Relativistic Harmonic Oscillator Research Articles

Effect of the Applied Electric Field on the Thermal Properties of the Relativistic Harmonic Oscillator in One Dimension

We study the relativistic harmonic oscillators (Dirac and Klein–Gordon ones) in a constant external electric field. The solutions obtained are exact. These solutions allowed us to focus on the effect of the external electric field on the thermal properties of such oscillators. These properties are calculated by means of the Zeta-based method. Some figures have been built to show the mentioned effect.

The generalized relativistic harmonic oscillator with the Snyder-de Sitter algebra

The Snyder-de Sitter (SdS) algebra is a model of non-commutative space–time admitting three fundamental parameters: the speed of light, the Planck mass and the cosmological constant, and therefore can be seen as an example of triply special relativity. In this paper, we first examine some aspects of the free Dirac equation, notably the Zitterbewegung (ZB) phenomenon, within the (SdS) algebra. Then, we investigate the effect of this algebra on the dynamics of the (1+1)-dimensional Generalized Relativistic Harmonic Oscillator (GRHO). Using the momentum representation, the oscillator equation is solved analytically: the exact normalized wave functions are expressed in terms of Gegenbauer polynomials and the corresponding energy equation is obtained. It is shown that the presence of the minimal length and the minimal momentum can drastically affect the spectroscopy of the system. The Dirac oscillator and other important particular cases are also discussed.

On the Relativistic Harmonic Oscillator

The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) → 1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.

ON RELATIVISTIC HARMONIC OSCILLATOR

A relativistic quantum harmonic oscillator in 3+1 dimensions is derived from a quaternionic non-relativistic quantum harmonic oscillator. This quaternionic equation also yields the Klein-Gordon wave equation with a covariant (space-time dependent) mass. This mass is quantized and is given by $m_{*n}^2=m_\omega^2\left(n_r^2-1-\beta\,\left(n+1\right)\right)\,,$ where $m_\omega=\frac{\hbar\omega}{c^2}\,,$ $\beta=\frac{2mc^2}{\hbar\,\omega}\, $, $n$, is the oscillator index, and $n_r$ is the refractive index in which the oscillator travels. The harmonic oscillator in 3+1 dimensions is found to have a total energy of $E_{*n}=(n+1)\,\hbar\,\omega$, where $\omega$ is the oscillator frequency. A Lorentz invariant solution for the oscillator is also obtained. The time coordinate is found to contribute a term $-\frac{1}{2}\,\hbar\,\omega$ to the total energy. The squared interval of a massive oscillator (wave) depends on the medium in which it travels. Massless oscillators have null light cone. The interval of a quantum oscillator is found to be determined by the equation, $c^2t^2-r^2=\lambda^2_c(1-n_r^2)$, where $\lambda_c$ is the Compton wavelength. The space-time inside a medium appears to be curved for a massive wave (field) propagating in it.

Quantum mechanics with spatial non-local effects and position-dependent mass

A new higher order Schrödinger equation characterized by a position-dependent mass is introduced based on long-range spatial kernel effects and von Roos arguments. The extended Schrödinger equation depends on the sign of the moments M k , k = 0 , 1 , 2 , … and a stabilized quantum dynamic is realized for M 2 > 0 and M 4 > 0 . We have discussed its implications in several quantum mechanical systems where more than a few were raised, mainly the emergence of the quantum Pais-Uhlenbeck and the relativistic quantum harmonic oscillators besides the complex periodic potential characterized by a PT symmetry.

The generalized relativistic harmonic oscillator with a point interaction

We study the perturbation of the one-dimensional generalized relativistic harmonic oscillator (GRHO) by a Lorentz scalar delta-shaped interaction. By exactly solving the implied Dirac equation, we show that the presence of the singular potential brings about drastic changes in the structure of the energy spectrum of the system. Particularly, an apparent anomaly of doubly degenerate energy levels is noted when the strength of the local term becomes infinite and energy eigenvalues in the range [−mc2,mc2] are obtained for some negative values of the delta-coupling and for all settings of the oscillator parameters.

