Harmonic Oscillator Potential Research Articles

Benchmarking DFT methods on linear and nonlinear electric properties of spatially confined molecules

AbstractThe accurate evaluation of molecular electric properties still constitutes a challenge for density functional theory (DFT) methods. Moreover, some new issues appear, within this area of research, that require analysis. One of them concerns the electric properties of confined matter. This study aims at a quantitative evaluation of the performance of DFT framework in the description of dipole moment, polarizability, and first hyperpolarizability of the isolated and spatially confined molecules. The calculations were performed using a set of 35 exchange‐correlation functionals. The reference data were determined employing the highly accurate CCSD(T) method. To mimic the effect of spatial confinement the cylindrically symmetric harmonic oscillator potential was applied. Based on the conducted research it was found that the behavior of the considered functionals substantially differs depending on the studied electric property and molecule. Moreover, it was observed that the strength of spatial confinement strongly influences the performance of DFT methods.

Optimal control of dark solitons

We consider optimal control methods for dark solitons and their higher-order states based on the completely integrable nonautonomous nonlinear Schrödinger equation (NLSE) models with varying gravitational-like and harmonic oscillator potentials, and in the presence of gain or absorption. In view of the fact that the time-dependence of the dark soliton phase (chirp) represents a major difference between bright and dark solitons, our main goal here is to gain a deeper understanding at a fundamental level based on the Lax pairs of how this time-dependent phase appears in the framework of the nonautonomous NLSE models with varying dispersion and nonlinearity. Specifically, we consider dynamics of dark solitons and their higher-order states in a variety of different optical scenarios. The main features of dark solitons are supplied by numerical calculations performed both for the exactly integrable and nonintegrable cases. We reveal nontrivial dynamics of black and gray solitons under the optimal control conditions. Since the experimental realization of dark solitons is possible only if a finite background pedestal is used, we consider dynamics of dark solitons and their higher-order states superimposed on the ground state of nonlinear harmonic oscillator. Direct numerical experiments performed in the framework of the Gross-Pitaevskii (GP) model in a broad range of parameters reveal many fundamental features of the dark solitons dynamics in confining harmonic oscillator potential, including the background BEC cloud spreading near the Feshbach resonance (when repulsive nonlinearity has a dispersive form). One of the most important effects is related to the “saturation” of the maximum value of self-compression near the Feshbach resonance. This effect exists both for the central black soliton and for the gray solitons formed during the higher-order dark soliton oscillations in the parabolic trap. The soliton becomes black at the turning points in the harmonic potential and gray at the trap center. This makes possible to observe the nontrivial dynamics of the periodic transformation of the black soliton to the gray one.

Lattice and quintic nonlinearity induced stripe phase in Bose–Einstein condensate under non-inertial and inertial motion

We consider a parametrically forced Bose–Einstein condensate in the combined presence of an optical lattice and harmonic oscillator potential in the mean field approach. A spatial symmetry broken Bose-condensed phase in non-inertial and inertial frame yields a stripe phase in the presence of both cubic and quintic nonlinearities. We show that the existence of such stripe phase solely depends on the interplay between the quintic nonlinearity and the lattice potential. Furthermore, we observe that a time-dependent harmonic oscillator frequency destroys such stripe ordering. A linear stability analysis of the obtained solution is performed and we found that the solution is stable. In order to gain a better understanding of the underlying physics, we compute the energy, showing nonlinear compression of the condensate in some parameter domain.

Finite difference calculation of optical properties of hydrogenic impurity in spherical quantum dot with parabolic confinement

The optical properties of an electron with an impurity in a spherical quantum dot under parabolic confinement are studied and energies, wave functions, binding energies, radial matrix elements, polarizability, susceptibility and oscillator strength have been evaluated. The numerical method used is the finite difference method in the framework of the effective mass approximation. The variation of the energy levels and radial matrix elements have been studied as function of the radius of the GaAs sphere and also as function of the frequency of the harmonic oscillator potential or parabolic potential. In addition we have studied how polarizability, susceptibility and the oscillator strength vary as a function of dot radius and at different parabolic potential frequencies.

