Harmonic Oscillator Potential Research Articles

Matter density distribution and longitudinal form factors for the ground and excited states of 17Ne exotic nucleus

The two-frequency shell model approach is used to calculate theground state matter density distribution and the corresponding rootmean square radii of the two-proton17Ne halo nucleus with theassumption that the model space of 15O core nucleus differ from themodel space of extra two loosely bound valence protons. Twodifferent size parameters bcore and bhalo of the single particle wavefunctions of the harmonic oscillator potential are used. Thecalculations are carried out for different configurations of the outerhalo protons in 17Ne nucleus and the structure of this halo nucleusshows that the dominant configuration when the two halo protons inthe 1d5/2 orbit (15O core plus two protons halo in pure 1d5/2 orbit). Thecalculated matter density distribution in terms of the two-frequencyshell model is compared with the calculated one in terms one sizeparameter for all orbits to illustrate the effect of introducing one ortwo size parameters in calculations. The longitudinal form factors forelastic C0 and inelastic C2 electron scattering from 17Ne nucleus arecalculated for the considered configurations and for three states ofeach configuration which are the ground state ( JT  1 2 3 2 ) andthe first two excited states ( JT  3 2 3 2 ) and ( JT  5 2 3 2 ).The electric transition strengths B(C2) are calculated for the excitedstates and for the effective nucleon charges which are used in thiswork and compared with the experimental values.

Jahn-Teller effect in density-functional theory

Jahn and Teller used degenerate perturbation theory to prove that open shell molecules with symmetry Jahn-Teller distort. In that vein, we have developed a perturbative approach to computing Jahn-Teller distortions in Kohn-Sham density-functional theory, starting from a symmetrized molecule with an electronic shell closed by fractional occupation numbers. The resulting highly symmetric state is an energy extremum but not necessarily a minimum. Using second-order perturbation theory, we find the changes in geometry that occur when the symmetry of the electron density is broken to form a state with integer occupation numbers. This methodology allows us to retain many of the computational benefits of working in higher symmetry. As a demonstration, we solve the resulting equations for ten electrons in a superatomlike harmonic-oscillator potential.

Study of charge density distributions, elastic charge form factors and root-mean square radii for 4He, 12C and 16O nuclei using Woods- Saxon and harmonic-oscillator potentials

The nuclear charge density distributions, form factors andcorresponding proton, charge, neutron, and matter root mean squareradii for stable 4He, 12C, and 16O nuclei have been calculated usingsingle-particle radial wave functions of Woods-Saxon potential andharmonic-oscillator potential for comparison. The calculations for theground charge density distributions using the Woods-Saxon potentialshow good agreement with experimental data for 4He nucleus whilethe results for 12C and 16O nuclei are better in harmonic-oscillatorpotential. The calculated elastic charge form factors in Woods-Saxonpotential are better than the results of harmonic-oscillator potential.Finally, the calculated root mean square radii usingWoods-Saxonpotentials how overestimation in comparison with experimental dataon contrary to the results of harmonic-oscillator potential.

Core-polarization effect in longitudinal electron scattering form factors of 65Cu nucleus

Inelastic longitudinal electron scattering form factors to 2+ and 4+ states in 65Cu nucleus has been calculated in the (2p3/2 1f 5/2 2p1/2) shell model space with the F5PVH effective interaction. The harmonic oscillator potential has been applied to calculate the wave functions of radial single-particle matrix elements. Two shell model codes, CP and NUSHELL are used to obtain results. The form factor of inelastic electron scattering to 1/21−, 1/22−, 3/22−, 3/23−, 5/21−, 5/22− and 7/2- states and finding the transition probabilities B (C2) (in units of e2 fm4) for these transitions and B (C4) (in units of e2 fm8) for the transition 7/2-, and comparing them with experimental data. Both the form factors and reduced transition probabilities with core-polarization effects gave a reasonable description of the experimental data.

Study of nuclear structure of <SUP align=right>58,62</SUP>Ni isotopes using the F5PVH effective interaction

The nuclear structure of 58,62Ni nuclei has been calculated using F5PVH as effective interaction for 1f5/2 2p3/2 2p1/2 shell model wave functions. Skyrme-Hartree Fock (SKX), Harmonic Oscillator (HO) and Woods-Saxon (WS) potentials were used to calculate the single particle wave function. The core-polarisation effects have been adopted using the shape of Tassie model. The level schemes are compared with the experimental data up to 3.42 MeV and 4.018 MeV for 58Ni and 62Ni, respectively. In this study, very good agreements are obtained for all nuclei. Results from C2 form factor and Charge Density Distribution (CDD) calculations give good agreement with the experimental data with no adjustable parameters.

