavy (complex) nuclear system (the 208Pb nucleus), but the new approach may effectively be applied in other complex atomic, nuclear and molecular systems. Program summaryProgram Title: DiracSolverProgram Files doi: http://dx.doi.org/10.17632/7hv9mtdvmp.1Licensing provisions: GPLv3Programming language: GNU-C++Nature of problem: The software tackles the problem of solving the Dirac equation using stochastic optimization methods.Solution method: The software utilizes three stochastic optimization methods for the solution of Dirac equation in the form of neural networks. The used methods are: genetic algorithm, particle swarm optimization and simulated annealing.

Small amplitude solitary waves in the Dirac-Maxwell system

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form \begin{document}$φ(x,ω)e^{-iω t}$\end{document} , with \begin{document}$ω∈(-m,ω_ *)$\end{document} for some \begin{document}$ω_ *>-m$\end{document} . The solutions satisfy \begin{document}$φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$\end{document} , and are small amplitude in the sense that \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$\end{document} and \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$\end{document} . The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.

Relativistic multireference coupled-cluster theory based on a B -spline basis: Application to atomic francium

In this paper, we report the relativistic Fock space multireference coupled-cluster method for atomic structure calculations. We use the no-pair Dirac-Coulomb-Breit Hamiltonian, together with a finite $B$-spline basis set to expand the large and small components of the Dirac wave function. Our method is applied to calculate ionization energies, reduced matrix elements, lifetimes, and polarizabilities for many states of atomic francium. To evaluate uncertainties of our results and investigate the role of electron correlation effects, we carry out calculations using approximated models at different levels. The quality of our calculations is assessed by comparing with available experimental results, showing a good agreement. In addition, the tune-out wavelengths of the ground state in the range of 340--800 nm, the magic wavelengths for the transition $7s\ensuremath{-}8s$ in the range of 800--1500 nm and the transition $7s\ensuremath{-}7p$ in the range of 600--1500 nm are determined by evaluating the dynamic polarizabilities of the $7s, 8s$, and $7p$ states for a linearly polarized light. These tune-out and magic wavelengths may be useful for laser cooling and trapping of the Fr atom, and for related high-precision trapping measurements.

On the Hamiltonian and energy operators in a curved spacetime, especially for a Dirac particle

The definition of the Hamiltonian operator H for a general wave equation in a general spacetime is discussed. We recall that H depends on the coordinate system merely through the corresponding reference frame. When the wave equation involves a gauge choice and the gauge change is time-dependent, H asan operator depends on the gauge choice. This dependence extends to the energy operator E, which is the Hermitian part of H. We distinguish between this ambiguity issue of E and the one that occurs due to a mere change of the “representation” (e.g. transforming the Dirac wave function from the “Dirac representation” to a “Foldy-Wouthuy senre presentation”). We also assert that the energy operator ought to be well defined in a given reference frame at a given time, e.g. by comparing the situation for this operator with the main features of the energy for a classical Hamiltonian particle.

Recent progresses on the pseudospin symmetry in single particle resonant states

The pseudospin symmetry (PSS) is a relativistic dynamical symmetry directly connected with the small component of the nucleon Dirac wave function. Much effort has been made to study this symmetry in bound states. Recently, a rigorous justification of the PSS in single particle resonant states was achieved by examining the asymptotic behaviors of the radial Dirac wave functions: The PSS in single particle resonant states in nuclei is conserved exactly when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign. Several issues related to the exact conservation and breaking mechanism of the PSS in single particle resonances were investigated by employing spherical square well potentials in which the PSS breaking part can be well isolated in the Jost function. A threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners were found when the depth of the square well potential varies from zero to a finite value.

Wave functions and eigenvalues of charge carriers in a nanotube in a neighborhood of the Dirac point in the presence of a longitudinal electric field

Based on the Hamiltonian for charge carriers in carbon nanotubes with finite lengths, we obtain eigenvalues and eigenfunctions in a neighborhood of the Dirac points (wave functions written analogously to the two-component Dirac wave function are expressed in terms of Hermite polynomials, and the spectrum is equidistant) in the presence of a longitudinal electric field. We express the solution in terms of the Hermite functions in the case of carbon nanotubes with infinite lengths. Based on the obtained wave function for an elongated nanotube, we consider the problem of determining the coefficient of charge carrier transport through the nanotube. The results of finding the transport coefficient can also be applied to other nanoparticles, in particular, to carbon chains and nanotapes. We propose to use the eigenvalues and eigenfunctions of nanotubes with finite lengths to consider the problem of radiation generation in a nonlinear medium based on an array of such noninteracting nanotubes.

