Nonrelativistic Approximation Research Articles

A Nonrelativistic Approximation for Energy Spectrum of a Vector Particle with Anomalous Magnetic Moment in the Coulomb Field

Recently, transition to the nonrelativistic limit was performed for the simplified description of a vector particle with anomalous magnetic moment in the Coulomb field. The problem was reduced to a single radial second order equation in the case of a minimal value of the total angular momentum. However, the exact energy spectrum was not obtained. In the present paper, the quantization rule within the framework of the WKB approximation modified by Langer is used. This approximation gives the exact expression for the energy in the Coulomb problem without anomalous magnetic moment. The dependence of discrete energy levels on anomalous magnetic moment is presented.

Momentum: QFT, Quantum Black Holes, and Some Cosmological Implications

The paper studies the general Pauli-like equation for a Dirac fermions doublet on the background of an external non-Abelian monopole field. The variables separation has been fulfilled, the non-relativistic approximation for the radial systems has been derived. For the case of a minimal value of the conserved quantum number j = 0, the Pauli equation has been obtained in the form of one second-order differential equation. In the case j > 0, the problem has been reduced to the system of two coupled second order equations. In Bogomol'nyi-Prasad-Sommerfeld approximation, this system of equations has been solved in terms of hypergeometric functions.

Electron-impact excitation of the 5varvec{^2textbf{S}_{1/2} rightarrow 5^2textbf{P}_{1/2}} and 5varvec{^2textbf{P}_{3/2}} transitions in rubidium by 40 eV electrons: theory and experiment

We report on a series of detailed Breit-Pauli and Dirac B-spline R-matrix (DBSR) differential cross section (DCS) calculations for excitation of the 5,^2textrm{S}_{1/2} rightarrow 5,^2textrm{P}_{1/2} and 5,^2textrm{S}_{1/2}rightarrow 5,^2textrm{P}_{3/2} states in rubidium by 40 eV incident electrons. The early BP computations shown here were carried out with both 5 states and 12 states, while the DBSR models coupled 150 and 325 states, respectively. We also report corresponding results from a limited set of DCS measurements on the unresolved 5,^2textrm{P}_{1/2,3/2} states, with the experimental data being restricted to the scattered electron angular range 2–10^circ . Typically, good agreement is found between our calculated and measured DCS for excitation of the unresolved 5,^2textrm{P}_{1/2,3/2} states, with best accord being found between the DBSR predictions and the measured data. The present theoretical and experimental results are also compared with predictions from earlier 40 eV calculations using the nonrelativistic Distorted-Wave Born Approximation and a Relativistic Distorted-Wave model.Graphic abstract

Adiabatic evolution of the self-interacting axion field around rotating black holes

Abstract Ultra-light axion fields, motivated by string theory, form a large condensate (axion cloud) around rotating black holes through superradiant instability. Several effects due to the axion cloud, such as the spin-down of black holes and the emission of monochromatic gravitational waves, open a new window to search for axions by astrophysical observations. When the axion self-interaction is considered, the evolution of clouds is altered significantly, and an explosive phenomenon called a bosenova can happen. Thus, it is necessary to understand the precise evolution of self-interacting clouds for the detection of axions by astrophysical observations. In this paper, we propose a new method to track the whole process of the growth of self-interacting axion clouds employing the adiabatic approximation. We emphasize that our method relies neither on the non-relativistic approximation nor on perturbative treatment of the self-interaction, which is often used in the literature. Our main finding is that the evolution of clouds in the strongly self-interacting regime depends on the strength of the gravitational coupling between the axion and the black hole. For a large coupling, the cloud evolves into a quasi-stationary state where the superradiant energy gain is balanced with the energy dissipation to infinity by the self-interaction. On the other hand, when one decreases the size of the coupling, clouds become unstable at some energy, which would be interpreted as the onset of a bosenova.

