Nonrelativistic Energy Research Articles

Energy depletion and re-acceleration of driver electrons in a plasma-wakefield accelerator

For plasma-wakefield accelerators to fulfill their potential for cost effectiveness, it is essential that their energy-transfer efficiency be maximized. A key aspect of this efficiency is the near-complete transfer of energy, or depletion, from the driver electrons to the plasma wake. Achieving full depletion is limited by the process of re-acceleration, which occurs when the driver electrons decelerate to nonrelativistic energies, slipping backward into the accelerating phase of the wakefield and being subsequently re-accelerated. Such re-acceleration is unambiguously observed here for the first time. At this re-acceleration limit, we measure a beam driver depositing (57 ± 3)% of its energy into a 195-mm-long plasma. This suggests that the energy-transfer efficiency of plasma accelerators could approach that of conventional accelerators. Published by the American Physical Society 2024

A new model describing improved bound-state and statistical properties in the framework of deformed Schrödinger equation for the improved Coshine Yukawa potential in 3D-NR(NCPS) symmetries

In this paper, within the three-dimensional non-relativistic non-commutative phase–space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schrödinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) ([Formula: see text]) and other terms produced from the effect of phase–space deformation ([Formula: see text] and [Formula: see text]). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp’s shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase–space are dependent on the discrete atomic quantum numbers ([Formula: see text],m), the dissociation energy [Formula: see text], the equilibrium bond length the screening parameter [Formula: see text], the deformation phase parameters ([Formula: see text]) and the deformation space parameters ([Formula: see text]). Additionally, the thermal properties of the CYP and ICYP with 3D Schrödinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase–space. This study has multiple applications in various domains, including atomic and molecular physics.

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

Non-relativistic and relativistic energy of molecules in external fields with time-dependent moving boundaries

Non-relativistic and relativistic energy of molecules in external fields with time-dependent moving boundaries

Elucidate the non-relativistic and relativistic energy density functional from momentum space to coordinate space within local density approximation

Elucidate the non-relativistic and relativistic energy density functional from momentum space to coordinate space within local density approximation

Non-relativistic energy spectra of and diatomic molecules confined in a modified Scarf potential via supersymmetric WKB approach

In this research, the energy equation of the Schrodinger equation with the modified Scarf potential has been obtained using the supersymmetric WKB (SWKB) approach. The Pekeris approximation has been applied to enable the analytical solution. The energy equation was applied to determine the rovibrational states of two diatomic molecules such as O 2 + ( X 2 Π g ) and N 2 ( X 1 Σ g + ) . The average kinetic and potential energies were computed via the Hellman-Feynman theorem. The analytical result of the energy equation obtained coincides with the work reported in earlier literature where the authors had applied a different approach. The energy eigenvalues for the studied molecules were found to agree with the ones obtained using the Teitz-Hua molecular potential, the deformed modified Rosen-Morse (DMRM) potential, the Morse potential and the Rydberg-Klein-Rees data points. For the energy spectra, we found average absolute deviations of 0.129515% for O 2 + ( X 2 Π g ) and 1.834993% for N 2 ( X 1 Σ g + ) molecules respectively. The energy spectra increase as the quantum numbers increase up to the dissociation limit before decreasing. For the molecules, the atoms were found to possess an oscillatory-type motion where an increase in the mean kinetic energy resulted in a decrease in mean potential energy and vice-visa.

Dirac fermions in a spinning conical Gödel-type spacetime

Abstract In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in (2+1)-dimensions, where we work with the Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels, where such levels are quantized in terms of the quantum numbers n and m j, and explicitly depends on the spin parameter s, spinorial parameter u, curvature and rotation parameters α and β, and on the vorticity parameter Ω. In particular, the quantization is a direct result of the existence of Ω (i.e., Ω acts as a kind of ``external field or potential''). We see that for m j>0, the energy levels do not depend on s and u; however, depend on n, m j, α, and β. In this case, α breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit 4Ωβ/α→1. Already for m j<0, we see that the energy levels depend on s, u and n; however, it no longer depends on m j, α and β. In this case, it is as if the fermion ``lives only in a flat Gödel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω, α, and β for three different values of n.

