Relativistic Harmonic Oscillator Research Articles

A New Relativistic Extension of the Harmonic Oscillator Satisfying an Isochronicity Principle

A new relativistic extension of the one-dimensional simple harmonic oscillator is presented. The equations of motion express the proportionality between the relativistic force acting on a particle and the distance to the oscillation center measured by a comoving observer with the particle, instead of the distance measured by a certain inertial observer. The resulting trajectory is really harmonic, i.e., the ‘proper period’ of the oscillations does not depend on the amplitude, against the anharmonicity of the standard relativistic harmonic oscillator.

Multiquark cluster form factors in the relativistic harmonic oscillator model

A QCD multiquark cluster system is studied in the relativistic harmonic oscillator potential model (RHOPM), and the electromagnetic form factors of the pion, proton and deuteron in the RHOPM are predicted. The calculated theoretical results are then compared with existing experimental data, finding very good agreement between the theoretical predictions and experimental data for these three target particles. We claim that this model can be applied to study QCD hadronic properties, particularly neutron properties, and to find six-quark cluster and/or nine-quark cluster probabilities in light nuclei such as helium 3He and tritium 3H. This is a problem of particular importance and interest in quark nuclear physics.

Examination of the pseudospin symmetry for the relativistic harmonic oscillator with the similarity renormalization group

The similarity renormalization group is used to explore the origin and breaking mechanism of the pseudospin symmetry for the relativistic harmonic oscillator. The calculated energies from the diagonal Dirac operator ${H}_{P}$ are shown to be in agreement with the exact results. In comparison with the contribution of every term to the pseudospin splitting, the quality of the pseudospin symmetry is revealed from the competition of the dynamical effect and the spin-orbit interaction. The shape of the potentials also has an important influence on the pseudospin symmetry. The dependence of the pseudospin symmetry on the shape of $\ensuremath{\Sigma}(r)$ is dominated by the nonrelativistic term, while that of $\ensuremath{\Delta}(r)$ is governed by the dynamical term and the spin-orbit interaction.

Wigner's pseudo-particle relativistic dynamics in external potential field

Abstract The Wigner's pseudo-particle formalism has been generalized to describe quantum dynamics of relativistic particle in external potential field. As a simplest application of the developed formalism the time evolution of the 1D relativistic quantum harmonic oscillator been considered. Due to the complex structure of the evolution equation for Wigner function, the only numerical treatment is possible by combining Monte Carlo and molecular dynamics methods. Relativistic dynamics results in appearance of the new physical effects as opposed to non-relativistic case. Interesting is the complete changing of the shape of the momentum and coordinate distribution functions as well as formation of ‘unexpected’ protuberances. To analyze the influence of relativistic effects on average values of quantum operators, the dependencies on time of average momentum, position, their dispersions and energy have been compared for the non-relativistic and relativistic dynamics.

Non-linear resonance in relativistic preheating

Inflation in the early Universe can be followed by a brief period of preheating, resulting in rapid and non-equilibrium particle production through the dynamics of parametric resonance. However, the parametric resonance effect is very sensitive to the linearity of the reheating sector. Additional self-interactions in the reheating sector, such as non-canonical kinetic terms like the DBI Lagrangian, may enhance or frustrate the parametric resonance effect of preheating. In the case of a DBI reheating sector, preheating is described by parametric resonance of a damped relativistic harmonic oscillator. In this paper, we illustrate how the non-linear terms in the relativistic oscillator shut down the parametric resonance effect. This limits the effectiveness of preheating when there are non-linear self-interactions.

An easy trick to a periodic solution of relativistic harmonic oscillator

In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is investigated by Homotopy perturbation method. Selection of a linear operator, which is a part of the main operator, is one of the main steps in HPM. If the aim is to obtain a periodic solution, this choice does not work here. To overcome this lack, a linear operator is imposed, and Fourier series of sines will be used in solving the linear equations arise in the HPM. Comparison of the results, with those of resulted by Differential Transformation and Harmonic Balance Method, shows an excellent agreement.