Predictions of exclusive ϒ photoproduction at the LHC and future colliders

The cross section for exclusive $\mathrm{\ensuremath{\Upsilon}}$ ultraperipheral photoproduction at present and future colliders is determined using the low $x$ gluon parton distribution function extracted from an analysis of exclusive $J/\ensuremath{\psi}$ measurements performed at the Hadron-Electron Ring Accelerator and the Large Hadron Collider. Predictions are given at next-to-leading order in collinear factorization over a wide $\ensuremath{\gamma}p$ centre-of-mass energy range, calculated assuming the nonrelativistic approximation for the $\mathrm{\ensuremath{\Upsilon}}$ wave function, and with skewing corrections incorporated.

Examining a Fifth Force Application by Using Dilaton Model and Padmanabhan Inflaton Scalar Field in Early Universe to Generate GW and Gravitons

On page 17 of a book on Modified Gravity by Li and Koyama, there is a discussion of how to obtain a Fifth force by an allegedly non-relativistic approximation with a force proportional to minus the spatial derivative of a scalar field. If the scalar field says for an inflaton, as presented by Padmanabhan only depends upon time, of course, this means that no scalar field contributing to a fifth force our proposal in the neighborhood of Planck time is to turn the time into being equal to r/[constant times c]. This is in the neighborhood of Planck time. Then having done that, consider the initially Plank regime inflaton field as being spatially varying and from there apply a fifth force as a way to help initiate black hole production and possibly Gravitons.

Fresh look at the effects of gravitational tidal forces on a freely-falling quantum particle

In this paper, we take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schrödinger equation for the particle by starting from the fully relativistic Klein–Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schrödinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepacket’s width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose–Einstein condensate or a superfluid in free fall using the nonlinear Gross–Pitaevskii equation.

Spin 3/2 particle: Fradkin theory, nonrelativistic approximation

The well-known relativistic wave equation for a spin 3/2 particle proposed by Pauli and Fierz is based on the use of the wave function with the transformation properties of vector-bispinor. Less known is the Fradkin theory based on the vector-bispinor wave function as well. At the vanishing Fradkin parameter Λ, this equation reduces to the Pauli – Fierz equation. To clarify the physical meaning of the additional parameter, in the present paper the nonrelativistic approximation in the Fradkin equation is studied, at this we take into account the presence of external electromagnetic fields. With the use of the technique of projective operators, we decompose the wave function into big and small constituents, and then derive a generalized nonrelativistic equation for a 16-component wave function. It is shown that when preserving only the terms of first order in the Fradkin parameter Λ after transition to 4 independent components of the nonrelativistic wave function there arises the ordinary nonrelativistic equation for the Pauli – Fierz theory without any additional interaction with electromagnetic fields. When preserving the terms of second order in parameter Λ, we obtain a 4-component nonrelativistic equation with additional interaction; however, only with the magnetic field. This interaction is quadratic in magnetic field components and governed by six 4-dimensional matrices. So the Fradkin theory may be understood as relevant to a particle with magnetic quadrupole moment.

Relativistic Calculations of the Chemical Properties of the Superheavy Element with Z = 119 and Its Homologues

Relativistic calculations of the electronic structure of the superheavy element of the eighth period $-$ eka-francium ($Z=119$) and its homologues, which form the group of alkali metals, are performed in the framework of the configuration-interaction method and many-body perturbation theory using the basis of the Dirac-Fock-Sturm orbitals (DFS). The obtained values of the ionization potentials, electron affinities, and root-mean-square radii are compared with the corresponding values calculated within the non-relativistic approximation. A comparison with the available experimental data and the results of previous theoretical calculations is given as well. The analysis of the obtained results indicates a significant influence of the relativistic effects for the francium and eka-francium atoms, which leads to a violation of the monotonic behaviour of the listed above chemical properties as a function of the alkaline-element atomic number. In addition, the quantum electrodynamics corrections to the ionization potentials are evaluated by employing the model Lamb-shift operator (QEDMOD).