Complex partition function of 1-dimensional Dirac particle confined in Woods-Saxon interaction

We present the bound state solutions of one-dimensional Schrödinger and Dirac equations for a particle confined in Woods-Saxon potential. In each case the wavefunctions and the energy eigenvalues have been obtained in terms of the Jacobi polynomials by using the Nikiforov-Uvarov method. The complex eigenenergy obtained from the Dirac equation is explicitly shown to contain the nonrelativistic energies plus complex relativistic correction terms The quantum partition functions obtained using the relativistic and the nonrelativistic energies have been used to generate important thermodynamic properties of the system in the high temperature limit. In particular, the complex partition function obtained in the relativistic regime is taken to indicate some important physical effects in the system.

Path integral formulation for Dunkl-Dirac oscillator in (1+1)-dimensional space-time coordinates

In this paper, we extend the path integral formalism for the Dirac oscillator in (1+1) dimension by replacing the spatial derivative with the Dunkl derivative. Utilizing representations in position space-time coordinates, we precisely calculate the propagator, expressed in terms of generalized Hermite polynomials. The energy eigenvalues of the electron, along with their corresponding wave functions, are determined. In special cases, we can precisely evaluate the non-relativistic energy eigenvalues and wave functions, even in the absence of Dunkl parameters.

Derivation of the Schrödinger equation from QED

The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.

Non-relativistic energy equations for diatomic molecules constrained in a deformed hyperbolic potential function.

In this paper, the approximate analytical energy equations for the deformed hyperbolic potential have been obtained for arbitrary parameters of the potential. The potential function was transformed to a molecular potential by subjecting it to the Varshni conditions which allows for the determination of the energy levels of diatomic molecules. The molecular vibrational energy spectra for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] diatomic molecules were obtained and found to match with the results obtained with another analytical approach, potential functions, and experimental data. The noticeable slight differences in the approximate energy spectra obtained in this work and existing literature may be ascribed to the analytical method, computational approach, and the accuracy of the molecular potential functions. The obtained energy equations were used to determine the energy of a particle for arbitrary parameters of the potential function. The obtained energy is bounded and increases with the increase in the quantum numbers. The results conformed to the ones obtained via the path integral approach and numerical solutions obtained via the MATHEMATICA program. The energy spectra equations were obtained via the Nikiforov-Uvarov approach and semi-classical WKB approximation. The Pekeris approximation has been applied to resolve the difficulty in solving the complete energy spectrum of the non-relativistic wave equation for the potential function. The numerical data of the energy spectra was obtained using the MATHEMATICA program.

Energy densities in quantum mechanics

Quantum mechanics does not provide any ready recipe for defining energy density in space, since the energy and coordinate do not commute. To find a well-motivated energy density, we start from a possibly fundamental, relativistic description for a spin-12 particle: Dirac's equation. Employing its energy-momentum tensor and going to the non-relativistic limit we find a locally conserved non-relativistic energy density that is defined via the Terletsky-Margenau-Hill quasiprobability (which is hence selected among other options). It coincides with the weak value of energy, and also with the hydrodynamic energy in the Madelung representation of quantum dynamics, which includes the quantum potential. Moreover, we find a new form of spin-related energy that is finite in the non-relativistic limit, emerges from the rest energy, and is (separately) locally conserved, though it does not contribute to the global energy budget. This form of energy has a holographic character, i.e., its value for a given volume is expressed via the surface of this volume. Our results apply to situations where local energy representation is essential; e.g. we show that the energy transfer velocity for a large class of free wave-packets (including Gaussian and Airy wave-packets) is larger than its group (i.e. coordinate-transfer) velocity.

Bayesian model averaging for nuclear symmetry energy from effective proton-neutron chemical potential difference of neutron-rich nuclei

The data-driven Bayesian model averaging is a rigorous statistical approach to combining multiple models for a unified prediction. Compared with the individual model, it provides more reliable information, especially for problems involving apparent model dependence. In this work, within both the non-relativistic Skyrme energy density functional and the nonlinear relativistic mean field model, the effective proton-neutron chemical potential difference Δμpn⁎ of neutron-rich nuclei is found to be strongly sensitive to the symmetry energy Esym(ρ) around 2ρ0/3, with ρ0 being the nuclear saturation density. Given discrepancies on the Δμpn⁎-Esym(2ρ0/3) correlations between the two models, we carried out a Bayesian model averaging analysis based on Gaussian process emulators to extract the symmetry energy around 2ρ0/3 from the measured Δμpn⁎ of 5 doubly magic nuclei 48Ca, 68Ni, 88Sr, 132Sn and 208Pb. Specifically, the Esym(2ρ0/3) is inferred to be Esym(2ρ0/3)=25.6−1.3+1.4MeV at 1σ confidence level. The obtained constraints on the Esym(ρ) around 2ρ0/3 agree well with microscopic predictions and results from other isovector indicators.