Study of nuclear structure of 58,62Ni isotopes using the F5PVH effective interaction

The nuclear structure of 58,62Ni nuclei has been calculated using F5PVH as effective interaction for 1f5/2 2p3/2 2p1/2 shell model wave functions. Skyrme-Hartree Fock (SKX), Harmonic Oscillator (HO) and Woods-Saxon (WS) potentials were used to calculate the single particle wave function. The core-polarisation effects have been adopted using the shape of Tassie model. The level schemes are compared with the experimental data up to 3.42 MeV and 4.018 MeV for 58Ni and 62Ni, respectively. In this study, very good agreements are obtained for all nuclei. Results from C2 form factor and Charge Density Distribution (CDD) calculations give good agreement with the experimental data with no adjustable parameters.

Electron propagator with vector and scalar energy-dependent potentials in (2+1)-dimensional space–time

We have studied the effect of energy-dependent potentials for the relativistic spinning particle using the formalism for supersymmetric path integrals. That leaves behind a new normalization of wave function, which is examined via the Dirac equation and can be confirmed by Feynman’s path integral method. Based on two important examples, Coulomb and Harmonic oscillator potentials, we find that the frequency and the Coulomb’s constant are dependent on spectral parameters. The propagator is calculated and the energy eigenvalues with their corresponding eigenfunctions are deduced.

An efficient method for obtaining the ground and excited states mass spectrum of doubly heavy $ \Omega$ Ω baryons

In this paper, we obtained the ground state masses and the positive and negative parity excited state masses of doubly heavy $ \Omega$ baryons. For this goal we have analytically solved the radial Schrodinger equation for three identical particles with the hypercentral potential by using the ansatz method. In this study the hypercentral potential is regarded as a combination of the color Coulomb plus linear confining term and the six-dimensional harmonic oscillator potential which has a two-body character and turns out to be exactly hypercentral. We also added the first-order correction and the spin-dependent part contains three types of interaction terms (the spin-spin term, spin-orbit term and tensor term) to the hypercentral potential. Our calculations have been performed for the radial excited states as well as orbital excited states for $ \Omega_{cc}$ , $ \Omega_{bb}$ and $ \Omega_{bc}$ systems. The masses obtained are compared with other theoretical reports, which could be a benefit tool for the interpretation of experimentally unknown doubly heavy baryons spectrum. The doubly heavy baryons masses are experimentally unknown so that the Regge trajectories are plotted using calculated masses to assign the quantum numbers of baryons unknown states.

The study of nuclear structure for some nuclei

An analytical form of the ground state charge density distributionsfor the low mass fp shell nuclei ( 40  A  56 ) is derived from asimple method based on the use of the single particle wave functionsof the harmonic oscillator potential and the occupation numbers ofthe states, which are determined from the comparison between theoryand experiment.For investigating the inelastic longitudinal electron scattering formfactors, an expression for the transition charge density is studiedwhere the deformation in nuclear collective modes is taken intoconsideration besides the shell model space transition density. Thecore polarization transition density is evaluated by adopting theshape of Tassie model together with the derived form of the groundstate charge density distribution. In this work, we devote ourinvestigation on 0 3 2 3 1 1   transition of Ti 50 , 0 1 2 1 1 1   transitionof Cr 50 and 0 2 2 2 1 1   of Cr 52 nuclei. It is found that the corepolarization effects, which represent the collective modes, areessential for reproducing a remarkable agreement between thecalculated inelastic longitudinal C2 form factors and those ofexperimental data.

Probing quasi-integrability of the Gross–Pitaevskii equation in a harmonic-oscillator potential

Previous simulations of the one-dimensional Gross–Pitaevskii equation (GPE) with repulsive nonlinearity and a harmonic-oscillator trapping potential hint towards the emergence of quasi-integrable dynamics—in the sense of quasi-periodic evolution of a moving dark soliton without any signs of ergodicity—although this model does not belong to the list of integrable equations. To investigate this problem, we replace the full GPE by a suitably truncated expansion over harmonic-oscillator eigenmodes (the Galerkin approximation), which accurately reproduces the full dynamics, and then analyze the system’s dynamical spectrum. The analysis enables us to interpret the observed quasi-integrability as the fact that the finite-mode dynamics always produces a quasi-discrete power spectrum, with no visible continuous component, the presence of the latter being a necessary manifestation of ergodicity. This conclusion remains true when a strong random-field component is added to the initial conditions. On the other hand, the same analysis for the GPE in an infinitely deep potential box leads to a clearly continuous power spectrum, typical for ergodic dynamics.

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solutions are obtained for any oscillation frequency considering both the 1∕r and ln r Ansätze for inter-electronic Coulombic-like potentials in 2D. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.