Gravity and the Spin-2 Planar Schrödinger Equation

A Schrödinger equation proposed for the Girvin-MacDonald-Platzman gapped spin-2 mode of fractional quantum Hall states is found from a novel nonrelativistic limit, applicable only in 2+1 dimensions, of the massive spin-2 Fierz-Pauli field equations. It is also found from a novel null reduction of the linearized Einstein field equations in 3+1 dimensions, and in this context a uniform distribution of spin-2 particles implies, via a Brinkmann-wave solution of the nonlinear Einstein equations, a confining harmonic oscillator potential for the individual particles.

Harmony in a many-particle quantum problem

The model quantum problem of a harmonically trapped cloud of identical nonrelativistic particles interacting via a harmonic oscillator potential has an exact solution that can be used to assess the efficiency of approximate methods standardly used in more realistic many-particle calculations.

Non-Standard and Numerov Finite Difference Schemes for Finite Difference Time Domain Method to Solve One- Dimensional Schrödinger Equation

<span>The purpose of this paper is to show some improvements of the finite-difference time domain (FDTD) method using Numerov and non-standard finite difference (NSFD) schemes for solving the one-dimensional Schr</span><span>ö</span><span>dinger equation. Starting with results of the unmodified FDTD method, Numerov-FD and NSFD are applied iteratively to produce more accurate results for eigen energies and wavefunctios. Three potential wells, infinite square well, harmonic oscillator and Poschl-Teller, are used to compare results of FDTD calculations. Significant improvements in the results for the infinite square potential and the harmonic oscillator potential are found using Numerov-NSFD scheme, and for Poschl-Teller potential are found using Numerov scheme.</span>

One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential.

We propose a new mechanism for the stabilization of confined modes in lasers and semiconductor microcavities holding exciton-polariton condensates, with spatially uniform linear gain, cubic loss, and cubic self-focusing or defocusing nonlinearity. We demonstrated that the commonly known background instability driven by the linear gain can be suppressed by a combination of a harmonic-oscillator trapping potential and effective diffusion. Systematic numerical analysis of one- and two-dimensional (1D and 2D) versions of the model reveals a variety of stable modes, including stationary ones, breathers, and quasi-regular patterns filling the trapping area in the 1D case. In 2D, the analysis produces stationary modes, breathers, axisymmetric and rotating crescent-shaped vortices, stably rotating complexes built of up to 8 individual vortices, and, in addition, patterns featuring vortex turbulence. Existence boundaries for both 1D and 2D stationary modes are found in an exact analytical form, and an analytical approximation is developed for the full stationary states.

Comment on: “The kinetic energy partition method applied to quantum eigenvalue problems with many harmonic-oscillator potentials” by Y.-H. Chen and S. D. Chao

We show that the recently proposed kinetic energy partition method for the approximate solution of the Schrodinger equation is a particular case of the well known Rayleigh–Ritz variational method with an extremely restricted implementation. In addition to it, the quantum-mechanical example chosen by the authors to illustrate their approach is unsuitable for the proposed application.

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Nuclear Electric Quadrapole Moments (Q) in <sup>58</sup>Ni

Nuclear Electric quadrapole moments Q in 58Ni for some selected levels have been investigated and calculated through Nuclear shell model and considering of 56Ni as an inert core with two active neutrons in a model space (2p3/2, 1f5/2 and 2p1/2) and the configuration mixing of the original states is also done. F5Pvh interaction has been utilized as a two body interaction to generate model space vectors with harmonic oscillator potential as a single particle wave function. OXBASH code is used to carry this calculations and the program of Core, Valence, Tassie (CVT) written in FORTRAN go language to calculate the Electric quadrapole moments between excited states themselves. All of these calculations have been carried through model space vectors only. One body density matrix elements (OBDM) for ground and Excited states is calculated in order to carry the calculations using single particle Transition matrix elements between excited states theme selves.