The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

Fermi motion and medium effects on reaction cross sections of 13,14,15B + C, Al at intermediate and high energies

Analysis of reaction cross sections of [Formula: see text] are performed with Glauber model (GM) including medium modifications arising from Fermi motion and in-medium nucleon–nucleon (NN) total reaction cross section. The density distributions of Boron isotopes are described by a deformed-average Fermi shape, whose proton and neutron radii and deformation parameters are derived from relativistic mean field (RMF), harmonic oscillator (HO) and two-parameter Fermi (2pF). These densities can describe the reaction cross section of Boron isotopes except for [Formula: see text] which is better described by an HO core plus extended Gaussian (HOG) or HO plus a modified Yukawa (HOMY) tail, which depends on the 1n separation energy. It is found that Fermi motion effects dominated at intermediate energy, and they are important to describe the data in addition to in-medium effects. In-medium effects, which are incorporated locally, reduced the reaction cross section by about 3–5% at all energies. [Formula: see text] is confirmed as a one-neutron halo nucleus, where a large extended tail is observed in the extracted HOMY and HOG model densities as well as in the extracted radii, where the matter radius of [Formula: see text] is found to be about 2.92 fm, which is larger than that of [Formula: see text] (2.5 fm) and [Formula: see text] (2.82 fm) predicted using the GM model including Fermi motion and in-medium effects.

Noncovariant Lagrangians Are Presented Which Yield Two-Component Equations of Motion for a Class of Relativistic Mechanical Systems in 1 + 1 Dimensions Including the Harmonic Oscillator

First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previous discussions of this system! The second result is that this Oscillator Langrangian generalizes to Langrangians for a class of particles in 1 + 1 dimensions subject to an arbitrary potential V which is space dependent only.

Sensitivity of Beam-Target Polarized Response Functions in Elastic Electron–Deuteron Scattering to Nucleon Structure and Modern N N Potentials

Polarized response functions for beam and target polarization in the elastic electron-deuteron scattering process are investigated. We report numerical results for all the non-vanishing polarized response functions as functions of the four-momentum transfer square Q2. The sensitivity of the predicted results to the nucleon form factors is investigated. For the neutron and proton form factors, the standard dipole fit, the modified dipole fit 1, the modified dipole fit 2, and the relativistic harmonic oscillator model are used. In addition, the dependence of the results on the deuteron wave functions obtained from different modern N N potentials is studied. For this purpose, the realistic and high-precision Argonne υ18, chiral N3LO, Nijmegen-I, and CD-Bonn N N potentials are used. We found that all the non-vanishing polarized response functions are sensitive to the choices of nucleon form factors, whereas they are slightly dependent on the N N potential model adopted for the deuteron wave function. The results for all polarized response functions using the relativistic harmonic oscillator model for the nucleon form factors are found to be very small in comparison to the results of other models.

Reestablishment of SO(4) symmetry in the relativistic Coulomb problem from SYM

We obtain a scalar field theory by applying a specific Higgs mechanism on SYM and imposing a constraint. We see that the resulting relativistic equation describes a charged particle non-minimally coupled to the Coulomb potential. The resulting non-minimal coupling allows us to build a relativistic Runge-Lenz vector that generates, together with the angular momentum, the SO(4) algebra. We also construct a Kustaanheimo-Stiefel duality between this modified relativistic Coulomb problem and the modified relativistic harmonic oscillator and see how the integrals of motion of both problems are mapped.

Lagrange-Mesh Method for Deformed Nuclei With Relativistic Energy Density Functionals

The application of relativistic energy density functionals to the description of nuclei leads to the problem of solving self-consistently a coupled set of equations of motion to determine the nucleon wave functions and meson fields. In this work the Lagrange-mesh method in spherical coordinates is proposed for the numerical calculation. The essential field equations are derived from the relativistic energy density functional and the basic principles of the Lagrange-mesh method are delineated for this particular application. The numerical accuracy is studied for the case of a deformed relativistic harmonic oscillator potential with axial symmetry. Then the method is applied to determine the point matter distributions and deformation parameters of self-conjugate even-even nuclei from 4He to 40Ca.

