Nonrelativistic Treatment Research Articles

A relativistic approach to teaching electrodynamics: Deriving Maxwell's equations from first principles

This paper presents a novel methodology for teaching classical electrodynamics based on special relativity theory. Traditional approaches often rely heavily on empirical laws, potentially obscuring the fundamental unity of electromagnetic phenomena. We demonstrate how the core principles of electrodynamics, including Maxwell's equations, can be derived from just two fundamental postulates: the principle of relativity and Coulomb's law. By consistently applying relativistic reasoning, we show how key concepts such as the Lorentz force, Biot-Savart law, and electromagnetic induction emerge naturally. This approach provides a logically coherent framework for understanding electrodynamics, emphasizes its inherently relativistic nature, and resolves apparent paradoxes that arise in non-relativistic treatments. We discuss the pedagogical advantages of this method, including its potential to foster deeper conceptual understanding and develop students' theoretical reasoning skills. The paper also addresses practical aspects of implementing this approach in higher education physics curricula and its implications for physics education research.

Toward local Madelung mechanics in spacetime

It has recently been shown that relativistic quantum theory leads to a local interpretation of quantum mechanics wherein the universal wavefunction in configuration space is entirely replaced with an ensemble of local fluid equations in spacetime. For want of a fully relativistic quantum fluid treatment, we develop a model using the nonrelativistic Madelung equations, and obtain conditions for them to be local in spacetime. Every particle in the Madelung fluid is equally real, and has a definite position, momentum, kinetic energy, and potential energy. These are obtained by defining quantum momentum and kinetic energy densities for the fluid and separating the momentum into average and symmetric parts, and kinetic energy into classical kinetic and quantum potential parts. The two types of momentum naturally give rise to a single classical kinetic energy density, which contains the expected kinetic energy, even for stationary states, and we define the reduced quantum potential as the remaining part of the quantum kinetic energy density. We treat the quantum potential as a novel mode of internal energy storage within the fluid particles, which explains most of the nonclassical behavior of the Madelung fluid. For example, we show that in tunneling phenomena, the quantum potential negates the barrier so that nothing prevents the fluid from flowing through. We show how energy flows and transforms in this model, and that enabling local conservation of energy requires defining a quantum potential energy current that flows through the fluid rather than only flowing with it. The nonrelativistic treatment generally contains singularities in the velocity field, which undermines the goal of local dynamics, but we expect a proper relativistic treatment will bound the fluid particle velocities at c.

A Parametric Study of the SASI Comparing General Relativistic and Nonrelativistic Treatments* * Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or

We present numerical results from a parameter study of the standing accretion shock instability (SASI), investigating the impact of general relativity (GR) on the dynamics. Using GR hydrodynamics with GR gravity, and nonrelativistic (NR) hydrodynamics with Newtonian gravity, in an idealized model setting, we vary the initial radius of the shock, and by varying its mass and radius in concert, the proto-neutron star compactness. We investigate four compactnesses expected in a post-bounce core-collapse supernova (CCSN). We find that GR leads to a longer SASI oscillation period, with ratios between the GR and NR cases as large as 1.29 for the highest-compactness suite. We also find that GR leads to a slower SASI growth rate, with ratios between the GR and NR cases as low as 0.47 for the highest-compactness suite. We discuss implications of our results for CCSN simulations.

Non-relativistic treatment of q-deformed modified Pöschel Teller potential via path integral approach

Non-relativistic treatment of q-deformed modified Pöschel Teller potential via path integral approach

Non-relativistic treatment of generalised inverse quadratic Yukawa potential via path integral approach

Non-relativistic treatment of generalised inverse quadratic Yukawa potential via path integral approach

Non-Relativistic Treatment of the 2D Electron System Interacting via Varshni–Shukla Potential Using the Asymptotic Iteration Method

The nonrelativistic treatment of the Varshni–Shukla potential (V–SP) in the presence of magnetic and Aharanov–Bohm fields is carried out using the asymptotic iteration method (AIM). The energy equation and wave function are derived analytically. The energy levels are summed to obtain the partition function, which is employed to derive the expressions for the thermomagnetic properties of the V–SP. These properties are analyzed extensively using graphical representations. It is observed that in the various settings of the analysis, the system shows a diamagnetic characteristic, and the specific heat capacity behavior agrees with the recognized Dulong–Petit law, although some slight anomaly is observed. This irregular behavior could be attributed to a Schottky anomaly. Our findings will be valuable in a variety of fields of physics, including chemical, molecular and condensed matter physics, where our derived models could be applied to study other diatomic molecules and quantum dots, respectively.