A wavefunction description for a localized quantum particle in curved spacetimes

We reduce Dirac’s spinor formalism for a spin 1/2 particle to a complex wavefunction description in curved spacetimes. We consider a localized fermionic particle in curved spacetimes and perform an expansion in terms of the acceleration and curvature around the center of mass of the system, generalizing the results of Parker (1980 Phys. Rev. Lett. 44 1559), Parker (1980 Phys. Rev. D 22 1922–1934). Under a non-relativistic approximation, one obtains a quantum description in a Hilbert space of complex wavefunctions defined in the rest space of the system. The wavefunction of the particle then evolves according to a modified Schrödinger equation associated with a symmetric Hamiltonian. When compared to the standard Schrödinger equation for a wavefunction, we obtain corrections in terms of the acceleration of the system’s center of mass and curvature of spacetime along its trajectory. In summary, we provide a formalism for the use of a complex wavefunction to describe a localized quantum particle in curved spacetimes.

Decoherence from general relativity

It is of great interest to explore matter in nontrivial quantum arrangements, including Schrodinger cat-like states. Such states are sensitive to decoherence from their environment. Recently, in Ref. [1] we computed the rate of decoherence of a piece of superposed matter that primarily only interacts gravitationally, a dark-matter-Schrodinger-cat-state (DMSCS), within the nonrelativistic approximation. In this work we improve this to a general relativistic analysis. We firstly derive a single particle relativistic Schrodinger equation for a probe particle that passes through the DMSCS; the interaction is provided by the weak field metric of general relativity from the source. For a static DMSCS we find a neat generalization of our previous results. We then turn to the interesting new case of a time dependent DMSCS, which can be provided by a coherently oscillating axion field leading to superposed time dependent oscillations in the metric; a truly quantum-general relativistic phenomenon. We use scattering theory to derive the decoherence rate in all these cases. When the DMSCS is in a superposition of distinct density profiles, we find that the decoherence rate can be appreciable. We then consider the novel special case in which the density is not in a superposition, but the phase of its field oscillation is; this is a property that cannot be decohered within the nonrelativistic framework. We find that if the probe particle and/or the DMSCS's velocity dispersion is slow, then the rate of decoherence of the phase is exponentially suppressed. However, if both the probe and the DMSCS's velocity dispersion are relativistic, then the phase can decohere more rapidly. As applications, we find that diffuse galactic axions with superposed phases are robust against decoherence, while dense boson stars and regions near black hole horizons are not, and we discuss implications for experiment.

Finding Bc(3S) states via their strong decays

The experimentally known Bc states are all below open bottom-charm threshold, which experience three main decay modes, and all induced by weak interaction. In this work, we investigate the mass spectrum and strong decays of the Bc(3S) states, which just above the threshold, in the Bethe-Salpeter formalism and P03 model. The numerical estimation gives M(Bc(31S0))=7273MeV, M(Bc⁎(33S1))=7304MeV, Γ(Bc(31S0)→B⁎D)=26.02−2.21+2.33MeV, Γ(Bc⁎(33S1)→BD)=3.39−0.26+0.27MeV, Γ(Bc⁎(33S1)→B⁎D)=14.77−1.33+1.40MeV and Γ(Bc⁎(33S1)→BD⁎)=6.14−0.54+0.58MeV. Compared with previous studies in non-relativistic approximation, our results indicate that the relativistic effects are notable in Bc(3S) exclusive strong decays. According to the results, we suggest to find the Bc(3S) states in their hadronic decays to B and D mesons in experiment, like the LHCb.

