Harmonic Oscillator Wave Functions Research Articles

Five-body Moshinsky brackets

In variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for five-body systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.

On the harmonic oscillator wavefunction modified to account for variations in the width of a trapped Bose gas

We propose to augment the harmonic oscillator wavefunction so that it can describe changes in the width of a trapped Bose gas under variations of temperature and interactions. This is achieved by the inclusion of a variational parameter which describes the width of the Bose gas. The goal is to enable a simple evaluation of the thermodynamic properties by the application of the second quantized many-body Hamiltonian. It is demonstrated that the width of this wavefunction decreases with increasing temperature until the zeroth harmonic oscillator state vanishes completely close to the transition temperature. This is a manifestation of the depletion of the harmonic oscillator ground state. Beyond the critical temperature of the Bose gas, the width asymptotically approaches a constant value because the system cannot shrink below a certain limit due to the barrier of repulsive interactions between the bosons. Some previous work having made use of the standard harmonic oscillator wavefunction to describe strongly interacting systems at finite T is mildly criticized. This is because the harmonic oscillator wavefunction in its common form does not allow changes in the width of a trapped Bose gas in response to changes in the interactions and/or temperature.

Two-body strong decay of light mesons with Gaussian expansion method

In this paper, we survey the radial and orbital excitations of I = 0 and [Formula: see text] systems anticipated up to 2.0 GeV in the framework of the chiral quark model. The Schrödinger equation is solved by using Gaussian expansion method. And using the wave functions obtained, instead of the simple harmonic oscillator wave functions, we study systematically the two-body strong decay (limited to S + S modes) of the systems. By comparing with the experimental data, we try to identify the normal states and exotic states.

Electroexcitation of Low-Lying Particle-Hole RPA States of 16O with WBP Interaction

The nuclear structure of 16O is studied in the framework of the particle-hole random phase approximation (ph RPA). The Hamiltonian is diagonalized within a model space with particle orbits {1d5/2,1d3/2, and 2s1/2} and the hole orbits {1p3/2 and 1p1/2} using Warburton and Brown interaction WBP. The ph RPA calculations are tested, by comparing the electron scattering form factors with the available experimental data. The results of electron scattering form factors and reduced transition strength for the states: 1−, T = 0 (7.116 MeV); 2−, T = 1 (12.968 MeV); 2−, T = 1 (20.412 MeV); and 3−, T = 0 (6.129 MeV) are interpreted in terms of the harmonic-oscillator (HO) wave functions of size parameter b. The occupation probabilities of the single particle and hole orbits are calculated. The spurious states are removed by adding the center of mass (CM) correction to the nuclear Hamiltonian. A comparison with the available experiments data is presented.

Electromagneticρ-meson form factors in point-form relativistic quantum mechanics

The relativistic point-form formalism that we proposed for the study of the electroweak structure of few-body bound states is applied to calculate the elastic form factors of spin-1 mesons, such as the rho meson, within constituent-quark models. We treat electron-meson scattering as a Poincare-invariant coupled-channel problem for a Bakamjian-Thomas mass operator and extract the meson current from the resulting invariant 1-photon-exchange amplitude. Wrong cluster properties inherent in the Bakamjian-Thomas framework are seen to cause spurious contributions in the current. These contributions, however, can be separated unambiguously from the physical ones and we end up with a meson current with all required properties. Numerical results for the rho-meson form factors are presented assuming a simple harmonic-oscillator bound-state wave function. The comparison with other approaches reveals a remarkable agreement of our results with those obtained within the covariant light-front scheme proposed by Carbonell et al. [Phys. Rep. 300, 215 (1998)].

Transitions ofBc→ψ(1S,2S)and the modified harmonic oscillator wave function in the light front quark model

The LHC$_b$ collaboration has systematically measured the rates of $B_c\to J/\psi K$, $B_c\to J/\psi D_s$, $B_c\to J/\psi D_s^*$ and $B_c\to \psi(2S) \pi$. The new data enable us to study relevant theoretical models and further determine the model parameters. In this work, We calculate the form factors for the transitions $B_c\to J/\psi$ and $B_c\to \psi(2S)$ numerically, then determine the partial widths of the semi-leptonic and non-leptonic decays. The theoretical predictions on the ratios of $\Gamma(B_c\to J/\psi K)/\Gamma(B_c\to J/\psi \pi)$, $\Gamma(B_c\to J/\psi D_s)/\Gamma(B_c\to J/\psi \pi)$ and $\Gamma(B_c\to J/\psi D_s^*)/\Gamma(B_c\to J/\psi \pi)$ are consistent with data within only 1$\sigma$. Especially, for calculating $\Gamma(B_c\to \psi(2S)X)$ the modified harmonic oscillator wave function (HOWF) which we developed in early works is employed and the results indicate that the modified harmonic oscillator wave function works better than the traditional HOWF.

