We modify a non-perturbative approach based on the Principle of Minimal Sensitivity to calculate the periods of relativistic oscillators with even polynomial potentials of general form. The optimization of the variational parameter is adapted by using the ultrarelativistic limit of the period as a boundary condition, which determines the values of additional free parameters introduced. Resulting general approximations prove to be accurate over the whole solution domain and for the whole class of potentials.
The operator algebras of a new family of relativistic geometric models of the relativistic oscillator are studied. It is shown that, generally, the operator of number of quanta and the pair of the shift operators of each model are the generators of a non-unitary representation of the $so(1,2)$ algebra, except a special case when this algebra becomes the standard one of the non-relativistic harmonic oscillator.
A family of relativistic geometric models is defined as a generalization of the actual anti-de Sitter (1+1) model of the relativistic harmonic oscillator. It is shown that all these models lead to the usual harmonic oscillator in the non-relativistic limit, even though their relativistic behavior is quite different. Among quantum models we find a set of models with countable energy spectra, and another one having only a finite number of energy levels and in addition a continuous spectrum.
Exactly solvable $N$-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational $\vec r_N$-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the $N$-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.
The phase-space representation for a relativistic linear oscillator in a homogeneous external field expressed through the finite-difference equation is constructed. Explicit expressions of the relativistic oscillator Wigner quasi-distribution function for the stationary states as well as of states of thermodynamical equilibrium are obtained and their correct limits are shown.
The two-dimensional relativistic configurational $\vec r$-space is proposed and the exactly solvable finite-difference model of the harmonic oscillator in this space is constructed. The wave functions of the stationary states and the corresponding energy spectrum are found for the model under consideration. It is shown that, they have correct non-relativistic limit. Explicit form of the plane momentum operator is determined.
We show the existence of Lorentz invariant Berry phases generated, in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They are computed numerically by using time independent perturbation theory and the definition of the Berry phase generalized to the framework of SHP relativistic quantum theory.
The frequency of a classical periodic system can be obtained using action variables without solving the dynamical equations. We demonstrate the construction of two equivalent forms of the action variable for a one dimensional relativistic harmonic oscillator and obtain its energy dependent frequency. This analysis of oscillation is compared with the traditional solution of the problem which requires the use of hypergeometric series.
An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schrödinger-like equation. We have found finite-difference raising and lowering operators, which are with the Hamiltonian operator form the close Lie algebra of the $SU(1, 1)$ group.
In this paper we study the $(2+1)$-dimensional Dirac-Dunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external magnetic field by changing the partial derivatives by the Dunkl derivatives. We solve the Dunkl-Klein-Gordon-type equations in polar coordinates in a closed form. The angular part eigenfunctions are given in terms of the Jacobi-Dunkl polynomials and the radial functions in terms of the Laguerre functions. Also, we compute the energy spectrum of this problem and show that, in the non-relativistic limit,…
Exactly-solvable model of the linear singular oscillator in the relativistic configurational space is considered. We have found wavefunctions and energy spectrum for the model under study. It is shown that they have correct non-relativistic limits.
Following to the lines drawn in my previous paper about the S=0 relativistic oscillator I build up an oscillatorlike system which can be named as the S=1 Proca oscillator. The Proca field function is obtained in the framework of the Bargmann-Wigner prescription and the interaction is introduced similarly to the S=1/2 Dirac oscillator case regarded by Moshinsky and Szczepaniak. We obtained the intriguing rule of quantization: E = \hbar ω/2 for the parity states (-1)^j and E = \pm \hbar ω(j+1/2) for the parity states -(-1)^j. There are no radial excitations. Finally, I apply the above-mentioned …
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 non-harmonic distortion of the spatial and momentum distributions due to the intrinsic non-linear nature of the relativistic contributions are discussed. We analyze the relativistic dynamics i…
A quantum realization of the Relativistic Harmonic Oscillator is realized in terms of the spatial variable $x$ and ${\d\over \d x}$ (the minimal canonical representation). The eigenstates of the Hamiltonian operator are found (at lower order) by using a perturbation expansion in the constant $c^{-1}$. Unlike the Foldy-Wouthuysen transformed version of the relativistic hydrogen atom, conventional perturbation theory cannot be applied and a perturbation of the scalar product itself is required.
The relativistic quantum dynamics of an electrically charged particle subject to the Klein-Gordon oscillator and the Coulomb potential is investigated. By searching for relativistic bound states, a particular quantum effect can be observed: a dependence of the angular frequency of the Klein-Gordon oscillator on the quantum numbers of the system. The meaning of this behaviour of the angular frequency is that only some specific values of the angular frequency of the Klein-Gordon oscillator are permitted in order to obtain bound state solutions. As an example, we obtain both the angular frequency…
Multiparametric quantum $gl(2)$ algebras are presented according to a classification based on their corresponding Lie bialgebra structures. From them, the non-relativistic limit leading to quantum harmonic oscillator algebras is implemented in the form of generalized Lie bialgebra contractions.