Remedies for the Inconsistences in the Times of Execution of the Unsorted Database Search Algorithm within the Wave Approach

The typical semiclassical wave version of the unsorted database search algorithm based on a system of coupled simple harmonic oscillators does not consider an important ingredient of Grover’s original algorithm as it is quantum entanglement. The role of entanglement in the wave version of the unsorted database search algorithm is explored and contradictions with the time of execution of Grover’s algorithm are found. We remedy the contradictions by employing two arguments, one of them qualitative and the other quantitative. For the qualitative argument we employ the probabilistic nature of a legitimate quantum algorithm and remedy the above inconsistence. Within the quantitative argument we identify a parameter in the wave version of the unsorted database search algorithm which is related to entanglement. The contradiction with the time of execution of Grover’s algorithm is solved by choosing an appropriate values of such a parameter which incorporates entanglement to the wave version of the unsorted database search algorithm. The utility of the present arguments are evident if the wave version of the unsorted data base search algorithm is experimentally implemented through a system of N quantum dots with a harmonic oscillator potential as a confinement potential for each of the quantum dots. Each of the above N vibrating quantum dots must be coupled∗vlkmanuel@uaemex.mx1to an extra single vibrating quantum dot which entangles to all of them. In order to obtain optimal results, the coupling constants of the mentioned quantum dots should be adjusted in the way described in the present work.

Solutions of the Schrödinger equation with the harmonic oscillator potential (HOP) in cylindrical basis

Solutions of the Schrödinger equation with the harmonic oscillator potential (HOP) in cylindrical basis

Экситоны и биэкситоны в сфероидальных квантовых точках А-=SUB=-2-=/SUB=-B-=SUB=-6-=/SUB=-

AbstractIn 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 (1 P _3/2) and even (1 S _3/2) 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 1 S _3/2 and 1 P _3/2 levels can be observed at values of the ratio of the effective masses of the light and heavy holes β = m _lh/ m _hh ≈ 0.14. The type of the trial wave functions of the hole for the state 1 P _3/2 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.

Large-scale shell model calculations of the <SUP align="right">25,26</SUP>Mg, <SUP align="right">27</SUP>Al and <SUP align="right">19</SUP>F nucleus

Shell model calculations were performed to study the energy levels for the 25,26Mg and 27Al nuclei by employing the sdpfnow effective interaction with the large-scale sdpf model space by using the shell model code [email protected] for Windows. The electron scattering form factors for 19F nucleus also have been studied with and without effective charge on the sdpf-model space and Tassie model. The Harmonic Oscillator and Skyrme potentials have been used to calculate the wave functions of radial single-particle matrix elements. The level schemes are compared with the experimental data. Coulomb and magnetic form factors in the present work include the transitions from ground state (1/2+ 1/2) to the (7/2+ 1/2), (9/2+ 1/2), (3/2− 1/2), and (11/2− 1/2) states in 19F. Good agreements were obtained for all nuclei under study for energy levels and form factors comparing with the available experimental data.

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.

Asymptotic results in solvable two-charge models

In [16], a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential is introduced. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied in [17, 10]. In this paper we explore the sharp asymptotic behavior of the number of unit charge particles on the line and on the circle, as the total charge goes to infinity. We establish extended central limit theorems, Berry-Esseen estimates and precise moderate deviations using the machinery of the mod-Gaussian convergence developed in [6, 15, 14, 4, 7]. Also a large deviation principle is derived using the Gärtner-Ellis theorem.

Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics.

Langevin dynamics has become a popular tool to simulate the Boltzmann equilibrium distribution. When the repartition of the Langevin equation involves the exact realization of the Ornstein-Uhlenbeck noise, in addition to the conventional density evolution, there exists another type of discrete evolution that may not correspond to a continuous, real dynamical counterpart. This virtual dynamics case is also able to produce the desired stationary distribution. Different types of repartition lead to different numerical schemes, of which the accuracy and efficiency are investigated through studying the harmonic oscillator potential, an analytical solvable model. By analyzing the asymptotic distribution and characteristic correlation time that are derived by either directly solving the discrete equations of motion or using the related phase space propagators, it is shown that the optimal friction coefficient resulting in the minimum characteristic correlation time depends on the time interval chosen in the numerical implementation. When the recommended "middle" scheme is employed, both analytical and numerical results demonstrate that, for good numerical performance in efficiency as well as accuracy, one may choose a friction coefficient in a wide range from around the optimal value to the high friction limit.