Charged-current quasielastic neutrino scattering cross sections on12C with realistic spectral and scaling functions

Charge-current quasielastic (CCQE) (anti)neutrino scattering cross sections on a $^{12}$C target are analyzed using a spectral function $S(p,{\cal E})$ that gives a scaling function in accordance with the ($e,e'$) scattering data. The spectral function accounts for the nucleon-nucleon (NN) correlations, it has a realistic energy dependence and natural orbitals (NO's) from the Jastrow correlation method are used in its construction. In all calculations the standard value of the axial mass $M_A= 1.032$ GeV/c$^2$ is used. The results are compared with those when NN correlations are not included, as in the Relativistic Fermi Gas (RFG) model, or when harmonic-oscillator (HO) single-particle wave functions are used instead of NO's. The role of the final-state interactions (FSI) on the theoretical spectral and scaling functions, as well as on the cross sections is accounted for. A comparison of the results for the cases with and without FSI, as well as to results from the phenomenological scaling function obtained from the superscaling analysis (SuSA) is carried out. Our calculations based on the impulse approximation (IA) underpredict the MiniBooNE data, but agree with the data from the NOMAD experiment. The possible missing ingredients in the considered theoretical models are discussed.

Study of Electromagnetic Multipole Transition Half-Lives of One-Hole 15O–15N and One-Particle 17O–17F Mirror Nuclei

Energy spectrum and wave functions are obtained numerically with a potential consisting of Woods-Saxon, Coulomb, and spin-orbit coupling parts for the nuclei 15O, 15N, 17O, and 17F. The radial parts of the wave functions are used to calculate some matrix elements of electromagnetic transitions. These results are applied to calculate half-lives of low-lying exited states in the one-particle 17O and 17F as well as in the one-hole 15O and 15N isotopes. The calculated half-lives are compared with available experimental and theoretical results based on harmonic oscillator wave functions and Weisskopf units. In comparison with the results calculated from the other methods, our results based on the Woods-Saxon potential indicate a satisfactory agreement with accessible experimental data.

Harmonic Oscillator Wave Functions of a Self-Assembled InAs Quantum Dot Measured by Scanning Tunneling Microscopy

InAs quantum dots embedded in an AlAs matrix inside a double barrier resonant tunneling diode are investigated by cross-sectional scanning tunneling spectroscopy. The wave functions of the bound quantum dot states are spatially and energetically resolved. These bound states are known to be responsible for resonant tunneling phenomena in such quantum dot diodes. The wave functions reveal a textbook-like one-dimensional harmonic oscillator behavior showing up to five equidistant energy levels of 80 meV spacing. The derived effective oscillator mass of m* = 0.24m0 is 1 order of magnitude higher than the effective electron mass of bulk InAs that we attribute to the influence of the surrounding AlAs matrix. This underlines the importance of the matrix material for tailored QD devices with well-defined properties.

Unified treatment of Franck–Condon Factors over harmonic oscillator wave function using binomial expansion theorems

In this article an alternative analytical formula is proposed for Franck–Condon (FC) factors. The proposed method enables us to accurately determine the various level intensities in the spectrum of dia- and polyatomic molecules. This method is based on the use of binomial expansion theorem for the analytical representation of the FC factors. As an example of the effectiveness of the method, we present the calculation results of the FC factors of the molecules AlO, CeO, CrH, CrO, LaO, and NiH. The calculated results are in good agreement with other calculations obtained by theoretical and experimental methods.

CALCULATION OF THE ELASTIC p6He AND p8He SCATTERING IN GLAUBER APPROXIMATION

A series of calculations of differential cross-sections (DCSs) and analyzing powers (Ay) of the elastic proton scattering on isotopes 6 He and 8 He were presented in the framework of the theory of Glauber multiple diffraction scattering. The three-body α–n–n and harmonic oscillator wave functions (WFs) were used for 6 He , for 8 He — the function of density distribution in LSSM. For p6 He scattering, the expansion of the Glauber operator into a series of multiple-scattering is written in the form consistent with the picture of weakly bound clusters in halo-nuclei, while for p8 He scattering the members of single and double collisions are included in the operator. We have done the comparison of our results with available experimental data (including new data of Ay for p6 He scattering at the E = 71 MeV/nucleon) and with the calculations' results in the other approaches (high-energy, optical model, RMF). Also, a comparison of the characteristics of two isotopes 6 He and 8 He was made, which showed that they correlate quite well with each other.