Experimental realization of a relativistic harmonic oscillator

We report the experimental study of a harmonic oscillator in the relativistic regime. The oscillator is composed of Bose-condensed lithium atoms in the third band of an optical lattice, which have an energy–momentum relation nearly identical to that of a massive relativistic particle, with an effective mass reduced below the bare value and a greatly reduced effective speed of light. Imaging the shape of oscillator trajectories at velocities up to 98% of the effective speed of light reveals a crossover from sinusoidal to nearly photon-like propagation. The existence of a maximum velocity causes the measured period of oscillations to increase with energy; our measurements reveal beyond-leading-order contributions to this relativistic anharmonicity. We observe an intrinsic relativistic dephasing of oscillator ensembles, and a monopole oscillation with exactly the opposite phase of that predicted for non-relativistic harmonic motion. All observed dynamics are in quantitative agreement with longstanding but hitherto-untested relativistic predictions.

Dirac and Klein-Gordon oscillators on anti-de Sitter space

The Dirac and Klein-Gordon oscillators on anti-de Sitter space were considered in a space with deformed commutation relations. Anti-de Sitter commutation relations give rise to the appearance of minimal uncertainty in the momentum. Using the position space representation, we determine the energy eigenvalues and the eigenfunctions for both cases. The wave functions can be given in terms of Gegenbauer polynomials. The high-temperature thermodynamic properties of the relativistic harmonic oscillators are then analyzed.

Generalized relativistic harmonic oscillator in minimal length quantum mechanics

We solve the generalized relativistic harmonic oscillator in 1 + 1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state solutions for fermions and antifermions. Furthermore, we also find an isolated solution from the Sturm–Liouville scheme. All cases already analyzed in the literature are obtained as particular cases.

Second quantization of a covariant relativistic spacetime string in Steuckelberg–Horwitz–Piron theory

A relativistic 4D string is described in the framework of the covariant quantum theory first introduced by Stueckelberg [Helv. Phys. Acta 14, 588 (1941)], and further developed by Horwitz and Piron [Helv. Phys. Acta 46, 316 (1973)], and discussed at length in the book of Horwitz [Relativistic Quantum Mechanics, Springer (2015)]. We describe the space-time string using the solutions of relativistic harmonic oscillator [J. Math. Phys. 30, 66 (1989)]. We first study the problem of the discrete string, both classically and quantum mechanically, and then turn to a study of the continuum limit, which contains a basically new formalism for the quantization of an extended system. The mass and energy spectrum are derived. Some comparison is made with known string models.

Quark mean field model with pion and gluon corrections for Λ and Ξ0 hypernuclei and neutron stars

Properties of $\Lambda,~\Xi^0$ hypernuclei and neutron stars are investigated in quark mean field model with pion and gluon corrections. Firstly, $u, ~d$, and $s$ quarks are confined by relativistic harmonic oscillator potentials to generate the baryons, like nucleon, $\Lambda, ~\Sigma$, and $\Xi$ hyperons. The effects of pion-quark coupling and one-gluon exchange are considered perturbatively. Then, the baryons interact with each other through exchanging $\sigma, ~\omega,$ and $\rho$ mesons between quarks in hypernuclei and nuclear matter. The strengths of confinement potentials for $u, ~d,$ and $s$ quarks are determined by the masses and radii of free baryons. The coupling constants between the quarks and mesons are fixed by the ground-state properties of several nuclei and single-hyperon potentials at nuclear saturation density, which arises three parameter sets for the coupling constants between mesons and quarks, named QMF-NK1S, QMF-NK2S, and QMF-NK3S. Compared to the results of previous quark mean field model without pion and gluon corrections, it is found that properties of $\Lambda$ hypernuclei, i.e. the single $\Lambda$ energies, are more consistent with the experimental observables. Properties of $\Xi^0$ hypernuclei are also calculated and compared with the results in previous quark mean field model. With these three parameter sets, the neutron stars containing hyperons are investigated through solving the Tolman-Oppenheimer-Volkoff equation. Maximum masses of neutron stars approach $2.1M_\odot$ with hyperons and corresponding radii are around $13$ km.

The invariant Berry phase of the quantum relativistic four-dimensional harmonic oscillator

The invariant Berry phase of the quantum relativistic four-dimensional harmonic oscillator

Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method

Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by means of a combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). The results that we have obtained, a series of powers of functions, have never been reported and show a very good match when compared with other approximate solutions, obtained by different methods. The method here proposed works with high degree of accuracy and because it requires less computational effort, it is very convenient to solve this kind of nonlinear differential equations.

Modified Parker-Sochacki method for solving nonlinear oscillators

ABSTRACTA new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.