Study of the Electric Quadrupole Moments for some Scandium Isotopes Using Shell Model Calculations with Different Interactions

The electric quadrupole moments for some scandium isotopes (41, 43, 44, 45, 46, 47Sc) have been calculated using the shell model in the proton-neutron formalism. Excitations out of major shell model space were taken into account through a microscopic theory which is called core polarization effectives. The set of effective charges adopted in the theoretical calculations emerging about the core polarization effect. NushellX@MSU code was used to calculate one body density matrix (OBDM). The simple harmonic oscillator potential has been used to generate the single particle matrix elements. Our theoretical calculations for the quadrupole moments used the two types of effective interactions to obtain the best interaction compared with the experimental data. The theoretical results of the quadrupole moments for some scandium isotopes performed with FPD6 interaction and Bohr-Mottelson effective charge agree with experimental values.

Lagrange-Mesh Method for Deformed Nuclei With Relativistic Energy Density Functionals

The application of relativistic energy density functionals to the description of nuclei leads to the problem of solving self-consistently a coupled set of equations of motion to determine the nucleon wave functions and meson fields. In this work the Lagrange-mesh method in spherical coordinates is proposed for the numerical calculation. The essential field equations are derived from the relativistic energy density functional and the basic principles of the Lagrange-mesh method are delineated for this particular application. The numerical accuracy is studied for the case of a deformed relativistic harmonic oscillator potential with axial symmetry. Then the method is applied to determine the point matter distributions and deformation parameters of self-conjugate even-even nuclei from 4He to 40Ca.

Busch-Anglin effect for matter-wave and optical dark solitons in external potentials

Discovered by Busch and Anglin, the most important and puzzling nonlinear effect for solitons in external potentials lies in the fact that in a harmonic oscillator potential, a dark soliton oscillates with a frequency that is smaller by a factor of 2 in comparison with the ground state oscillations frequency. We review the main features of the Busch-Anglin effect and show that the physical nature of the self-defocusing nonlinearity can radically alter the Busch-Anglin dynamics. To bridge the gap between the matter-wave and optical dark solitons, we consider both the impact of linear and nonlinear absorption and the crucial role of nonlocality (or nonstationarity) of the nonlinear response in the triggering of the Busch-Anglin effect. We conclude that nonlocality (or nonstationarity) brings into existence the self-induced Busch-Anglin effect. Both linear and nonlinear losses cannot radically affect the period of dark solitons oscillations. We establish that from the physical point of view, it is best to redefine the specific dark soliton angle between the asymptotic phases of non-vanishing boundary conditions in accordance with Faddeev and Takhtajan's mathematical arrangement based on the phase shift across a dark soliton “core” controlling the soliton “blackness”. Among other things, we show that the effects analogous to the self-induced Busch-Anglin effect exist for bright solitons as well. Namely, triggered by nonlocal (or time-delayed) self-focusing nonlinearity, the bright solitons oscillate in the parabolic trap with a shifted equilibrium position and a frequency that exactly coincides with the oscillator frequency.

Excitons and Biexcitons in Spheroidal Quantum Dots A2B6

In the limit of strong quantum confinement the lower energy states of excitons and biexcitons in spheroidal quantum dots of semiconductors with a fourfold degenerate vertex of the valence band, which are active in the dipole approximation at one- and two-photon excitation, have been considered. The comparative analysis of the order of energy levels of the hole in the potentials of the infinitely deep quantum well and a three-dimensional harmonic oscillator taking into account the axial anisotropy of the quantum dot (QD) shape is carried out. It is shown that the anisotropy of the QD shape can lead to the opposite sign of splitting with respect to angular momentum projection ±3/2, ±1/2 for spatially odd (1P3/2) and even (1S3/2) levels of the hole. At the same time, in the case of the potential of an infinitely deep quantum well, an inversion of the order of 1S3/2 and 1P3/2 levels can be observed at values of the ratio of the effective masses of the light and heavy holes β = mlh/mhh ≈ 0.14. The type of the trial wave functions of the hole for the state 1P3/2 in the potential of an isotropic three-dimensional harmonic oscillator depending on β is proposed. The dependence of the binding energy of excitons in the considered potentials on β is presented and the possibility of formation of various biexcitonic states is considered.

A New Model for Calculating the Ground and Excited States Masses Spectra of Doubly Heavy Ξ Baryons

Since the doubly heavy baryons masses are experimentally unknown (except Ξcc+ and Ξcc++), we present the ground state masses and the positive and negative parity excited state masses of doubly heavy Ξ baryons. For this purpose, we have solved the six-dimensional hyperradial Schrödinger equation analytically for three particles under the hypercentral potential by using the ansatz approach. In this paper, the hypercentral potential is regarded as a combination of the color Coulomb plus linear confining term and the six-dimensional harmonic oscillator potential. We also added the first-order correction and the spin-dependent part contains three types of interaction terms (the spin-spin term, spin-orbit term, and tensor term) to the hypercentral potential. Our obtained masses for the radial excited states and orbital excited states of Ξccd, Ξccu, Ξbbd, Ξbbu, Ξbcd, and Ξbcu systems are compared with other theoretical reports, which could be a beneficial tool for the interpretation of experimentally unknown doubly heavy baryons spectrum.