Discrete Spherical Harmonic Oscillator Transforms on the Cartesian Grids Using Transformation Coefficients

The analog harmonic oscillators are well-studied in quantum physics, including their energy states, wavefunctions, orthogonal properties, and eigenfunctions of the Fourier transform. In addition, the continuous solutions in different dimension and coordinate systems are known. Some discrete equivalents of the 1D wavefunctions were also studied. However, in the 3D spherical coordinate system, the discrete equivalents of the 3D wavefunctions are not established. In this paper, we focus on the spherical harmonic oscillator wavefunctions (SHOWs) the spherical harmonic oscillator transforms (SHOTs), and their discrete implementation. The SHOWs can be synthesized by linear combinations of the Hermite Gaussian functions with proper transformation coefficients. We find that computing the coefficients can be speeded up using the fast Fourier transforms or some recursive relations. These coefficients relate the Hermite transforms with the SHOTs. Some applications of the discrete SHOWs and the discrete SHOTs are introduced. First, the SHOWs are exactly the eigenfunctions of the 3D DFT. Also, the SHOTs can be used to derive the spherical harmonic oscillator descriptor (SHOD), which is a rotational invariant descriptor. We find that the SHOD is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3D Cartesian sampled, bandlimited input data. Besides, the SHOTs can be used to decompose 3D signals into spherical components. Hence, 3D signal reconstruction is done using partially chosen spherical component, and 3D data compression for MRI data is demonstrated using SHOTs for medical applications.

Properties of bottomonium in a semi-relativistic model

Using a semi-relativistic potential model we investigate the spectra and decays of the bottomonium (bb̄) system. The Hamiltonian of our model consists of a relativistic kinetic energy term, a vector Coulomb-like potential and a scalar confining potential. Using this Hamiltonian, we obtain a spinless wave equation, which is then reduced to the form of a single particle Schrodinger equation. The spin dependent potentials are introduced as a perturbation. The three-dimensional harmonic oscillator wave function is employed as a trial wave function and the bb̄ mass spectrum is obtained by the variational method. The model parameters and the wave function that reproduce the the bb̄ spectrum are then used to investigate some of their decay properties. The results obtained are then compared with the experimental data and with the predictions of other theoretical models.

Dynamics of Particle in Confined-Harmonic Potential in External Static Electric Field and Strong Laser Field

Dynamics of a particle in confined-harmonic potential, subjected to external static electric and time-dependent laser fields is studied. The energy levels and wave functions of unperturbed harmonic oscillator are evaluated using B-polynomial Galerkin method. Matrix formulation is used throughout the procedure. This procedure is very simple and efficient in comparison with other methods. Modifications of wave functions and energy levels due to static electric field are also calculated. Finally, absorption spectra of such a driven oscillator are studied and explained.

Electroweak form factors of heavy-light mesons: A relativistic point-form approach

We present a general relativistic framework for the calculation of the electroweak structure of heavy-light mesons within constituent-quark models. To this aim the physical processes in which the structure is measured, i.e., electron-meson scattering and semileptonic weak decays, are treated in a Poincar\'e-invariant way by making use of the point form of relativistic quantum mechanics. The electromagnetic and weak meson currents are extracted from the $1\mathrm{\text{\ensuremath{-}}}\ensuremath{\gamma}$- and $1\mathrm{\text{\ensuremath{-}}}W$-exchange amplitudes that result from a Bakamjian-Thomas type mass operator for the respective systems. The covariant decomposition of these currents provides the electromagnetic and weak (transition) form factors. Problems with cluster separability, which are inherent in the Bakamjian-Thomas construction, are discussed and it is shown how to keep them under control. It is proved that the heavy-quark limit of the electroweak form factors leads to one universal function, the Isgur-Wise function, confirming that the requirements of heavy-quark symmetry are satisfied. A simple analytical expression is given for the Isgur-Wise function and its agreement with a corresponding front-form calculation is verified numerically. Electromagnetic form factors for ${B}^{\ensuremath{-}}$ and ${D}^{+}$ and weak $B\ensuremath{\rightarrow}{D}^{(*)}$ decay form factors are calculated with a simple harmonic-oscillator wave function and heavy-quark symmetry breaking due to finite masses of the heavy quarks is discussed.

