Nonrelativistic Energy Research Articles

Analysis of Non-Relativistic Energy, Wave Function, and Black Hole’s Application with Minimal Length

Analysis of Non-Relativistic Energy, Wave Function, and Black Hole’s Application with Minimal Length

A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries

In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.

Satellite lines of helium-like ions in strongly coupled plasma environment

We report the transition energies and probabilities of satellite lines due to 2p2(3P0,1,2)→1s2p(3P0,1,2) transitions of He-like C,Ne,Al and Ar ions embedded in strongly coupled plasma (SCP) environment. The SCP environment is represented by the ion-sphere (IS) model and the IS radius is varied in such a way that plasma electron densities (∼1020−1026 per cm3) cover a wide range of experimental interest. Both relativistic and non-relativistic energies of 2p2(3P) and 1s2p(3P) states are reported. The relativistic MCDF results are obtained by modified GRASP2K code and the non-relativistic variational results are done by using explicitly correlated Hylleraas basis. The relativistic results are in good agreement with experimental data available in literature. The positions of intercombination lines in presence of dense plasma can be calculated from present results which are comparable with experimental measurements. The partial pressures experienced by the plasma-embedded ions are analyzed in the purview of available experimental observations. An empirical relation is reported to estimate partial pressure on ions in dense plasma environment.

2D relativistic oscillators with a uniform magnetic field in anti-de Sitter space

We study analytically the two-dimensional deformed bosonic oscillator equation for charged particles (both spin 0 and spin 1 particles) subject to the effect of an uniform magnetic field. We consider the presence of a minimal uncertainty in momentum caused by the anti-de Sitter model and we use the Nikiforov–Uvarov (NU) method to solve the system. The exact energy eigenvalues and the corresponding wave functions are analytically obtained for both Klein–Gordon and scalar Duffin–Kemmer–Petiau (DKP) cases and we find that the deformed spectrum remains discrete even for large values of the principal quantum number. For spin 1 DKP case, we deduce the behavior of the DKP equation and write the nonrelativistic energies and we show that the space deformation adds a new spin-orbit interaction proportional to its parameter. Finally, we study the thermodynamic properties of the system and here we find that the effects of the deformation on the statistical properties are important only in the high-temperature regime.

Directed flow of D mesons at RHIC and LHC: non-perturbative dynamics, longitudinal bulk matter asymmetry and electromagnetic fields

We present a study of the directed flow v1 for D mesons discussing both the impact of initial vorticity and electromagnetic field. Recent studies predicted that v1 for D mesons is expected to be surprisingly much larger than that of light charged hadrons; we clarify that this is due to a different mechanism leading to the formation of a directed flow with respect to the one of the bulk matter at both relativistic and non-relativistic energies. We point out that the very large v1 for D mesons can be generated only if there is a longitudinal asymmetry between the bulk matter and the charm quarks and if the latter have a large non-perturbative interaction in the QGP medium. A quite good agreement with the data of STAR and ALICE is obtained if the diffusion coefficient able to correctly predict the RAA(pT), v2(pT) and v3(pT) of D meson is employed. Furthermore, the mechanism for the build-up of the v1(y) is associated to a quite small formation time that can be expected to be more sensitive to the initial high-temperature dependence of the charm diffusion coefficient.We discuss also the splitting of v1 for D0 and {overline{D}}^0 due to the electromagnetic field that is again much larger than the one observed for charged particles and in agreement with the data by STAR that have however still error bars comparable with the splitting itself, while at LHC standard electromagnetic profile assuming a constant conductivity is not able to account for the huge splitting observed.

Approximate non-relativistic s-wave energy spectra with non-polynomial potentials within the framework of the WKB approximation

Approximate non-relativistic s-wave energy spectra with non-polynomial potentials within the framework of the WKB approximation

Effect of relativistic kinematics on the stability of multiquarks

We discuss whether the bound nature of multiquark states in quark models could benefit from relativistic effects on the kinetic energy operator. For mesons and baryons, relativistic corrections to the kinetic energy lead to lower energies, and thus call for a retuning of the parameters of the model. For multiquark states, as well as their respective thresholds, a comparison is made of the results obtained with non-relativistic and relativistic kinetic energy. It is found that the binding energy is lower in the relativistic case. In particular, $QQ\bar q\bar q$ tetraquarks with double heavy flavor become stable for a larger ratio of the heavy to light quark masses; and the all-heavy tetraquarks $QQ\bar Q\bar Q$ that are not stable in standard non-relativistic quark models remain unstable when a relativistic form of kinetic energy is adopted