Non-uniform magnetic field as a booster for quantum speed limit: faster quantum information processing

We probe the quantum speed limit (QSL) of an electron when it is trapped in a non-uniform magnetic field. We show that the QSL increases to a large value, but within the regime of causality, by choosing a proper variation in magnetic fields. We also probe the dependence of QSL on spin of electron and find that it is higher for spin-down electron in the relativistic regime. This can be useful in achieving faster speed of transmission of quantum information. Further, we use the Bremermann–Bekenstein bound to find a critical magnetic field that bridges the gap between non-relativistic and relativistic treatments and relates to the stability of matter. An analytical framework is developed. We also provide a plausible experimental design to supplement our theory.

Magneto-transport and thermal properties of the Yukawa potential in cosmic string space-time

Yukawa potential has garnered remarkable attention since its proposition. It is applied in condensed matter physics, plasma physics, high energy physics and related areas. Based on this widespread applicability, this study focused on the analysis of the effects of topological defect on the thermal and magnetic properties of this system. Non-relativistic treatment of this potential in the presence of magnetic fields and Aharonov–Bohm is performed using the functional analytical approach (FAA). The energy equation and the wavefunction are analytically derived. The accessible energy levels are summed up to obtain to get the partition function that is used to derive the expressions for the thermo-magnetic properties of the Yukawa potential. These properties are analysed in detail by means of graphical representations. It is observed that in the various settings of the analysis, the system has a diamagnetic characteristic and the behaviour of the heat capacity is in accordance with the recognized law of Dulong–Petit, although some anomalies are observed. This irregular behaviour could be attributed to Schottky anomaly. The results of our research will be helpful in many fields of physics: condensed matter physics, high energy physics, etc.

Dynamical friction of black holes in ultralight dark matter

In this work we derive simple closed-form expressions for the dynamical friction acting on black holes moving through ultralight (scalar field) dark matter, covering both non-relativistic and relativistic black hole speeds. Our derivation is based on long known scattering amplitudes in black hole spacetimes, it includes the effect of black hole spin and can be easily extended to vector and tensor light fields. Our results cover and complement recent numerical and previous non-relativistic treatments of dynamical friction in ultralight dark matter.

Axion as a fuzzy dark matter candidate: proofs in different gauges

Axion as a coherently oscillating massive scalar field is known to behave as a zero-pressure irrotational fluid with characteristic quantum stress on a small scale. In relativistic perturbation theory, the case was proved in the axion-comoving gauge up to fully nonlinear and exact order. Our basic assumption is that the field is oscillating with Compton frequency and the Compton wavelength is smaller than the horizon scale. Here, we revisit the relativistic proof to the linear order in the other gauge conditions. We show that the same equation for density perturbation known in the non-relativistic treatment can be derived in two additional gauge conditions: the zero-shear gauge and the uniform-curvature gauge. The uniform-expansion gauge fails to get the aimed equation, and the quantum stress term is missing in the synchronous gauge. For comparison, we present the relativistic density perturbation equations in the zero-pressure fluid in these gauge conditions. Except for the comoving and the synchronous gauge, the equations strikingly differ from the axion case. We clarify that the relativistic analysis based on time averaging is valid for scales larger than the Compton wavelength. Below the Compton wavelength, the field is not oscillating, and our oscillatory ansatz does not apply. We suggest an equation valid in all scales in the comoving gauge. For comparison, we review the non-relativistic quantum hydrodynamics and present the Schrödinger equation to first-order post-Newtonian expansion in the cosmological context.

