Terms Of Harmonic Oscillators Research Articles

The Heavy Baryon Masses and Spin-Isospin Dependence

Heavy flavour baryons containing one charm (beauty) quark with light flavour combinations are studied using the hypercentral approach. The considered potential is a combination of Coulombic, linear confining and harmonic oscillator terms. An improved form of the hyperfine interaction and isospin dependent quark potential is introduced. The ground state masses of heavy baryons are computed. Also the magnetic moments of heavy baryons are computed using the effective mass of the confined quarks. The obtained results are very close to the ones obtained in experiments or in other works.

Quantization in terms of symplectic groups: The harmonic oscillator as a generic example

The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space Sq,p ≊ ℝ2. In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space Sϕ,I corresponds to the multiply connected punctured plane ℝ2 – {0}, on which the 3-dimensional symplectic group Sp(2, ℝ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U(1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator Ko of U(l) represents the classical action variable I. It has the possible spectra n + k, n = 0,1,…; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K0 etc. of Sp(2, ℝ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.

Órbitas fechadas e o potencial harmônico de Manev

Estudamos condições para a existência de órbitas fechadas em três potenciais distintos. Primeiramente, confirmamos condições já conhecidas para o problema de Manev. Em um segundo passo, inspirados pelo teorema de Bertrand, construímos o potencial harmônico de Manev substituindo o termo do oscilador harmônico no lugar do termo Newtoniano no potencial de Manev. Assim como no potencial de Manev, encontramos condições similares para a existência de órbitas fechadas que são relacionadas com valores restritos do momento angular. Analisamos também um potencial Newtoniano corrigido com o termo harmônico.

A study of the confined 2D isotropic harmonic oscillator in terms of the annihilation and creation operators and the infinitesimal operators of the SU(2) group

The eigenspectral properties of the 2D isotropic harmonic oscillator, centrally enclosed in the symmetric box with impenetrable walls, are studied for the first time using the annihilation and creation operators and the infinitesimal operators of the SU(2) group. It is shown explicitly how the imposition of the Dirichlet boundary condition at a certain uniquely prescribed confinement radius leads to the energy difference of two harmonic oscillator units between all successive pairs of the confined states, defined by the projection angular momentum quantum numbers [m, m ± 2] such that the lowest energy state corresponding to the chosen m is excluded in the first pair.

Large curvature tunnelling on the reaction path

The reaction path Hamiltonian [J. Chem. Phys., 1980, 72(1), 99] has found few if any successful applications to the explicit treatment of hydrogen tunnelling dynamics in the limit of large path curvature which is typical for hydrogen exchange tunnelling. I argue hat this is due to fundamental limitations of the reaction path formulation. A two-dimensional model mimicking the intramolecular hydrogen transfer in malonaldehyde shows that commonly used approximations can produce misleading results. As a solution to the problem I propose a non-orthogonal representation of the Hamiltonian in terms of local harmonic oscillators distributed along the reaction path. In contrast to the usual reaction path Hamiltonian it has an exact variational limit for all curvatures. This distributed harmonic oscillator approach has the added advantages of stability against variations of the reaction path as well as the ability to treat bifurcating paths.

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola–Kanai oscillator.

A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

Relativistic versus nonrelativistic solution of the N-fermion problem in a hyperradius-confining potential

We investigate the quantum system of N identical fermions in the relativistic limit. In this article the considered potential is a combination of Coulombic, linear confining and harmonic oscillator terms. By using Jacobi coordinates and introducing the hyperradius quantity we obtain the wave functions of the system as well as the corresponding energy eigenvalues. Assuming that all particles are confined within a hypersphere we calculate the corresponding x bag . In particular we consider the case N = 3 which corresponds to baryonic systems. By using the experimental values of the charge radius of each baryon we calculate the potential coefficients. Within our treatment the results of the MIT bag model are recovered for N = 1. Finally we compare the results obtained by the Dirac equation with the corresponding results of the Schrodinger equation and we find that the energy spectra obtained by the former are much closer to experimental values.

Noncommutative induced gauge theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ4-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

Quantum collapse of dust shells in 2 + 1 gravity

This paper considers the quantum collapse of infinitesimally thin dust shells in 2+1 gravity. In 2+1 gravity a shell is no longer a sphere but a ring of matter. The classical equation of motion has been considered by Peleg and Steif and Cristosomo and Olea. The minisuperspace quantum problem can be reduced to that of a harmonic oscillator in terms of the curvature radius of the shell, allowing the use of well-known methods to find the motion of coherent wave packets that give the quantum collapse of the shell. Classically, as the radius of the shell falls below a certain point, a forms. In the quantum problem one can define various quantities that give indications of formation. Without proper definitions of a horizon in quantum gravity, these can be nothing but indications.