Solutions of the Schrödinger equation and thermodynamic properties of a combined potential

The solution of the radial Schrödinger equation was obtained using the methodology of supersymmetric approach with a combination of modified generalized Pöschl-Teller potential and inversely quadratic Yukawa potential model. The non-relativistic ro-vibrational energy spectra and the corresponding wave functions were obtained and numerical results were generated for some states. The variation of energy of the combined potential and the subsets potentials with the screening parameter for various quantum number were graphically studied. The effect of the potential parameters on the energy for different states was also studied numerically. For more usefulness and applications of the work, the vibrational partition function and the various thermal properties like mean energy, Helmholtz energy, heat capacity and entropy were calculated. The behaviour of the thermodynamic properties with respect to temperature change for various quantum number and maximum quantum states were examined in detail. The temperature has positive effect on all the thermal properties except the free energy.

Bound and scattering states solutions of the Klein–Gordon equation with generalized Mobius square potential in D-dimensions

In this study, the Klein–Gordon equation (KGE) was solved with the generalized Mobius square (GMS) potential using the functional analysis approach (FAA) in D-dimensions. By employing the Pekeris-type approximation scheme, the relativistic and nonrelativistic bound state energies were obtained in closed form. Also, the expression for the scattering state phase shift of GMS potential was obtained in D-dimensions. The effects of the vibrational and rotational quantum number on the vibrational energies and the scattering state phase shift of nitrogen monoiodide (NI) diatomic molecule were studied numerically and graphically at different dimensions. An interesting result of this study is the inter-dimensional degeneracy symmetry for scattering phase shift of the NI diatomic molecule. Hence, this concept is applicable in the areas of chemical physics, nuclear and particle physics. In this study, the relativistic and nonrelativistic bound state energies of KGE with the GMS potential were obtained in D-dimensions using the FAA. In addition, the scattering state phase shift of GMS potential was obtained in D-dimensions. The effects of the vibrational and rotational quantum number on the vibrational energies and the scattering state phase shift of NI diatomic molecule were studied. The inter-dimensional degeneracy symmetry for scattering phase shift of the NI diatomic molecule was obtained at unique quantum states.

One-Hadron-Exchange K−p interaction model in nonrelativistic energy region

A nonrelativistic K−p interaction model is derived as one-hadron-exchanges. The exchanged particles are the scalar meson, the vector mesons, the hyperons, and the resonances Λ*(1600),Σ*(1385). The models parameters are determined by fitting to experimental data of spin-averaged differential cross section for kaon laboratory energy range of 51.27 MeV up to 376.87 MeV. We use a 3D technique without partial wave expansion to calculate K−p scattering.

Proton density polarization of the doubly magic 40Ca core in 48Ca and EoS parameters

Abstract The proton density polarization in ${\rm ^{48}Ca}$ by the eight neutrons added on the ${\rm ^{40}Ca}$ core is discussed. We adopt the proton density distributions determined by electron scattering and the neutron density distributions determined by proton elastic scattering in ${\rm ^{40}Ca}$ and ${\rm ^{48}Ca}$. As a measure of the proton density polarization effect due to the neutrons, a ratio $R$ of the peak values of the neutron density difference $\Delta\rho_{n,{\rm peak}}$ between ${\rm ^{48}Ca}$ and ${\rm ^{40}Ca}$ to the proton one $\Delta\rho_{p,{\rm peak}}$ is introduced and compared with the Hartree–Fock and Hartree calculations by non-relativistic and relativistic energy density functionals (EDFs), respectively. The value $R$ shows clear correlations with the symmetry energy parameters $J$ and $L$, but not with the isoscalar incompressibility $K_\infty$. The experimental value $R$ can be reproduced by theoretical calculations using the EDFs with $L=25$–50 MeV.

Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential

In this research article, the modified approximation to the centrifugal barrier term is applied to solve an approximate bound state solutions of Dirac equation for spin and pseudospin symmetries with hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential using parametric Nikiforov–Uvarov method. The energy eigen equation and the unnormalised wave function were presented in closed and compact form. The nonrelativistic energy equation was obtain by applying nonrelativistic limit to the relativistic spin energy eigen equation. Numerical bound state energies were obtained for both the spin symmetry, pseudospin symmetry and the non relativistic energy. The screen parameter in the potential affects the solutions of the spin symmetry and non-relativistic energy in the same manner but in a revised form for the pseudospin symmetry energy equation. In order to ascertain the accuracy of the work, the numerical results obtained was compared to research work of existing literature and the results were found to be in excellent agreement to the existing literature. The partition function and other thermodynamic properties were obtained using the compact form of the nonrelativistic energy equation. The proposed potential model reduces to Hulthen and exponential inversely quadratic potential as special cases. All numerical computations were carried out using Maple 10.0 version and Matlab 9.0 version softwares respectively.

Varying $$\boldsymbol{\Lambda}$$ Theory Revisited

We revisit the variable $$\Lambda$$ theory recently developed by Alexander et al. [1, 2]. We implement it to explain the current expansion of the Universe. The varying $$\Lambda$$ theory reduces one degree of freedom. The varying $$\Lambda$$ term (cosmological term) follows the energy density of the Universe. We consider such a theory which does not respect self-dual symmetry. The Bianchi identity enforces the $$\Lambda$$ term to be a constant for a theory without torsion. However, torsion can incorporate a varying $$\Lambda$$ term in the action. We consider a form of the action which has such a varying $$\Lambda$$ term, and the presence of only nonrelativistic matter energy density. We fit the luminosity distance according to the model and compare it with that in the $$\Lambda\textrm{{CDM}}$$ model and find their close behavior. However, we find a current value of matter energy density $$\rho_{m0}$$ larger than the critical energy density. Using the supernova data from Union 2.1 compilation, we further consider a different set of parameters, for which $${\kappa\rho_{m0}}/{3H_{0}^{2}}$$ becomes close to unity but, however, greater. For this set of parameters, we find a negative value in the effective equation of state, however, the model has a very different pattern of the Hubble parameter from the standard scenario.

Trajectory in 2D plot of isoscalar and isovector densities of Ca48 and Pb208 , and symmetry energy

We propose a new two-dimensional (2D) plotting method of isoscalar-isovector (IS-IV) densities of $^{48}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$ to extract not only empirical symmetry energy coefficients $J, L$, and ${K}_{\mathrm{sym}}$ but also the asymmetric isospin term ${K}_{\ensuremath{\tau}}$ in the nuclear incompressibility. We adopted several modern relativistic and nonrelativistic energy density functional (EDF) for the theoretical calculations of IS and IV densities to compare with experimental neutron and proton density distributions determined by recent proton elastic scatterings. We found very strong correlations between the trajectory of the 2D density at the density $\ensuremath{\rho}=0.1\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ and the symmetry energy coefficients $J, L$, and also ${K}_{\ensuremath{\tau}}$. For ${K}_{\mathrm{sym}}$, the Skyrme EDF SAMi-J family shows a strong correlation in the trajectory of 2D density, while the RMF Lagrangian DDME-J family shows a peculiar quadratic curve and does not provide the experimental value. The symmetry energy term ${K}_{\ensuremath{\tau}}$ of nuclear incompressibility is also discussed in term of the slope of 2D trajectory.

An interacting fermion-antifermion pair in the spacetime background generated by static cosmic string

We consider a general relativistic fermion antifermion pair that they interact via an attractive Coulomb type interparticle interaction potential in the 2+1 dimensional spacetime background spanned by cosmic string. By performing an exact solution of the corresponding fully-covariant two body Dirac Coulomb type equation we obtain an energy spectrum that depends on angular deficit parameter of the static cosmic string spacetime background for such a composite system. We arrive that the influence of cosmic string spacetime topology on the binding energy of Positronium-like atoms can be seen in all order of the coupling strength constant, even in the well-known non-relativistic binding energy term (∝αc2). We obtain that the angular deficit of the static cosmic string spacetime background causes a screening effect. For a predicted value of angular deficit parameter, α∼1−10−6, we apply the obtained result to an ortho-positronium, which is an unstable atom formed by an electron and its antimatter counterpart a positron, and then we determine the shift in the ground state binding energy level as 27,2μeV. We also arrive that, in principle, the shift in ground state binding energy of ortho-positronium can be measured even for α∼1−10−11 value with current techniques in use today. Moreover, this also gives us an opportunity to determine the altered total annihilation energy transmitted by the annihilation photons. The yields also impose that the total lifetime of an ortho-positronium can be changed by the topological feature of static cosmic string spacetime background. In principle, we show that an ortho-positronium system has a potential to prove the existence of such a spacetime background.