Single- Ξ− hypernuclei within a quark mean-field model

The single-${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ hypernuclei are studied using a quark mean-field model. At the quark level, the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ hyperon is composed of one $d$ quark and two $s$ quarks, which are confined by the harmonic oscillator potentials. In the case of hadrons, the baryons interact with each other by exchanging $\ensuremath{\sigma},\ensuremath{\omega}$, and $\ensuremath{\rho}$ mesons between quarks in different baryons. The single-${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ binding energies of $\mathrm{\ensuremath{\Xi}}$ hypernuclei are investigated from $\mathrm{Be}$ to $\mathrm{Tl}$ using different parameter sets, which are determined by the ground-state properties of several stable nuclei and the empirical values of the single-$\mathrm{\ensuremath{\Lambda}}$ and single-$\mathrm{\ensuremath{\Xi}}$ potentials at the nuclear saturation density. For the bound states of ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}+\phantom{\rule{0.16em}{0ex}}^{14}\mathrm{N}$ (i.e., $\mathrm{C}$) system named as KISO event in the KEK-E373 experiment, it is found that the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ binding energies are around $5.61--5.89$ MeV for $1s$ state and $0.94--1.21$ MeV for $1p$ state with QMF-NK1S, QMF-NK2S, and QMF-NK3S parameter sets, whose single-$\mathrm{\ensuremath{\Xi}}$ potentials are $\ensuremath{-}12$ MeV. These results and those from cluster models with the Gaussian expansion method concerning on the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}+\phantom{\rule{0.16em}{0ex}}^{12}\mathrm{Be}$ show that in the KISO event, the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ hyperon may occupy the $1p$ state. Furthermore, the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ binding energies are achieved around $27\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$ for the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}+\phantom{\rule{0.16em}{0ex}}^{207}\mathrm{Pb}$ (i.e., $\mathrm{Tl}$) system. The energies were nearly comparable to the single-$\mathrm{\ensuremath{\Lambda}}$ binding energy of $\mathrm{Pb}$ observed by experiments. It demonstrates that the $\mathrm{\ensuremath{\Lambda}}$ and ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ hyperons seem to appear simultaneously in a neutron star.

Dynamics of binary Bose–Einstein condensate via Ehrenfest like equations: appearance of almost shape invariant states

We derive Ehrenfest like equations for the coupled Gross Pitaevskii equations (CGPE) which describe the dynamics of the binary Bose-Einstein condensate (BBEC) both in the free particle regime and in the regime where condensate is well trapped. Instead of traditional variational technique, we propose a new Ehrenfest based approach to explore so far unrevealed dynamics for CGPE and illustrate the possibility of almost shape invariant states in both the regimes. In absence of trapping potential, when all the interactions present in the system are attractive, it is possible for an initially mixed Gaussian state to propagate with almost no change in width if the proper initial condition is satisfied. Even for repulsive intra-atomic and attractive inter-atomic interaction ($g_{\alpha\beta}$) one can tune $|g_{\alpha\beta}|$ such that the width of the propagating wave packet remains bounded within almost about $10\%$. We also discuss the dynamics of the initially phase separated condensate and have shown the breakdown of Gaussian nature of the wave packets due to collisions. However, when BEC is trapped in simple harmonic oscillator(SHO) potential, for $g_{\alpha\beta}>0$, it is possible for an initially overlapping state to retain its initial shape if $g_{\alpha\beta}$ is less than a critical value ($g_{\alpha\beta}^c$). If $g_{\alpha\beta}$ exceeds $g_{\alpha\beta}^c$, an overlapping state can become phase separated while keeping its shape unchanged.