Simple Harmonic Oscillator Potential Research Articles

Radial excitation of light mesons and the γγ*→π0 transition form factor

We investigate radial excitation of the quark-antiquark pair in the ${\ensuremath{\pi}}^{0}$ meson and its effects on the $\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor in the framework of light cone perturbative QCD. The existing constraints on the light cone wave function of the lowest Fock state $|q\overline{q}⟩$ in the ${\ensuremath{\pi}}^{0}$ meson allow a sizeable radial excitation of the quark-antiquark pair. We construct the light cone wave function for the quark-antiquark pair in the first radially excited state (the 2S state) using a simple harmonic oscillator potential. The distribution amplitude obtained for the 2S state has two nodes in $x$ at low scale of $Q$ and thereby has a much strong scale dependence than the 1S state. Contributions from this radial excitation to the $\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor exhibit different ${Q}^{2}$-dependence behavior from the ground state and thus can modify the prediction for the transition form factor in the medium-large region of ${Q}^{2}$.

A simple superconductor quantum interference device for testing gravity

A simple tabletop setup based on a superconductor quantum interference device is proposed to test the gravitational interaction. A D-shaped superconducting loop has the straight segment immersed inside a massive sphere while the half-circle segment is wrapped around the sphere. The superconducting condensate within the straight arm of the loop thus bathes inside a gravitational simple harmonic oscillator potential while the condensate in the half-circle arm bathes in the constant gravitational potential around the sphere. The resulting phase difference at the Josephson junctions on both sides of the straight arm induces a sinusoidal electric current that has a frequency determined by the precise gravitational potential due to the massive sphere.

Microscopic calculations of the electric Quadrupole transition strengths of Be isotopes (9, 10, 12, 14)

Electric Quadrupole transitions are calculated for beryllium isotopes (9, 10, 12 and 14). Calculations with configuration mixing shell model usually under estimate the measured E2 transition strength. Although the consideration of a large basis no core shell model with 2ℏtruncations for 9,10,12 and14 where all major shells s, p, sd are used, fail to describe the measured reduced transition strength without normalizing the matrix elements with effective charges to compensate for the discarded space. Instead of using constant effective charges, excitations out of major shell space are taken into account through a microscopic theory which allows particle–hole excitations from the core and model space orbits to all higher orbits with 2ℏw excitations which are called core-polarization effects. The two body Michigan sum of three ranges Yukawa potential (M3Y) is used for the core-polarization matrix element. The simple harmonic oscillator potential is used to generate the single particle matrix elements of all isotopes considered in this work. The b value of each isotope is adjusted to reproduce the experimental matter radius, These size parameters of the harmonic oscillator almost reproduce all the root mean square (rms) matter radii for 9,10,12,14Be isotopes within the experimental errors. Almost same effective charges are obtained for the neutron- rich Be isotopes which are smaller than the standard values. The major contribution to the transition strength comes from the core polarization effects. The present calculations of the neutron-rich 12,14Beisotopes show a deviation from the general trends in accordance with experimental and other theoretical studies. The configurations arises from the shell model calculations with core-polarization effects reproduce the experimental B(E2) values.

Quadrupole moment of 14B exotic nucleus


 
 
 
 The quadrupole moment of 14B exotic nucleus has been calculated using configuration mixing shell model with limiting number of orbital's in the model space. The core- polarization effects, are included through a microscopic theory which considers a particle-hole excitations from the core and the model space orbits into the higher orbits with 6ħω excitations using M3Y interaction. The simple harmonic oscillator potential is used to generate the single particle wave functions. Large basis no-core shell model with (0+2)ћω truncation is used for 14B nucleus. The effective charges for the protons and neutrons were calculated successfully and the theoretical quadrupole moment was compared with the experimental data, which was found to be in a good agreement.
 
 
 

Quadrupole moments of exotic carbon isotopes (9C, 11C, 17C and 19C) including core polarization effect

Quadrupole Q moments and effective charges are calculated for 9C, 11C, 17C and 19C exotic nuclei using shell model calculations. Excitations out of major shell space are taken into account through a microscopic theory which are called core-polarization effects. The simple harmonic oscillator potential is used to generate the single particle matrix elements of 9,11,17,19C. The present calculations with core-polarization effects reproduced the experimental and theoretical data very well.

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.

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.

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.

Bose–Einstein condensation transition studies for atoms confined in Laguerre–Gaussian laser modes

Multiply-connected traps for cold, neutral atoms fix vortex cores of quantum gases. Laguerre–Gaussian laser modes are ideal for such traps due to their phase stability. We report theoretical calculations of the Bose–Einstein condensation transition properties and thermal characteristics of neutral atoms trapped in multiply connected geometries formed by Laguerre–Gaussian (LG p l ) beams. Specifically, we consider atoms confined to the anti-node of a LG 0 1 laser mode detuned to the red of an atomic resonance frequency, and those confined in the node of a blue-detuned LG 1 1 beam. We compare the results of using the full potential to those approximating the potential minimum with a simple harmonic oscillator potential. We find that deviations between calculations of the full potential and the simple harmonic oscillator can be up to 3%–8% for trap parameters consistent with typical experiments.

Particle-vibrational doorway states

The escape width ( Gamma d spin up), damping width ( Gamma d spin down) and the average interaction potential (Vd) between a nucleon in a doorway state and a nucleon in more complicated states of the nucleus are discussed as a function of 2 pi ( Gamma mu )/D. It is found that the probability with which a doorway state is present in a compound state is equal to (nd/n)2 and that (Vd) resembles closely a simple harmonic-oscillator potential. The magnitude of the potential is approximately 8 keV.