Nonrelativistic energies and fine-structure splittings for the Rydberg P states of lithium

The Schr\odinger equation for the $nP$ $(4\ensuremath{\le}n\ensuremath{\le}10)$ states of lithium is solved using the Rayleigh-Ritz variational method in Hylleraas coordinates, where the asymptotic behavior of the wave function that is specific to a Rydberg state is built in explicitly to improve the accuracy of the solution. The nonrelativistic variational energies of the $nP$ $(4\ensuremath{\le}n\ensuremath{\le}10)$ states reach an accuracy of ${10}^{\ensuremath{-}13}$ to ${10}^{\ensuremath{-}14}$. In addition, the fine-structure splittings of these states are also calculated and compared with experimental results.

The performance of CIPSI on the ground state electronic energy of benzene

Following the recent work of Eriksen et al. [J. Phys. Chem. Lett. 11, 8922 (2020)], we report the performance of the configuration interaction using a perturbative selection made iteratively method on the non-relativistic frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of -863.4 mEh, which agrees with the theoretical estimate of -863 mEh proposed by Eriksen et al. [J. Phys. Chem. Lett. 11, 8922 (2020)] using an extensive array of highly accurate new electronic structure methods.

Approximate non-relativistic energy expression and the rotational–vibrational constants of the Tietz–Hua potential: a semiclassical approach

The low- and high-lying rovibrational energy levels of the Schrodinger equation with the molecular Tietz–Hua potential are obtained via the Wentzel–Kramers–Brilluoin (WKB) quantization approach. The Pekeris-type approximation scheme is applied to deal with the orbital centrifugal term of the effective potential function. The obtained energy spectra and the rotational–vibrational (rovibrational) coefficients for [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] diatomic molecules were compared with the ones obtained by other analytical methods and available experimental data in the literature. The results revealed that the accuracy of the energy spectra for the high-lying rovibrational quantum states may depend on the rotational-vibrational constants.

Analytical Calculation of Cl$$^{15+}$$ Ion Immersed in Dense Plasmas

A method of analytically calculating energy levels of He-like ions in an environment of dense plasma is given by using the angular momentum coupling theory and irreducible tensor theory under Hartree–Fock approximation. In order to obtain higher calculation precision, relativistic correction terms of the non-relativistic energy including corrections caused by relativistic mass, one- and two-body Darwin effect, spin–spin contact interaction and orbit–orbit interaction, are calculated. The binding energies of the ground state $$1s^2$$ $$^1S$$ and excited state 1snp $$^1P$$ (n = 1 − 4) of He-like Cl ions and the transition energy between two energy levels in an environment of dense plasma are calculated. The scaling relationship between the energy shift of plasma and its temperature and density is given. According to our study, the energy shift of plasma conforms very well with the recent high-precision experimental result (Phys Rev A 100:012511, 2019).

Exponentially correlated Hylleraas-configuration interaction non-relativistic energy of the 1S ground state of the helium atom.

A generalization of the Hylleraas-Configuration Interaction method (Hy-CI) first proposed by Wang, et al., the Exponentially Correlated Hylleraas-Configuration Interaction method (E-Hy-CI) in which the single r ij of an Hy-CI wave function is generalized to a form of the generic type , is explored. This type of correlation, suggested by Hirshfelder in 1960, has the right behavior both in the vicinity of the rij cusp as rij goes to 0 and as rij goes to infinity; this work explores whether wave functions containing both linear and exponential r ij factors converge more rapidly than either one alone. The method of calculation of the two-electron E-Hy-CI kinetic energy and electron repulsion integrals in a stable and efficient way using recursion relations is discussed, and the relevant formulas are given. The convergence of the E-Hy-CI wave function expansion is compared with that of the Hy-CI wave function without exponential correlation factors, demonstrating both convergence acceleration and an improvement in the accuracy for the same basis. This makes the application of the E-Hy-CI method to systems with N > 4, for which this formalism with at most a single factor per term leads to solvable integrals, very promising. E-Hy-CI method variational calculations with up to 10080 expansion terms are reported for the ground 1 S state of the neutral helium atom, with a resultant nonrelativistic energy of -2.9037 2437 7034 1195 9831 1084 hartree for the best expansion.