The main pairs of leader operators of the quantum models of relativistic rotating oscillators in arbitrary dimensions are derived. To this end one exploits the fact that these models generate Pöschl-Teller radial problems with remarkable properties of supersymmetry and shape invariance.
A nonpolynomial one-dimensional quantum potential in the form of an isotonic oscillator (harmonic oscillator with a centripetal barrier) is studied. We provide the non-relativistic bound state energy spectrum E_{n} and the wave functions ψ_{n}(x) in terms of the associated Laguerre polynomials in the framework of the Nikiforov-Uvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding two-component upper- and lower-spinors of the Dirac particle are obtained, in closed form.
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 becau…
The dynamics of wave packets in a relativistic Dirac oscillator is compared to that of the Jaynes-Cummings model. The strong spin-orbit coupling of the Dirac oscillator produces the entanglement of the spin with the orbital motion similar to what is observed in the model of quantum optics. The collapses and revivals of the spin which result extend to a relativistic theory our previous findings on nonrelativistic oscillator where they were known under the name of `spin-orbit pendulum'. There are important relativistic effects (lack of periodicity, zitterbewegung, negative energy states). Many o…
The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum.
Exactly solvable model of the quantum isotropic three-dimensional singular oscillator in the relativistic configurational $\vec r$-space is proposed. We have found the radial wavefunctions, which are expressed through the continuous dual Hahn polynomials and energy spectrum for the model under consideration. It is shown that they have the correct non-relativistic limits.
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 fermion 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.
In this paper we present a straightforward systematic method for the exact and approximate calculation of integrals that appear in formulas for the period of anharmonic oscillators and other problems of interest in classical mechanics.
We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in compact form, i. e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.
The Dirac oscillator is an exactly soluble model recently introduced in the context of many particle models in relativistic quantum mechanics. The model has been also considered as an interaction term for modelling quark confinement in quantum chromodynamics. These considerations should be enough for demonstrating that the Dirac oscillator can be an excellent example in relativistic quantum mechanics. In this paper we offer a solution to the problem and discuss some of its properties. We also discuss a physical picture for the Dirac oscillator's non-standard interaction, showing how it arises …
We study analytically the two dimensional deformed bosonic oscillator equations for charged particles (both spin 0 and spin 1 particles) subject to the effect of a uniform magnetic field. We consider the presence of a minimal uncertainty in momentum caused by the Anti-deSitter model and we use the Nikiforov-Uvarov method to solve the system. The exact energy eigenvalues and the corresponding wave functions are analytically obtained for both Klein-Gordon and scalar Duffin-Kemmer-Petiau cases. For spin 1 DKP case, we deduce the behaviour of the DKP equation and write the non-relativistic energie…
We develop a simple method to obtain approximate analytical expressions for the period of a particle moving in a given potential. The method is inspired to the Linear Delta Expansion (LDE) and it is applied to a large class of potentials. Precise formulas for the period are obtained.
Comment on "Relativistic quantum oscillator model under the effects of the violation of Lorentz symmetry by an arbitrary fixed vector field'' by Faizuddin Ahmed
Facultad de Ciencias Naturales y Matemáticas, Universidad Nacional del Callao（卡亚俄国立大学自然科学与数学学院）；Universidade Federal do Maranhão（马拉尼昂联邦大学）；Universidad Tecnológica del Perú (UTP)（秘鲁理工大学）；Departamento de Ciencias, Universidad Privada del Norte (UPN)（北方私立大学科学系）
We construct coherent states through special superpositions of photon number states of the relativistic isotonic oscillator. In each superposition the coefficients are chosen to be L 2 eingenfunctions of a sigma weight Maass Laplacian on the Poincare disk, which are associated with discrete eigenvalues. For each nonzero m the associated coherent states transform constitutes the m true polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel Kampe de Feriet hypergeometric function. The obtained results coul…
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the wi…
Consider quantum harmonic oscillator, perturbed by an even almost-periodic complex-valued potential with bounded derivative and primitive. Suppose that we know the first correction to the spectral asymptotics $\{Δμ_n\}_{n=0}^\infty$ ($Δμ_n=μ_n-μ_n^0+o(n^{-1/4})$, where $μ_n^0$ and $μ_n$ is the spectrum of the unperturbed and the perturbed operators, respectively). We obtain the formula that recovers the frequencies and the Fourier coefficients of the perturbation.