Unified formalism for electromagnetic and gravitational probes: Densities

The use of light front coordinates allows a fully relativistic description of a hadron's spatial densities to be obtained. These densities must be two-dimensional and transverse to a chosen spatial direction. We explore their relationship to the three-dimensional, non-relativistic densities, with a focus on densities associated with the energy momentum tensor. The two-dimensional non-relativistic densities can be obtained from the light front densities through a non-relativistic limit, and can subsequently be transformed into three-dimensional non-relativistic densities through an inverse Abel transform. However, this operation is not invertible, and moreover the application of the inverse Abel transform to the light front densities does not produce a physically meaningful result. We additionally find that the Abel transforms of so-called Breit-frame densities generally differ significantly from the true light front densities. Numerical examples are provided to illustrate the various differences between the light front, Breit frame, and non-relativistic treatment of densities.

Nonrelativistic treatment of inversely quadratic Hellmann-Kratzer potential and thermodynamic properties

The study presents approximate analytical solutions of the Schrödinger equation with a newly proposed potential model called Inversely Quadratic Hellmann-Kratzer potential (IQHKP). This potential is a superposition of Inversely Quadratic Hellman potential and Kratzer potential. The energy eigenvalues and corresponding wavefunction are calculated via the formula method. We applied our results to evaluate thermodynamic functions such as vibrational free energy, F, vibrational internal energy, U, vibrational entropy, S, and vibrational specific heat, C. We also reported special cases of importance.

Analysis of the impact of external fields on the energy spectra and thermo-magnetic properties of N 2,I 2,CO,NO and HCl diatomic molecules

This research article is concerned with the non-relativistic treatment of the Hulthen-Kratzer potential (HKP) model. The two-dimensional Schrodinger equation is solved considering the presence of AB and magnetic fields with the HKP model using the functional analysis approach. The energy equation and wave function of the system are obtained as a function of the external fields and potential parameters. The analytic expression for the energy is used to obtain the partition function by means of the Boltzmann-Gibbs statistics and then thermal and magnetic properties of and diatomic molecules are obtained. The impact of these fields on the obtained energy spectra and thermo-magnetic properties of these molecules have been duly analyzed. It has been shown that to control certain behaviours of the system and molecules, external fields are required. Our results are fascinating and have potential applications in condensed matter physics, atomic physics, and chemical physics.

Nonrelativistic treatment of fully-heavy tetraquarks as diquark-antidiquark states

The goal of the present work is to obtain a reliable estimate of the masses of the ground and radially excited states of fully-heavy tetraquark systems. In order to do this, we use a nonrelativistic model of tetraquarks which are assumed to be compact and consist of diquark-antidiquark pairs. This nonrelativistic model is composed of Hulthen potential, a linear confining potential and spin-spin interaction. We computed ground, first, and second radially excited cc{bar{c}}{bar{c}} and bb{bar{b}}{bar{b}} tetraquark masses. It was found that predicted masses of ground states of cc{bar{c}}{bar{c}} and bb{bar{b}}{bar{b}} tetraquarks are significantly higher than the thresholds of the fall-apart decays to the lowest allowed two-meson states. These states should be broad and are thus difficult to observe experimentally. First radially excited states are considerably lower than their corresponding (2S-2S) two-meson thresholds. We hope that our study may be helpful to the experimental search for ground and excited cc{bar{c}}{bar{c}} and bb{bar{b}}{bar{b}} tetraquark states.

Speed limits for radiation-driven SMBH winds

Context.Ultra-fast outflows (UFOs) have become an established feature in analyses of the X-ray spectra of active galactic nuclei (AGN). According to the standard picture, they are launched at accretion disc scales with relativistic velocities, up to 0.3−0.4 times the speed of light. Their high kinetic power is enough to induce an efficient feedback on a galactic scale, possibly contributing to the co-evolution between the central supermassive black hole (SMBH) and the host galaxy. It is, therefore, of paramount importance to gain a full understanding of UFO physics and, in particular, of the forces driving their acceleration and the relation to the accretion flow from which they originate.Aims.In this paper, we investigate the impact of special relativity effects on the radiative pressure exerted onto the outflow. The radiation received by the wind decreases for increasing outflow velocity,v, implying that the standard Eddington limit argument has to be corrected according tov. Due to the limited ability of the radiation to counteract the black hole gravitational attraction, we expect to find lower typical velocities with respect to the non-relativistic scenario.Methods.We integrated the relativistic-corrected outflow equation of motion for a realistic set of starting conditions. We concentrated on a range of ionisations, column densities, and launching radii consistent with those typically estimated for UFOs. We explore a one-dimensional, spherical geometry and a three-dimensional setting with a rotating, thin accretion disc.Results.We find that the inclusion of special relativity effects leads to sizeable differences in the wind dynamics and thatvis reduced up to 50% with respect to the non-relativistic treatment. We compare our results with a sample of UFOs from the literature and we find that the relativistic-corrected velocities are systematically lower than the reported ones, indicating the need for an additional mechanism, such as magnetic driving, to explain the highest velocity components. Finally, we note that these conclusions, derived for AGN winds, are generally applicable.