Nonrelativistic bound-state energy spectrum and persistent current in the modified Rosen–Morse oscillator confined to a 2D electromagnetic potential field

Nonrelativistic bound-state energy spectrum and persistent current in the modified Rosen–Morse oscillator confined to a 2D electromagnetic potential field

New Eigensolution of the Klein--Gordon and Shrödinger equations for improved modified Yukawa-Kratzer potential and its applications using Boop's Shift method and standard perturbation theory in the 3D-ERQM and 3D-ENRQM symmetries

The deformed Klein-Gordonequation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the improved modified Yukawa-Kratzer potential (IMYKP) model under the influence of the deformation space-space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the IMYKP model it was shown to be sensitive to the atomic quantum numbers (j, l, s, m), mixed potential depths (V0, De, re), the screening parameter’s inverse α and noncommutativity parameters (Θ,τ ,χ). In addition, we analyzed the nonrelativistic energy values by applying the well-known transmission rules known in the literature. In addition, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the improved modified Kratzer potential, the improved generalized Kratzer potential, the improved Kratzer potential, the improved modified Kratzer plus screened Coulomb potential, the improved Hellmann potential, the improved Yukawa potential, and improved inversely square Yukawa potential. We noticed that these particular results are identical to our previous work and other known works in the literature. The study is further extended to calculate the mass spectra of mesons of charmonium (cc) and bottomonium (bb) within the framework of the IMYKP model in three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The deformed Dirac equation has been investigated, in the context of 3D-relativistic noncommutative space (3D-RNCS) symmetries, using the improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model under the conditions of spin symmetry and pseudospin symmetry. The ISDM(CP-CLTI) model is the combining the spatially dependent mass Coulomb potential with the Coulomb-like tensor interaction (CP-CLTI) and the two central terms that are generated to the topological defects of spacespace. Within the confines of the parametric Bopp shift method and conventional perturbation theory, the new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms ( H-atoms), such as He+,Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived. The novel values Encsp(n,C,m0,m1,H,Θ,τ,χ,j,l,s,m) and Encps(n,C,m0,m1,H,Θ,τ,χ,j,l,l̃,s̃,m̃) that we discovered examined to be dependent on the noncommutativity parameters (NP) (Θ,σ,χ), mixed potential depths C/q,m0,m1,H), and quantum atomic discrete quantum numbers (j,k,l,s,m,l̃,s̃,m̃). We have obtained several interesting special examples within the framework of relativistic extended quantum mechanics, which we believe will be of interest to the expert researcher. We were able to retrieve the typical results of relativistic and non-relativistic examples in the literature when we applied the three simultaneous constraints (Θ,σ,χ)→(0,0,0). Compared to previous models that are known from the literature, our new model had novel physical characteristics.

Convergence of the Laplace and the alternative multipole expansion approximation series for the Coulomb potential

Multipole expansion is a powerful technique used in many-body physics to solve dynamical problems involving correlated interactions between constituent particles. The Laplace multipole expansion series of the Coulomb potential is well established in literature. We compare its convergence with our recently developed perturbative and analytical alternative multipole expansion series of the Coulomb potential. In our working, we confirm that the Laplace and the analytical alternative multipole expansion series are equivalent as expected. In terms of performance, the perturbative alternative multipole expansion series underapproximate the expected results to some extent while the Laplace and the analytical alternative multipole expansion series yield results which are relatively accurate but oscillatory in nature even with a higher number of angular momentum terms employed. As a practical example, we have evaluated the Slater double integrals for two-electron systems using the multipole expansion techniques and a mean field approximation. The estimated results show that only spherical interactions are dominant while the higher-order interactions are negligible. To highlight the discrepancy in the application of each of the formulations of the multipole expansion series for the electron-electron interaction potential, an estimation of the non-relativistic groundstate energies of some helium-like systems, evaluated using the spherical approximation of the multipole potential, is provided. Our findings are likely to be useful in the treatment of the Coulomb potential in electronic structure calculations as well as in celestial mechanics.

New relativistic and non-relativistic investigation of deformed Klein–Gordon and Schrödinger equations for new improved screened Kratzer potential in the framework of non-commutative space

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text], control parameter [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text] that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.

Effect of the non-commutativity of space on the improved Mobius square plus generalized Yukawa potentials of the Klein–Gordon and Schrödinger equations in 3D-RNCQS and 3D-NRNCQS symmetries

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N2, K2, NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text], that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.