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…
We investigate the pseudospin symmetry case of a spin-1/2 particle governed by the generalized isotonic oscillator, by presenting quasi exact polynomial solutions to Dirac equation with pseudospin symmetry vector and scalar potentials. The resulting equation is found to be quasi-exactly solvable owing to the existence of a hidden $sl(2)$ algebraic structure. A systematic and closed form solution to the basic equation is obtained using the Bethe ansatz method. Analytic expression for the energy is obtain and the wavefunction is derived in terms of the roots to a set of Bethe ansatz equations.
Bound State Solutions of the Relativistic Finite-difference Equation for the Ring-shaped Quesne Oscillator Potential
Institute of Physics, Ministry of Science and Education（阿塞拜疆教育与科学部物理研究所）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；Sumgait State University（苏姆盖特国立大学）
Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and that is based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States which correspond to the representations with the lowest weights s=1/4 and s=3/4 with four possible (non-trivial) fractional representations for the group decomposition of the spin structure. From th…
SO(2,1) is the symmetry algebra for a class of three-parameter problems that includes the oscillator, Coulomb and Morse potentials as well as other problems at zero energy. All of the potentials in this class can be mapped into the oscillator potential by point canonical transformations. We call this class the "oscillator class". A nontrivial graded extension of SO(2,1) is defined and its realization by two-dimensional matrices of differential operators acting in spinor space is given. It turns out that this graded algebra is the supersymmetry algebra for a class of relativistic potentials tha…
We study the lowest energy E of a semirelativistic system of N identical massless bosons with Hamiltonian H= sum{i=1 to N} sqrt(p_i^2)+ sum{j>i=1 to N} g |r_i - r_j|^2, g > 0. We prove the inequalities A [g N^2 (N-1)^2]^{1/3} < E < B [g N^2 (N-1)^2]^{1/3}, where A = 2.33810741 and B = [81/(2 pi)]^{1/3} = 2.3447779. The average of these bounds determines E with an error less than 0.15% for all N > 1.
While internal space-time symmetries of relativistic particles are dictated by the little groups of the Poincaré group, it is possible to construct representations of the little group for massive particles starting from harmonic oscillator wave functions for the quark model. The resulting oscillator wave functions are covariant and can be Lorentz-boosted. It is thus possible to obtain the parton model by boosting the quark model. A review of Wigner's theory of the little groups is given. It is shown that the covariant oscillator wave functions become squeezed as as the system becomes boosted. …
We prove the reducibility of quantum harmonic oscillators in $\mathbb R^d$ perturbed by a quasi-periodic in time potential $V(x,ωt)$ with $\mathit{logarithmic~decay}$. By a new estimate built for solving the homological equation we improve the reducibility result by Grébert-Paturel(Annales de la Faculté des sciences de Toulouse : Mathématiques. $\mathbf{28}$, 2019).
An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the phenomenon of resonance may appear. We give a sufficient condition on the perturbation in order that resonance occurs when the period annulus of the isochronous oscillator is bounded. In this context, resonance means that all solutions escape from the period annulus.
A simple exact analytical solution of the relativistic Duffin-Kemmer-Petiau equation within the framework of the asymptotic iteration method is presented. Exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the Coulomb potentials. As a non-trivial example, the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method.
We study the relativistic version of the $d$-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension $d$. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors o…
The Killingbeck potential consisting of the harmonic oscillator-plus-Cornell potential, is of great interest in high energy physics. The solution of Dirac equation with the Killingbeck potential is studied in the presence of the pseudospin (p-spin) symmetry within the context of the quasi-exact solutions. Two special cases of the harmonic oscillator and Coulomb potential are also discussed.
An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral propertiessuch as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle…
We study the relativistic quantum dynamics of a DKP oscillator field subject to a linear interaction in cosmic string space-time in order to better understand the effects of gravitational fields produced by topological defects on the scalar field. We obtain the solution of DKP oscillator in the cosmic string background. Also, we solve it with an ansatz in presence of linear interaction. We obtain the eigenfunctions and the energy levels of the relativistic field in that background.
We study the lowest energy E of a relativistic system of N identical bosons bound by harmonic-oscillator pair potentials in three spatial dimensions. In natural units the system has the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i - r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N (m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A sharper lower bound is given by the function P = P(mu), where mu…
In recent years, an extensive survey on various wave equations of relativistic quantum mechanics with different types of potential interactions has been a line of great interest. In this regime, special attention has been given to the Dirac equation because the spin-1/2 fermions represent the most frequent building blocks of the molecules and atoms. Motivated by the considerable interest in this equation and its relativistic symmetries (spin and pseudospin) in the presence of solvable potential model, we attempt to obtain the relativistic bound states solution of the Dirac equation with double…