Nondipole effects in laser-assisted electron scattering

Laser-assisted electron scattering is often described using the electric dipole approximation for the interaction between the electron and the assisting electromagnetic field. Within the dipole approximation, theory predicts a vanishing scattering cross section for processes involving exchange of photons in critical geometries, where the electron momentum transfer vector is perpendicular to the linear polarisation of the external field. In a nonrelativistic spin-free treatment, we consider modifications of the scattering cross section by nondipole contributions to the first order in 1/c in the interaction between the laser field and the electron using the nondipole strong-field-approximation Hamiltonian (Jensen et al 2020 Phys. Rev. A 101 043408). The approach allows for an analytical solution for the wave function of a free electron in the laser field with a nondipole contribution. This wave function leads to an analytical formula for the laser-assisted scattering cross section including nondipole effects. The nondipole corrections depend on the propagation direction of the field relative to the electron momentum transfer vector. The approach gives a finite cross section in scattering geometries, where the dipole approach predicts a vanishing cross section. The theory is illustrated by application to scattering geometries similar to those considered in experiments where large deviations between experimental data and predictions based on the dipole approximation have been reported. In these scattering geometries, the nondipole strong-field-approximation approach suggests an increase in the cross section by orders of magnitude compared to results obtained within the dipole approximation.

Nonrelativistic treatment of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative quantum mechanics

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as [Formula: see text] and [Formula: see text] and neutral atoms such as ([Formula: see text] and [Formula: see text]) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp’s shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers [Formula: see text] in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with [Formula: see text]) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers [Formula: see text], two infinitesimal parameters [Formula: see text], which are induced by (position–position and phase–phase). In addition, the dimensional parameters [Formula: see text] of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms.

Eigensolution and various properties of the screened cosine Kratzer potential in D dimensions via relativistic and non-relativistic treatment

In this article, we proposed screened cosine Kratzer potential (SCKP). Using approximation suggested by Greene–Aldrich, the approximate bound-state solutions of the D-dimensional Klein–Gordon equation for SCKP have been obtained via the generalized Nikiforov–Uvarov method. The bound-state energy eigenvalues and the corresponding normalized wave functions expressed in terms of hypergeometric functions were obtained. From the proposed SCKP model, we recovered different potentials such as screened Kratzer potential, standard Kratzer potential, generalized cosine Yukawa potential, screened Coulomb potential, Coulomb potential, inversely quadratic Yukawa potential and its corresponding energy eigenvalues from obtained energy eigenvalues of the SCKP in both the relativistic and non-relativistic regime. We obtained rotational–vibrational energy for few heterogeneous (LiH, HCl, NO) and homogeneous (H $$_2$$ , I $$_2$$ , O $$_2$$ ) diatomic molecules in three dimensions. The numerical results obtained for LiH, HCl, NO and I $$_2$$ , O $$_2$$ diatomic molecules are in very good agreement with the results previously obtained by others. We obtained various properties of the SCKP such as thermodynamical properties (partition function, vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity and vibrational entropy, information energy $$E(\gamma )$$ , Tsallis entropy T(q), Renyi entropy R(q), Shannon entropy $$ S(\gamma )$$ and Fisher information entropy $$I(\gamma )$$ ) using energy eigenvalues and eigenfunctions, the expressions for the expectation values of the square of inverse of position $$1/r^2$$ , inverse of position 1/r, kinetic energy T, and square of momentum $$p^2$$ via Hellmann–Feynman theorem. We also presented quarkonium mass spectroscopy using energy eigenvalues of the SCKP.