Theoretical angular dependence of L3 sub-shell X-rays following photoionization

Angular distribution studies are made for L3 sub-shell X-rays of high Z elements; W, Pt, Hg, Pb, and U in the angular span 0°–180° at incident photon energies; near L3 edge, where considerable anisotropy is anticipated and at 60 keV, the energy of experimental interest from standard laboratory source Am-241. Angular distribution of X-rays originating from L3 sub-shell vacancies for each element is derived from theoretical alignment parameter already evaluated under non-relativistic dipole approximation (NDA) in a point Coulomb potential (A20) and in a screened Coulomb potential (A20*) that accommodates the effect of screened nuclear drag on electron wave function, and analytical perturbation theory (A20**). The normalized Lℓ and Lα intensity distributions show more anisotropy at L3 threshold than at 60 keV for all the elements and under all theoretical models. Besides this, the anisotropy for Lℓ is larger than Lα which comprises the unresolved Lα1 and Lα2 lines with opposite anisotropy. The angular distribution trends are almost similar for all the elements. The anisotropy values at L3 threshold energy are assuredly greater than the reported experimental uncertainties (5–8 %) of X-ray intensities. Anisotropy magnitude is found sensitive to the theoretical models. The present anisotropy trends resemble the results of Zhang et al (Journal of Quantitative Spectroscopy and Radiative Transfer, 244 (2020) 106844), however, the values are not quite same.

Релятивистские расчеты химических свойств сверхтяжелого элемента с Z=119 и его гомологов

Relativistic calculations of the electronic structure of the superheavy element of the eighth period - eka-francium (Z=119) and its homologues, which form the group of alkali metals, are performed in the framework of the configuration-interaction method and many-body perturbation theory using the basis of the Dirac-Fock-Sturm orbitals (DFS). The obtained values of the ionization potentials, electron affinities, and root-mean-square radii are compared with the corresponding values calculated within the non-relativistic approximation. A comparison with the available experimental data and the results of previous theoretical calculations is given as well. The analysis of the obtained results indicates a significant influence of the relativistic effects for the francium and eka-francium atoms, which leads to a violation of the monotonic behaviour of the listed above chemical properties as a function of the alkaline-element atomic number. In addition, the quantum electrodynamics corrections to the ionization potentials are evaluated by employing the model Lamb-shift operator (QEDMOD).

Pauli approximation for a vector particle with anomalous magnetic moment in an external Coulomb field

Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.

On the two-dimensional Schrödinger operator with an attractive potential of the Bessel–Macdonald type

We analyze the Schrödinger operator in two-dimensions with an attractive potential given by a Bessel–Macdonald function. This operator is derived in the non-relativistic approximation of planar quantum electrodynamics (QED3) models as a framework for evaluation of two-quasiparticle scattering potentials. The analysis is motivated keeping in mind the fact that parity-preserving QED3 models can provide a possible explanation for the behavior of superconductors. Initially, we study the self-adjointness and spectral properties of the Schrödinger operator modeling the non-relativistic approximation of these QED3 models. Then, by using Setô-type estimates, an estimate is derived of the number of two-particle bound states which depends directly on the value of the effective coupling constant, C, for any value of the angular momentum. In fact, this result in connection with the condition that guarantees the self-adjointness of the Schrödinger operator shows that there can always be a large number of two-quasiparticle bound states in planar quantum electrodynamics models. In particular, we show the existence of an isolated two-quasiparticle bound state if the effective coupling constant C∈(0,2) in case of zero angular momentum. To the best of our knowledge, this result has not yet been addressed in the literature. Additionally, we obtain an explicit estimate for the energy gap of two-quasiparticle bound states which might be applied to high-Tcs-wave Cooper-type superconductors as well as to s-wave electron–polaron–electron–polaron bound states (bipolarons) in mass-gap graphene systems.

Spin 3/2 particle: Puali – Fierz theory, non-relativistic approximation

The relativistic wave equation is well-known for a spin 3/2 particle proposed by W. E. Pauli and M. E. Fierz and based on the 16-component wave function with the transformation properties of the vector-bispinor. In this paper, we investigated the nonrelativistic approximation in this theory. Starting with the first-order equation formalism and representation of Pauli – Fierz equation in the Petras basis, also applying the method of generalized Kronecker symbols and elements of the complete matrix algebras, and decomposing the wave function into large and small nonrelativistic constituents with the help of projective operators, we have derived a Pauli-like equation for the 4-component wave function describing the non-relativistic particle with a 3/2 spin.