Ionization probabilities in low-energy heavy-ion collisions

Relativistic calculations of differential ionization probability in the U91+(1s)–U collision are performed in the monopole approximation. The method employs the active electron approximation, in which only the active electron of the H-like ion participates in the excitation and ionization processes while the passive electrons of the neutral atom provide a screening potential. The time-dependent Dirac wave function of the active electron is represented as a linear combination of the eigenstates of the unperturbed Hamiltonian localized at the H-like ion.

Algebra of physical space and the geometric spacetime solution of Dirac's equation

AbstractThe algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, the Dirac Hamiltonian can be presented as the scalar product of the electron's four‐momentum and Dirac's four‐vector of gamma matrices, $({\gamma_0},\vec \gamma)$, the latter of which is a Clifford algebra. We show here that a geometric spacetime or four‐space solution of Dirac's equation conforms to the principles of APS, an early example of which is Schroedinger's solution of Dirac's equation for a free electron, which exhibits Zitterbewegung. In a four‐space solution the spacetime coordinates, $\vec r$ and the scaled time ct, are treated on an equal footing as physical observables to avoid any suggestion of a preferred frame of reference. The geometric spacetime theory is studied here for the Coulomb problem. The positive‐energy spectrum of states is found to be identical within numerical error to that of standard Dirac's theory, but the wave function exhibits Zitterbewegung. It is shown analytically how the geometric spacetime solution can be reduced to the standard solution of Dirac's equation, in which Zitterbewegung is absent. The rigor of APS and of its conforming geometric spacetime solution provide strong support for further investigation into the physical interpretation of the geometric spacetime Dirac's wave function and Zitterbewegung. © 2012 Wiley Periodicals, Inc.

FOUR-VECTOR VERSUS FOUR-SCALAR REPRESENTATION OF THE DIRAC WAVE FUNCTION

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

Relativistic calculations of theK-Kcharge transfer andK-vacancy production probabilities in low-energy ion-atom collisions

The previously developed technique for evaluation of charge-transfer and electron-excitation processes in low-energy heavy-ion collisions [I.I. Tupitsyn et al., Phys. Rev. A 82, 042701(2010)] is extended to collisions of ions with neutral atoms. The method employs the active electron approximation, in which only the active electron participates in the charge transfer and excitation processes while the passive electrons provide the screening DFT potential. The time-dependent Dirac wave function of the active electron is represented as a linear combination of atomic-like Dirac-Fock-Sturm orbitals, localized at the ions (atoms). The screening DFT potential is calculated using the overlapping densities of each ions (atoms), derived from the atomic orbitals of the passive electrons. The atomic orbitals are generated by solving numerically the one-center Dirac-Fock and Dirac-Fock-Sturm equations by means of a finite-difference approach with the potential taken as the sum of the exact reference ion (atom) Dirac-Fock potential and of the Coulomb potential from the other ion within the monopole approximation. The method developed is used to calculate the K-K charge transfer and K-vacancy production probabilties for the Ne$(1s^2 2s^2 2p^6)$ -- F$^{8+}(1s)$ collisions at the F$^{8+}(1s)$ projectile energies 130 keV/u and 230 keV/u. The obtained results are compared with experimental data and other theoretical calculations. The K-K charge transfer and K-vacancy production probabilities are also calculated for the Xe -- Xe$^{53+}(1s)$ collision.

Dirac particle in a spherical scalar potential well

In this paper we investigate a solution of the Dirac equation for a spin-$\frac{1}{2}$ particle in a scalar potential well with full spherical symmetry. The energy eigenvalues for the quark particle in ${s}_{1/2}$ states (with $\ensuremath{\kappa}=\ensuremath{-}1$) and ${p}_{1/2}$ states (with $\ensuremath{\kappa}=1$) are calculated. We also study the continuous Dirac wave function for a quark in such a potential, which is not necessarily infinite. Our results, at infinite limit, are in good agreement with the MIT bag model. We make some remarks about the sharpness value of the wave function on the wall. This model, for finite values of potential, also could serve as an effective model for the nucleus where $U(r)$ is the effective single particle potential.