CHARMONIUM SPECTRA AND DECAYS IN A SEMIRELATIVISTIC MODEL

We investigate the spectra and decays of charmonium [Formula: see text] system in a semirelativistic potential model. The Hamiltonian of our model consists of a relativistic kinetic energy term, a vector Coulomb-like potential and a scalar confining potential. From this Hamiltonian a spinless wave equation is obtained. The wave equation is then reduced to the form of a single particle Schrödinger equation. The spin dependent potentials are introduced as a perturbation. The three-dimensional harmonic oscillator wave function is employed as a trial wave function and the [Formula: see text] mass spectra is obtained by the variational method. The model parameters and the wave function that reproduce the mass spectra of the [Formula: see text] mesons are then used to investigate some of the decay properties. The results obtained are then compared with the experimental data and with the predictions of other theoretical models. We also propose possible [Formula: see text] assignments for the recently observed charmonium or charmonium-like states.

Heavy quarkonium 2S states in the light-front quark model

We study the charmonium 2S states $\psi'$ and $\eta_c'$, and the bottomonium 2S states $\Upsilon'$ and $\eta_b'$, using the light-front quark model and the 2S state wave function of harmonic oscillator as the approximation of the 2S quarkonium wave function. The decay constants, transition form factors and masses of these mesons are calculated and compared with experimental data. Predictions of quantities such as Br$(\psi' \to \gamma \eta_c')$ are made. The 2S wave function may help us learn more about the structure of these heavy quarkonia.

ORBITALLY EXCITED STATES OF QUARKONIA IN A NONRELATIVISTIC MODEL

Having succeeded in predicting the S wave spectra and decays of [Formula: see text] and [Formula: see text] mesons, Bhaghyesh, K. B. Vijaya Kumar and A. P. Monteiro, J. Phys. G: Nucl. Part. Phys.38, 085001 (2011), in this article, we apply our nonrelativistic quark model to calculate the spectra and decays of the orbitally excited states (P- and D-waves) of heavy quarkonia. The full [Formula: see text] potential used in our model consists of a Hulthen potential and a confining linear potential. The spin hyperfine, spin-orbit and tensor interactions are introduced to obtain the masses of the P- and D-wave states. The three-dimensional harmonic oscillator wave function is employed as a trial wave function to obtain the mass spectra. The model parameters and the wave function that reproduce the mass spectra of [Formula: see text] and [Formula: see text] mesons are used to investigate their decay properties. The two-photon decay widths, two-gluon decay widths and E1 radiative decay widths are calculated. The obtained values are compared with the experimental results and those obtained from other theoretical models.

Non-analytic quantum oscillator image of complete replica symmetry breaking

We describe the effect of replica symmetry breaking in the field distribution function P(h) of the T = 0 Sherrington–Kirkpatrick (SK) model as the difference between a split Gaussian and the first excited state ψ1 of a weakly anharmonic oscillator with non-analytic shift by means of the analogy P(h) ↔ |ψ1(x)|. New numerical calculations of the leading 100 orders of replica symmetry breaking (RSB) were performed in order to obtain P(h), employing the exact mapping between the density of states ρ(E) of the fermionic SK model and P(h) of the standard model, as derived by Perez-Castillo and Sherrington. Fast convergence towards a fixed point function ρ(E) for infinite steps of RSB is observed. A surprisingly small number of harmonic oscillator wavefunctions suffices to represent this fixed point function. This allows us to determine an anharmonic potential V(x) with non-analytic shift, whose first excited state represents ρ(E) and hence P(h). The harmonic potential with unconventional shift V 2(x) ∼ (|x| − x 0)2 = (x − x 0 sign(x))2 already yields a very good approximation, since anharmonic couplings of V(x) − V 2(x) ∼ |x| m , m > 2, decay rapidly with increasing m. We compare the pseudo-gap-forming effect of replica symmetry breaking, hosted by the fermionic SK model, with the analogous effect in the Coulomb glass as designed by Davies, Lee and Rice, and described by Müller and Pankov.

The time-dependent forced anisotropic oscillator in noncommutative phase space

Wave functions of the time-dependent forced anisotropic harmonic oscillator in noncommutative phase space are derived using the linear transformation and unitary transformation methods. The energy spectrum is given for the stationary system. Further, quantum fluctuations and the squeezing effect are investigated. It is found that the anisotropic property of the harmonic oscillator in noncommutative space has the squeezing effect.