Reconstructing quantum geometry from quantum information: spin networks as harmonic oscillators

Loop quantum gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allows us to make a link with non-commutative geometry and to look at the issue of the semi-classical limit of loop quantum gravity from a new perspective. This work is thought of as part of a bigger project of describing quantum geometry in quantum information terms.

Transverse radiation realized as deformed harmonic oscillators

We present a theory of quantized radiation fields described in terms of q-deformed harmonic oscillators. The creation and annihilation operators satisfy deformed commutation relations and the Fock space of states is constructed in this formalism in terms of basic numbers familiar from the theory of quantum groups. Expressions for the Hamiltonian and momentum arising from deformed Heisenberg relations are obtained and their consequences investigated. The energy momentum properties of the vacuum state are studied. The commutation relation for the fields is shown to involve polarization sums more intricate than those encountered in standard quantum electrodynamics, thus requiring explicit representations of polarization vectors. The electric field commutation rules are investigated under simplifying assumptions of polarization states, and the commutator in the deformed theory in this case is shown to be reminiscent of the coordinate-momentum uncertainty relation in the theory of q-deformed quantum oscillators.

The air bubble: experiments on an unusual harmonic oscillator

We discuss a series of easily reproducible experiments with one and two air bubbles immersed in water, and we show that these systems can be analyzed in terms of harmonic oscillators.

Noncommutative 3D harmonic oscillator

We find transformation matrices allowing us to express a noncommutative three-dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following the 'philosophy of simplicity' approach. Noncommutative parameters have a physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.

Quantum Dynamics of Damped Harmonic Oscillator in Terms of Representations of the Group SU(1,1) and in Relation to Models of Continuous Measurement

Quantum Dynamics of Damped Harmonic Oscillator in Terms of Representations of the Group SU(1,1) and in Relation to Models of Continuous Measurement

Exact solutions to the Schrödinger equation for potentials with Coulomb and harmonic oscillator terms

Exact solutions to the Schrödinger equation for potentials containing Coulomb (∼1/ r) plus harmonic oscillator (∼ r 2) terms are found, subject to constraints on the ratio of the strengths of the Coulomb and harmonic oscillator terms. The solutions have the simple form of a product of exponential and polynomial functions.

GENERALIZED DEFORMED OSCILLATOR FOR VORTICES IN SUPERFLUID FILMS

The algebra of observables of a system of two identical vortices in a superfluid thin film is described as a generalized deformed oscillator with a structure function containing a linear (harmonic oscillator) term and a quadratic term. In contrast to the deformed oscillators occurring in other physical systems (correlated fermion pairs in a single-j nuclear shell, Morse oscillator), this oscillator is not amenable to perturbative treatment and cannot be approximated by quons. From the mathematical viewpoint, this oscillator provides a novel boson realization of the algebra su (1,1).

Field quantization in dielectric media and the generalized multipolar Hamiltonian.

Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.

Spinorial representations ofSU(3)from a factored harmonic-oscillator equation

We construct a seven-dimensional Dirac equation for quarks by extending the Klein - Gordon equation to seven dimensions, introducing a harmonic-oscillator term in the extra three dimensions, and taking the `square root' of the resulting equation. To facilitate solving the factored equation, we identify an SU(3) algebra in the higher dimensions and use the eigenstates of its Cartan subalgebra as basis functions. We find that the eigenstates of the extended Dirac equation are indeed exact representations of SU(3), one of which we identify with quarks. Indeed, a whole tower of SU(3) Dirac states appears. For principal quantum number , we find a singlet of mass zero; for , we find two 3s of mass , where is an adjustable constant; for , we find two 6s and two 3-bars, all of mass ; etc. Except for the ground state, all SU(3) representations appear in pairs. It is not clear that the pairs can be identified with weak-isospin doublets.

On some crystal-optic properties of γ1-(GaxIn1−x)2Se3 single crystals

The optical parameters of optically uniaxial γ 1 -(Ga x In 1-x ) 2 Se 3 single crystals are systematically measured and analyzed. The optical rotatory power along the optical c-axis is determined in visible spectral region at 77 K and at room temperature and complemented with theoretical estimates for other crystallographic directions. The g 33 component of the gyration tensor is calculated from the experimental data. The dichroism is studied in the vicinity of the absorption edges. The dispersion of ordinary and extraordinary refraction indices is determined at room temperature for the transparency window 0.57 to 10 μm. Up to 5 μm the dispersion can be interpreted in terms of noninteracting harmonic oscillators (single-term Sellmeier relation).