The effect of the three-body force on the β-stable matter in RLOCV framework

The symmetry energy of isospin asymmetric nuclear matter (IANM), the density dependence of symmetry energy and also the equation of state (EOS) of the β-stable matter (BSM) are considered within relativistic corrections in the lowest order constrained variation approach (RLOCV) by using AV18 potential, with and without including Urbana (UIX) three-body force (TBF). The relative proton abundance for a wide range of baryon number density is calculated by including TBF in RLOCV approach. It is observed that if the bare two-body force (2BF) interaction AV18 is used, the relative proton abundance (ρp/ρB) increases by increasing baryon number density which tends to be saturated at high densities in both relativistic and non-relativistic treatments, while including TBF to our potential produces the ρp/ρB with higher values at low densities, which is saturated at the moderate densities in both LOCV and RLOCV frameworks. Moreover, adding the relativistic corrections (RC) to the results of β-stable matter decreases the value of proton abundance. In contrast, including TBF increases the value of ρp/ρB at low baryonic densities, which is saturated at the moderate baryon number densities. The opposite behaviors are also found on the saturation properties of nuclear matter by including TBF and RC to our calculations. It is also shown that the effect of TBF is much larger than that of the RC. Our results agree with other interactions and methods.

BSHF: A program to solve the Hartree–Fock equations for arbitrary central potentials using a B-spline basis

BSHF solves the Hartree–Fock equations in a B-spline basis for atoms, negatively charged ions, and systems of N electrons in arbitrary central potentials. In the B-spline basis the Hartree–Fock integro-differential equations are reduced to a computationally simpler eigenvalue problem. As well as solving this for the ground-state electronic structure self-consistently, the program can calculate discrete and/or continuum excited states of an additional electron or positron in the field of the frozen-target N-electron ground state. It thus provides an effectively complete orthonormal basis that can be used for higher-order many-body theory calculations. Robust and efficient convergence in the self-consistent iterations is achieved by a number of strategies, including by gradually increasing the strength of the electron–electron interaction by scaling the electron charge from a reduced value to its true value. The functionality and operation of the program is described in a tutorial style example. Program summaryProgram Title: BSHFProgram Files doi:http://dx.doi.org/10.17632/fj3y6c58dy.1Code Ocean Capsule:https://doi.org/10.24433/CO.1226817.v2Licensing provisions: GPLv3Programming language: Fortran 90.External routines/libraries: LAPACK.Nature of problem: Self-consistent solution of electronic structure for atoms and electrons in arbitrary central potentials in the Hartree–Fock approximation.Solution method: A B-spline basis is employed that transforms the Hartree–Fock integro-differential equations to a computationally simpler eigenvalue problem. The eigenvalue problem is solved iteratively until self-consistency is achieved.Unusual or notable features:1. Robust and efficient convergence in the self-consistent iterations is achieved by gradually increasing the value of the electron charge from a reduced value to its true value, i.e., increasing the strength of the electron–electron interaction. In this way all orbitals are calculated simultaneously at every iteration of the self-consistency loop, i.e., without the need to successively fill the occupied shells from the core to valence (as most other Hartree–Fock codes require for convergence).2. In addition to atoms and negative ions, the program solves the Hartree–Fock equations for systems of N electrons confined in an arbitrary central potential specified by the user: thus the program can be used to e.g., calculate the structure of electrons confined in harmonic potentials, which are known to approximate the electron gas.3. In addition to calculating the ground state of the N-electron system, the program calculates the discrete and/or continuum excited states for an additional electron or positron in the field of the ‘frozen’ N-electron target. It thus provides an orthonormal basis that can be used as input for many-body theory calculations.Restrictions: The program solves non-relativistic Hartree–Fock equations for spherically symmetric systems only (in modelling systems other than atoms, e.g., harmonically confined electron gas or jellium-type models of clusters, relativistic effects can often be negligible and a non-relativistic treatment is perfectly suitable). For open-shell atoms the central-field approximation is used, i.e., the potential is angularly averaged.