Note on the harmonic oscillator calculation of two-nucleon transfer form factors

In a recent publication, Iano and Pinkston showed that the zero-range form factor for two-nucleon transfer reactions obtained through their shell model calculation was well approximated in the asymptotic region by the one calculated using the standard well-depth procedure. We wish to show that such an agreement is merely due to the restricted space which they have used. It is found that by including a larger working basis, the form factor may increase in the asymptotic region by as much as 50 % over the one obtained by the well-depth procedure; this in turn will bring the theoretical cross section to within $\text{2}\text{3}$ of the experimental one for the reaction ⁴⁰Ca(t, p)⁴²Ca.     

Nucleon form factors in a relativistic quark model

Electromagnetic form factors of protons and neutrons are investigated based on a relativistic quark model with the inclusion of a pion cloud. Pseudo-scalar π-quark interaction is employed to study the coupling between the nucleon and the π. The results show the important role of the pion cloud for the neutron charge form factor. Moreover, our numerical analysis indicates a difference between the relativistic and the nonrelativistic treatments. Received: 10 March 1999 / Revised version: 14 June 1999     

Pseudospin symmetry and the relativistic ring-shaped non-spherical harmonic oscillator

A generalized relativistic harmonic oscillator for spin 1/2 particles is studied. The eigenfunctions and eigenenergies are obtained for the ring-shaped non-spherical harmonic oscillator by solving Dirac equation with equal mixture of vector and scalar potentials in opposite signs, for which pseudospin symmetry is exact. Several particular cases such as the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.     

Multiquark cluster effect in a three-body system

The charge density of ³He was calculated in the hybrid quark-hadron model. The results clearly show that the multiquark cluster effect in ³He is most likely responsible for the observed central “hole” of ³He charge density.     

The relativistic bound states for a new ring-shaped harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.     

Quantum normal form and the harmonic oscillator withx 6 perturbation

The eigenvalue problem of the harmonic oscillator with thex ⁶ perturbation,H=1/2(p ²+x ²+x ⁶), is investigated using the method of quantum normal form. The energy eigenvalues are found to be in good agreement with the WKB results.     

Stable states of a relativistic harmonic oscillator imbedded in a random stochastic thermostat

An analysis in phase space of the behavior of a relativistic four-dimensional harmonic oscillator undergoing stochastic interactions shows that the group of linear canonical transformations in phase space which leaves invariant the Poisson brackets is anSp(12,4) group, withSp(12,4)U(6,2)SU(1,1)SO(6,2). The application of Cartan's treatment to its behavior implies a classification of its stable states characterized by a set of discrete numbers.     

The calculating formula for radial matrix elements of a relativistic harmonic oscillator

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.     

A relativistic oscillator model applied to lepton-nucleon reactions

A relativistic oscillator model is applied to lepton-nucleon reactions. It agrees quantitatively with data on hadron energy distribution, multiplicity, and $\text{π}{}^{\mathrm{+}}\text{π}{}^{\mathrm{?}}$ ratios.     

Radiative decays and transition form factors of strange mesons in the relativistic constituent quark model

Motivated by the recent lattice QCD results indicating that the topological charge contribution to the flavor singlet axial vector current can be traded off by the constituent quark masses, we investigate the radiative decays of pseudoscalar (π,K, η, η′), vector (ρ,K*, ω, ?) and axial vector (A ₁) mesons using a simple relativistic constituent quark model. For both simplicity and relativity, we take advantage of the distinguished features in the light-cone quantization method: (1) the Fock-state expansion of meson wavefunctions are not contaminated by the vacuum fluctuation, (2) the assignment of meson quantum numbers are given by the Melosh transformation. Except the well-known constituent quark masses of (u,d,s) quarks and the spin-averaged meson masses, the only parameter in the model is the gaussian parameter β which determines the broadness (or sharpness) of radial wavefunction. The computed decay widths and the transition form factors of ρ, ω → π(η)γ*,K* →Kγ* andA ₁ → πγ* at 0≤Q ²≤5 GeV² and π⁰(η) → γ*γ at 0≤Q ²≤3 GeV² are in a remarkably good agreement with the experimental data and the result forA ₁ ⁺ → π₊ γ* transition is quite consistent with the experiments of pion scattering on a nucleus using Primakoff effect. This model is potentially useful in the cocktail analyses of the dilepton productions in proton-proton, proton-nucleus and nucleus-nucleus collisions at SPS energies and a little above.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Electromagnetic form factors of nucleons in a relativistic square-root potential model of independent quarks

The nucleon electromagnetic form factorsG _E ^P (q²),G _M ^P (q²) and the axial-vector form factor G_A(q²) are studied in a relativistic model of independent quarks confined by an equally mixed scalar-vector square root potentialV _q(r)=1/2(1+γ ⁰)(ar ^1/2+ν ₀) taking into account the appropriate centre-of-mass corrections. The respective root-mean-square radii associated withG _E ^P (q²) and G _A (q²) come out as [〈r ²〉 _E ^P ]^1/2=0.86 fm and 〈r _A ² 〉^1/2=0.88 fm. Restoration of chiral symmetry in this model is discussed to derive the pion-nucleon form factorG _πNN(q²) and consequently the pion-nucleon coupling constant is obtained asg _πNN(q²)=12.81 as compared tog _πNN(q²)_exp⋍13.     

One-dimensional model of a quantum nonlinear harmonic oscillator

In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the λ → 0 limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.     

A time-discrete harmonic oscillator model of human car-following

A time-discrete stochastic harmonic oscillator is presented as a model of human car-following behaviour. This describes especially the non-continuous control of a human driver – acceleration changes from time to time at so called action-points and is kept constant in between. Analytical results can be derived which allow to classify the different types of motion possible within this approach. These results show that with weaker control by the human, unstable behaviour of the oscillator becomes more likely. This is in line with common understanding about the causes of accidents. Finally, since even the stochastic behaviour of this model is in parts analytically tractable, the width of the speed-difference and distance fluctuations can be expressed as function of the model’s parameter. This allows a fresh view on empirical car-following data and the identification of parameters from real data in the context of the theory presented here.     

U(3) and pseudo-U(3) symmetry of the relativistic harmonic oscillator

We show that a Dirac Hamiltonian with equal scalar and vector harmonic oscillator potentials has not only a spin symmetry but a U(3) symmetry and that a Dirac Hamiltonian with scalar and vector harmonic oscillator potentials equal in magnitude but opposite in sign has not only a pseudospin symmetry but a pseudo-U(3) symmetry. We derive the generators of the symmetry for each case.     

Meson spectrum in a relativistic harmonic model with instanton-induced interaction

On the basis of the phenomenological relativistic harmonic model for quarks, we have obtained the ground-state masses of the light pseudo-scalar and vector mesons. The full Hamiltonian used in the investigation has Lorentz scalar + vector confinement potential, along with one-gluon-exchange potential (OGEP) and the instanton-induced quark-antiquark interaction. A good agreement is obtained with the experimental masses. The respective role of instanton-induced interaction and OGEP for the determination of the meson masses is discussed.Received: 18 October 2002, Revised: 13 January 2003, Published online: 23 December 2003PACS: 12.39.Ki Relativistic quark model - 12.39.Pn Potential models - 14.40.Aq , K, and mesonsS. Pepin: Present address: rue de Sluse 13, B-4000 Liége, Belgium