On the Volterra and Other Nonlinear Models of Interacting Populations

A detailed investigation of various aspects of the populations of interacting biological species is made, using Volterra's and other models. For a system with two species, a detailed analysis is possible, and we plot the time variation of the population of two species for a set of values of the parameters. For a many-species system, it is shown that the time average of the population ${N}_{i}$ of the $i\mathrm{th}$ species is equal to its steady state value ${q}_{i}$. For time averages of other functions of ${N}_{i}$ and ${N}_{i}{N}_{j}$, only the equations satisfied by them are derived. The population growth of a species is studied by assuming that the effect of other species is to introduce a random function of time in the growth equation. The resulting Fokker-Planck equation is shown to have the same form as the Schr\odinger and Bloch equations. The of the Schr\odinger equation depends upon the form of the number-dependent term in the growth function. For the Gompertz form, it is a simple harmonic oscillator potential and, for the Verhulst form, the Morse potential. For both the forms, the Fokker-Planck equation is explicitly solved to obtain the probability distribution $P(N, t)$ of the population as a function of time. It is shown that considerable simplification is achieved in the calculation of the steady state concentrations (${q}_{i}'\mathrm{s}$) of the component species if Pfaffians are used. Since ${q}_{i}$ is equal to the time average of the population of the $i\mathrm{th}$ species, a great deal can be said about the stability of the population in the sense of the rarity of explosion or extinction of one or more species. If the interactions between various species and their growth coefficients (in the absence of interactions) are known, we show that a priori one can determine whether the population will be stable or not and, if not, which of the species will disappear. Also, the stability of the population is discussed when several new species are introduced. We show that the stability is dependent on how the newly introduced species interact with each other and with the population into which they are introduced. If the information about the detailed interactions between various species is absent, as is usually the case, then a statistical mechanical treatment of the population is desirable. We show that for small deviations from steady state populations, the necessary and sufficient condition for such a treatment is that the number of species is large. For arbitrary deviations the latter is a necessary condition and may not be sufficient. This statistical mechanical treatmetn provides an empirical method for calculating the interaction between two species and the stability of the population. A measure of the stability of an ecology is defined which could be used to compare the relative stabilities of two ecologies with the same macroscopic properties. The effect of changes in the interactions between var\'{\i}ous species, due to changes in temperature, humidity, age distribution, etc., is studied by assuming the rate constants to be random. A master equation for the probability distributions of ${N}_{i}'\mathrm{s}$ is derived. It is shown that the stationary population distribution is Poisson only if the variation in the rate constants is not too rapid. A brief outline of another stochastic model for the population growth in terms of the probabilities of birth and death of the individuals is given. Since the members of the population do not react instantaneously to any change in the environment, the prey-predator interaction does not affect the population of either prey or predator instantaneously, and the egg is not converted into an adult instantaneously, the effects of the time lags in the above processes on the behavior, in particular the stability, of the population are studied. Further generalizations of the Volterra model are discussed, and a brief review of other systems of interacting species, e.g., systems of biochemical oscillators, nervous systems, multimode lasers, systems of simultaneously growing bacteria, etc., is given. Finally, a sampling of the experiments which throw some light on the validity of Volterra's model and the statistical mechanical treatment is given.

Binding energy calculations for finite nuclei with tensor forces

The method of Nickel, Tomusiak and Peierls for computing nuclear binding energies is extended to deal with tensor forces. A second order calculation is carried out for 16O using the Hamada-Johnston potential. It is found that the method is not useful when the simple harmonic oscillator potential is used as the shell model potential of the unperturbed Hamiltonian. It is suggested that the convergence of the perturbation series can be improved if the single particle potential is modified so as to be approximately self-consistent.

Neutrons from Negative-Muon Capture

Average neutron-emission and neutron-multiplicity distributions from ${\ensuremath{\mu}}^{\ensuremath{-}}$ capture in Al, Si, Ca, Fe, Ag, I, Au, and Pb were measured. A high-efficiency (54.5% for fission neutrons) cadmium-loaded liquid-scintillator tank was used as the neutron detector. Simple nuclear models with Fermi-gas and Gaussian nucleon-momentum distributions were used to fit the experimental results. A reduced nucleon effective mass was employed to yield the observed average neutron multiplicity, and values were obtained as a function of the width of the assumed nucleon-momentum distributions. Use of the Gaussian momentum distribution obtained in experiments on quasi-elastic scattering gives effective masses ranging from \ensuremath{\cong}0.7 ${M}_{p}$ in the lighter nuclei to \ensuremath{\cong}0.45 ${M}_{p}$ in the heavier nuclei. An excited-Fermi-gas distribution gives larger effective masses for the lighter nuclei and smaller effective masses for the heavier nuclei than does the Gaussian model. Neither of these models gives good agreement with the neutron-multiplicity distributions, although calculations in which direct neutron emission and nucleon clustering on the nuclear surface are assumed improve the fit. For both the Fermi-gas and Gaussian models, the average nuclear excitation energy varies linearly with average neutron multiplicity and is relatively insensitive to the model parameters. When the average nuclear excitation is expressed in units of the muon mass reduced by its $K$-shell binding energy, the result (with the exception of calcium) is constant, 0.18\ifmmode\pm\else\textpm\fi{}0.01, over the wide range of atomic numbers covered in the experiment. For ${\mathrm{Ca}}^{40}$ we also made a shell-model calculation using a simple harmonic-oscillator potential. The agreement with the observed average neutron multiplicity is good although perhaps fortuitous.