Relativistic harmonic oscillator, the associated equations of motion, and algebraic integration methods

We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique is used to solve the associated Liouville equations, yielding the phase-space evolution of an ensemble of relativistic particles, subject to a ``harmonic'' potential. The nonharmonic distortion of the spatial and momentum distributions due to the intrinsic nonlinear nature of the relativistic contributions is discussed. We analyze the relativistic dynamics induced by two types of Hamiltonian, which can be ascribed to those of harmonic oscillator type. Finally, we briefly discuss the quantum aspects of the problem by considering possible strategies for the solution of the associated Salpeter equation.

Oscillations of a Punctual Charge in the Electric Field of a Charged Ring: A Comparative Study

We applied multiple parameters method (MPM) to obtain natural frequency of the nonlinear oscillator with rational restoring force. A frequency analysis is carried out and the relationship between the angular frequency and the initial amplitude is obtained in analytical/numerical form. This equation is analyzed in three cases: the relativistic harmonic oscillator, a mass attached of a stretched elastic wire and oscillations of a punctual charge in the electric field of charged ring. The three and four parameters solutions are obtained. The results obtained are compared with the numerical solution, showing good agreement.

Relativistic Two-Dimensional Harmonic Oscillator Plus Cornell Potentials in External Magnetic and AB Fields

The Klein-Gordon (KG) equation for the two-dimensional scalar-vector harmonic oscillator plus Cornell potentials in the presence of external magnetic and Aharonov-Bohm (AB) flux fields is solved using the wave function ansatz method. The exact energy eigenvalues and the wave functions are obtained in terms of potential parameters, magnetic field strength, AB flux field, and magnetic quantum number. The results obtained by using different Larmor frequencies are compared with the results in the absence of both magnetic field (ωL= 0) and AB flux field (ξ=0) cases. Effect of external fields on the nonrelativistic energy eigenvalues and wave function solutions is also precisely presented. Some special cases like harmonic oscillator and Coulombic fields are also studied.

Spectral properties of the massless relativistic harmonic oscillator

The spectral properties of the pseudo-differential operator(−d2/dx2)1/2+x2 are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.

DYNAMICAL SYMMETRIES OF DIRAC HAMILTONIAN

Several dynamical symmetries of the Dirac Hamiltonian are reviewed and the conditions under which such symmetries hold are considered. These include relativistic spin and orbital angular momentum symmetries, SO (4)× SU σ(2) symmetry for generalized relativistic hydrogen atom that includes an extra Lorentz scalar potential, SU (3)× SU σ(2) symmetry for the relativistic simple harmonic oscillator. The energy spectrum in each case is calculated from group-theoretic considerations.

Amplitude-frequency Relationship for the Relativistic Oscillator

The Hamiltonian approach and the variational approach are utilized to treat the relativistic harmonic oscillator for the amplitude-frequency relationship. The nice reliability is shown by the result comparison with that from open literature. The simplicity and efficiency of the methods are also disclosed for different range of the initial amplitude during looking for the amplitude-frequency relationship for the nonlinear relativistic harmonic oscillator.

Perturbative interpretation of relativistic symmetries in nuclei

Perturbation theory is used systematically to investigate the symmetries of the Dirac Hamiltonian and their breaking in atomic nuclei. Using the perturbation corrections to the single-particle energies and wave functions, the link between the single-particle states in realistic nuclei and their counterparts in the symmetry limits is discussed. It is shown that the limit of S-V=const and relativistic harmonic oscillator (RHO) potentials can be connected to the actual Dirac Hamiltonian by the perturbation method, while the limit of S+V=const cannot, where S and V are the scalar and vector potentials, respectively. This indicates that the realistic system can be treated as a perturbation of spin-symmetric Hamiltonians, and the energy splitting of the pseudospin doublets can be regarded as a result of small perturbation around the Hamiltonian with RHO potentials, where the pseudospin doublets are quasidegenerate.