Quantum pseudodots under the influence of external vector and scalar fields

We study the spherical quantum pseudodots in the Schrödinger equation by using the pseudo-harmonic plus harmonic oscillator potentials considering the effect of the external electric and magnetic fields. The finite energy levels and the wave functions are calculated. Furthermore, the behavior of the essential thermodynamic quantities such as, the free energy, the mean energy, the entropy, the specific heat, the magnetization, the magnetic susceptibility, and the persistent currents are also studied by using the characteristic function. Our analytical results are found to be in good agreement with the other works. The numerical results on the energy levels as well as the thermodynamic quantities have also been given.

On the quantization of the massive Bateman system

Bateman’s dual system (BDS) is a linear time-reversal invariant system that consists of damping and amplifying coordinate variables subjected to a harmonic oscillator potential. It is known that canonical quantization of the BDS in the underdamping region suffers from pathologies such as the non-stationary vacuum and the breakdown of Heisenberg’s uncertainty principle. In this paper, following a previous study that targeted the pathology-free massless BDS, the massive BDS is quantized by decomposing it into two independent effectively massless subsystems with reduced degrees of freedom. In each subsystem, the amplifying variable is the conjugate momentum of a damping variable that obeys a non-canonical quantization condition. By virtue of general scaling invariance including U(1), the variables in BDS are treated as non-self-adjoint without changing the total degrees of freedom. The physical states are constructed with time-reversal normalization and are shown to satisfy the time-independent uncertainty relation. The original massive BDS that fulfills the canonical quantization condition is reconstructed by superposing the two subsystems. The same method is also applied to the underdamped BDS to obtain the time-independent uncertainty relations. The expectation values of the coordinate operators represent the complex representation of the solution of the classical equation of motion. The scaling invariance may be broken by coupling with the environment.

Theoretical Study of Density Distributions and Size Radii of 8B and 17Ne

The proton, neutron and matter density distributions, the corresponding size radii and elastic electron scattering form factors of one-proton 8 B and two-proton 17 Ne halo nuclei are calculated. The theoretical technique used to fulfill calculations is by assuming that both nuclei under study are composed of two main parts; the first is the compact core and the second is the unstable halo part. The single-particle radial wavefunctions of harmonic-oscillator (HO) and Woods-Saxon (WS) potentials are used to study core and halo parts, respectively. And other approach is studied by using HO potential for both core and halo parts, but using two HO size parameters for both supposed parts. The long tail behavior which is the main characteristic of halo nuclei are well produced for both 8 B and 17 Ne. The calculated size radii are in general in good agreement with the available experimental data. The electron scattering form factors of the C0+C2 and C0 components are also calculated for 8 B and 17 Ne, respectively and compared with corresponding stable 10 B and 20 Ne nuclei. For 8 B calculations, the core-polarization (CP) effects are taken into account by using Tassie and Bohr-Mottelson models. The contribution from model-space (MS) part C2 component is taken through pwt interaction. The results of the calculated charge form factors are left for the planned electron-radioactive ion beam colliders where the study of skin or halo on the charge form factors are going to be studied.

Assessment of local density approximation based exchange–correlation functional for a two-dimensional spin polarized dipolar Fermi gas

In this paper we assess the accuracy of the local density approximation (LDA) based expressions for exchange and correlation functionals associated with a two-dimensional, spin-polarized, dipolar Fermi gas. For this purpose we consider and study an exactly solvable system of two interacting spin-polarized dipolar fermions confined in a two-dimensional harmonic oscillator potential. We determine the exact triplet ground state wavefunction, energy and the density by separating the time-independent Schrodinger equation in center-of-mass and relative coordinates and numerically solving the eigenvalue equation in relative coordinate. The exact Kohn–Sham orbitals, corresponding energies and exact Kohn–Sham potential are obtained from the exact density by inverting the Kohn–Sham equation. These results serve as benchmark data for assessing the accuracies of the LDA based expressions for a two-dimensional, spin-polarized, dipolar Fermi gas. We find that LDA results for total energy, interaction energy, and Kohn–Sham eigenvalues match well with the exact results for smaller values of dipole–dipole interaction strength. For higher values of interaction strength, LDA results for the total energy are underestimated. On the other hand, the LDA based interaction potentials overestimate the repulsion between the dipoles and their spatial profiles differ considerably from their exact counterparts and the discrepancy between them is higher for stronger interaction.