Spin 3/2 particle: Puali – Fierz and Fradkin models, interaction with external fields

In the frame of the general Gel’fand – Yaglom formalism, the Fradkin theory for a spin 3/2 particle in presence of external fields is investigated. Applying the standard requirements of relativistic invariance, P-symmetry, existence of a Lagrangian for the model, we derive a set of spinor equations, first in absence of external fields. The wave function consists of a bispinor and a vector-bispinor. It is shown that in absence of external fields the Fradkin model is reduced to the Pauli – Fierz theory. Taking into account the presence of external electromagnetic fields, the Fradkin theory can be turned to the minimal form of the equation for the main bispinor. This equation contains an additional interaction term governed by the electromagnetic tensor F αβ . Meanwhile, there appears a parameter in the Fradkin equation related to any characteristic of the particle additional to its charge. The theory is generalized for taking into account the pseudo-Riemannian space-time geometry. In this case, the Fradkin equation contains an additional interaction term, governed by the Ricci tensor R αβ . If the electric charge of the particle is zero, the Fradkin model remains correct and describes a neutral spin 3/2 particle of the Majorana type interacting nonminimally with the geometrical background through the Ricci tensor. To clarify the meaning of the additional physical characteristics underlying the Fradkin model in contrast to the Pauli – Fierz one we have considered nonrelativistic approximation for both theories in presence of an external uniform magnetic field, and found respective energy spectra. The structure of the ninrelativistic Fradkin equation permits to consider such an additional parameter as polarizability.

A plane wave theory of photoionization using the dipole acceleration form: predictions concerning circular dichroism in photoelectron angular distributions at high energies

Photoelectron angular distributions for randomly oriented atoms and molecules are characterized by the linear and circular dichroic parameters β and b 1, respectively. In the region where the photoelectron kinetic energy is several 100 eV, the values of b 1 and β are expected to be constants. In this study, we show that β for all hydrogen atom orbitals other than s orbitals approach values between 0 to 0.5 at the high-energy limit in the nonrelativistic dipole approximation and that the values can be obtained by the plane wave approximation using the dipole acceleration form. These limiting values are smaller than the widely accepted values for molecules. We discuss the implications of this result for the circular dichroism in photoelectron angular distributions of linear molecules.

The non-ideal theory of the Aharonov\u2013Bohm effect

Elay Shech and John Earman have recently argued that the common topological interpretation of the Aharonov–Bohm (AB) effect is unsatisfactory because it fails to justify idealizations that it presupposes. In particular, they argue that an adequate account of the AB effect must address the role of boundary conditions in certain ideal cases of the effect. In this paper I defend the topological interpretation against their criticisms. I consider three types of idealization that might arise in treatments of the effect. First, Shech takes the AB effect to involve an idealization in the form of a singular limit, analogous to the thermodynamic limit in statistical mechanics. But, I argue, the AB effect itself features no singular limits, so it doesn’t involve idealizations in this sense. Second, I argue that Shech and Earman’s emphasis on the role of boundary conditions in the AB effect is misplaced. The idealizations that are useful in connecting the theoretical description of the AB effect to experiment do interact with facts about boundary conditions, but none of these idealizations are presupposed by the topological interpretation of the effect. Indeed, the boundary conditions for which Shech and demands justification are incompatible with some instances of the AB effect, so the topological interpretation ought not justify them. Finally, I address the role of the non-relativistic approximation usually presumed in discussions of the AB effect. This approximation is essential if—as the topological interpretation supposes—the AB effect constrains and justifies a relativistic theory of the electromagnetic interaction. In this case the ends justify the means. So the topological view presupposes no unjustified idealizations.