The action variable and frequency of a relativistic harmonic oscillator

We present three series representations of the frequency of a relativistic harmonic oscillator. The first two representations use two equivalent forms of the action variable. The third representation involves determining its period by direct integration. The energy dependance of the oscillator frequency is manifestly seen in all three representations. We demonstrate that all three forms yield the same expression for the frequency in the case of the weakly relativistic oscillator and have an identical nonrelativistic limit.

RELATIVISTIC HARMONIC OSCILLATOR MODEL FOR QUARK STARS

The relativistic harmonic oscillator (RHO) model of hadrons is used to study quark stars. The mass–radius relationship is obtained and compared with bag model of quark star, using Tolman–Oppenheimer–Volkoff equation. In this model, the outward degenerate pressure due to discrete Landau levels and Landau degeneracy balances the inward gravitational pressure. Where as in bag model the degenerate pressure is due to the standard continuum levels which balances the combined inward pressure due to gravitation and bag pressure. So in RHO model, the confinement effect is included in the degenerate pressure. We found a qualitative similarity, but quantitative differences in mass–radius relationship of quark stars in these two models. Masses and radii are relatively larger and the central energy densities, required for stable quark stars, are lower in RHO model than that of bag model.

Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method

The relativistic harmonic oscillator equation is a nonlinear ordinary differential equation given by: x ¨ + ( 1 - x ˙ 2 ) 3 / 2 x = 0 . In this paper, the differential transformation method (DTM) and a relatively new technique, known as aftertreatment technique, are proposed to obtain new approximate periodic solutions for the relativistic harmonic oscillator equation under the initial conditions x ( 0 ) = 0 , x ˙ ( 0 ) = β .

He’s frequency formulation for the relativistic harmonic oscillator

In this paper, He’s frequency formulation is used to solve the nonlinear differential equation that governs the oscillation of the relativistic oscillator. The obtained solution with high accuracy shows that He’s frequency formulation is a simple but promising method without any requirement for advanced calculus.

Relativistic harmonic oscillator revisited

The familiar Fock space commonly used to describe the relativistic harmonic oscillator, for example as part of string theory, is insufficient to describe all the states of the relativistic oscillator. We find that there are three different vacua leading to three disconnected Fock sectors, all constructed with the same creation-annihilation operators. These have different spacetime geometric properties as well as different algebraic symmetry properties or different quantum numbers. Two of these Fock spaces include negative norm ghosts (as in string theory) while the third one is completely free of ghosts. We discuss a gauge symmetry in a worldline theory approach that supplies appropriate constraints to remove all the ghosts from all Fock sectors of the single oscillator. The resulting ghost free quantum spectrum in d+1 dimensions is then classified in unitary representations of the Lorentz group SO(d,1). Moreover all states of the single oscillator put together make up a single infinite dimensional unitary representation of a hidden global symmetry SU(d,1), whose Casimir eigenvalues are computed. Possible applications of these new results in string theory and other areas of physics and mathematics are briefly mentioned.

Intrinsically extended stochastic particles with internal de Sitter symmetry

We give the propagation of extended particles in a generic curved spacetime. The extended particle may be a scalar system of two quarks viewed as two quantum modes. Precisely, the first mode represents the global location of the extended particle in the curved spacetime and is quantized by the geometro-stochastic method which seems to be well suited for that purpose. The other mode, which represents a relative motion and is naturally confined in a de Sitter internal spacetime, is quantized by the method of induced representations. This corresponds to the relativistic harmonic oscillator in which the interaction has been replaced by a curvature of the internal space (relativistic rotator). States of the extended particle are then defined in a Hilbert bundle structure with the direct product of the external Poincaré and internal de Sitter symmetries playing the role of the structural group. Intertwining operators are used to define propagation in one fibre as a transition amplitude between the so-called local quantum frames. Parallel transport of these frames is used to define the total propagation of the extended particle as advocated by the geometro-stochastic theory.

Solution of the relativistic (an)harmonic oscillator using the harmonic balance method

The harmonic balance method is used to construct approximate frequency–amplitude relations and periodic solutions to the relativistic oscillator. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the square-root expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes, A, while the discrepancy between the second approximate frequency and the exact one never exceeds 0.82% and tends to 0.52% when A tends to infinity. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.