Isotropic Harmonic Oscillator Research Articles

Angular momentum on the lattice: The case of nonzero linear momentum

The irreducible representations (IRs) of the double cover of the Euclidean group with parity in three dimensions are subduced to the corresponding cubic space group. The reduction of these representations gives the mapping of continuum angular momentum states to the lattice in the case of nonzero linear momentum. The continuous states correspond to lattice states with the same momentum and continuum rotational quantum numbers decompose into those of the IRs of the little group of the momentum vector on the lattice. The inverse mapping indicates degeneracies that will appear between levels of different lattice IRs in the continuum limit, recovering the continuum angular momentum multiplets. An example of this inverse mapping is given for the case of the ``moving'' isotropic harmonic oscillator.

A numerical method that conserves the Runge-Lenz vector

An exhaustive discussion is carried out on isolating integrals and the trapezoidal formula which can conserve the Runge-Lenz vector. An isolating integral is an invariant that restricts the region of particle motion. The autonomous integrable Hamiltonian system with n degrees of freedom has only n mutually involutive independent isolating integrals, and the existence of other isolating integrals is meaningful to the particle motion. In the Kepler two-body system there exist the energy integral, the angular momentum integral and the Runge-Lenz vector. These correspond to 3 independent isolating integrals in the case of plane motion, and to 5 in the case of space motion. In the former, the integrals makes up the symmetry group SO (3) of the system, which can be transformed into the symmetry group of the two-dimensional isotropic harmonic oscillator through the Levi-Civita transformation, which is accurately conserved by the trapezoidal formula. On the other hand, in the case of space motion, the strict conservation of the energy and angular momentum inegrals and the Runge-Lenz vector by the trapezoidal formula is manifested in the 5 Kepler orbital elements a, e, i,and ω.

Analytic Coulomb matrix elements in a three-dimensional geometry

Using a complete basis set we have obtained an analytic expression for the matrix elements of the Coulomb interaction. These matrix elements are written in a closed form. We have used the basis set of the three-dimensional isotropic quantum harmonic oscillator in order to develop our calculations, which can be useful when treating interactions in localized systems.

Exactly solvable model of two three-dimensional harmonic oscillators interacting with the quantum electromagnetic field: The far-zone Casimir-Polder potential

We consider two three-dimensional isotropic harmonic oscillators with the same frequency and interacting with the quantum electromagnetic field in the Coulomb gauge and within dipole approximation. Using a Bogoliubov-type transformation, we can obtain transformed operators such that the Hamiltonian of the system, when expressed in terms of these operators, assumes a diagonal form. We are also able to obtain an expression for the energy shift of the ground state, which is valid at all orders in the coupling constant. From this energy shift, the nonperturbative Casimir--Polder potential energy between the two oscillators can be obtained. When approximated to the fourth order in the electric charge, the well-known expression of the far zone Casimir--Polder potential in terms of the polarizabilities of the oscillators is recovered.

Uniform semiclassical trace formula forU(3) →SO(3) symmetry breaking

We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term $\propto r^4$. This term breaks the U(3) symmetry of the HO, resulting in a spherical system with SO(3) symmetry. We first treat the anharmonic term in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbits over the manifold $\mathbb{C}$P$^2$ which characterizes their 4-fold degeneracy. Then we obtain an analytical uniform trace formula which in the limit of strong perturbations (or high energy) asymptotically goes over into the correct trace formula of the full anharmonic system with SO(3) symmetry, and in the limit $\epsilon$ (or energy) $\to 0$ restores the HO trace formula with U(3) symmetry. We demonstrate that the gross-shell structure of this anharmonically perturbed system is dominated by the two-fold degenerate diameter and circular orbits, and {\it not} by the orbits with the largest classical degeneracy, which are the three-fold degenerate tori with rational ratios $\omega_r:\omega_\phi=N:M$ of radial and angular frequencies. The same holds also for the limit of a purely quartic spherical potential $V(r)\propto r^4$.

Radial coherent states—From the harmonic oscillator to the hydrogen atom

We construct spectrum generating algebras of SO(2, 1) ∼ SU(1, 1) in arbitrary dimension for the isotropic harmonic oscillator and the Sturm-Coulomb problem in radial coordinates. Using these algebras, we construct the associated radial Barut-Girardello coherent states for the isotropic harmonic oscillator (in arbitrary dimension). We map these states into the Sturm-Coulomb radial coherent states and show that they evolve in a fictitious time parameter without dispersing.

Hamilton–Jacobi theory for Hamiltonian systems with non-canonical symplectic structures

A proposal for the Hamilton–Jacobi theory in the context of the covariant formulation of Hamiltonian systems is done. The current approach consists in applying Dirac’s method to the corresponding action which implies the inclusion of second-class constraints in the formalism which are handled using the procedure of Rothe and Scholtz recently reported. The current method is applied to the non-relativistic two-dimensional isotropic harmonic oscillator employing the various symplectic structures for this dynamical system recently reported.

GENERALIZED HYPERVIRIAL AND RECURRENCE RELATIONS FOR RADIAL MATRIX ELEMENTS IN ARBITRARY DIMENSIONS

In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = rk with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.

Uniform function constants of motion and their first-order perturbation

The main purpose of this work is to present some uniform function constants of motion rather than the well-known quantities arising from spacetime symmetries. These constants are usually associated with the intrinsic characteristics of the trajectories of a particle in a central potential field. We treat two cases. The first is the Lenz vector which sometimes is found in the literature [1, 2]; the other is associated with the isotropic harmonic oscillator, of relative importance in some simple models of the classical molecular interaction. The first example is applied to describe the perturbation of the trajectories in the Rutherford scattering and the precession of the Keplerian orbit of a planet. In the other case the conserved quantity is a symmetric tensor. We find the eigenvectors and eigenvalues of that tensor while at the same time we obtain the solution to the problem of calculating the rotation rate of the orbits in first order of a perturbation parameter in the potential energy, by performing a simple coordinate transformation in the Cartesian plane. We think that the present work addresses many aspects of mechanics with a didactical interest in other physics or mathematics courses.

Sensitivity tools vs. Poincaré sections

In this paper we introduce a modification of the fast Lyapunov indicator (FLI) denominated OFLITT2 indicator that may provide a global picture of the evolution of a dynamical system. Therefore, it gives an alternative or a complement to the pictures given by the classical Poincaré sections and, besides, it may be used for any dimension. We present several examples comparing with the Poincaré sections in two classical problems, the Hénon–Heiles and the extensible-pendulum problems. Besides, we show the application to Hamiltonians of three degrees of freedom as an isotropic harmonic oscillator in three dimensions perturbed by a cubic potential and non-Hamiltonian problems as a four-dimensional chaotic system. Finally, a numerical method especially designed for its computation is presented in the appendix.

REDUCTION AND UNFOLDING: THE KEPLER PROBLEM

In this paper, we show, in a systematic way, how to relate the Kepler problem to the isotropic harmonic oscillator. Unlike previous approaches, our constructions are carried over in the Lagrangian formalism dealing, with second order vector fields. We therefore provide a tangent bundle version of the Kustaanheimo-Stiefel map.

Genuine covariant description of Hamiltonian dynamics

After reviewing the covariant description of Hamiltonian dynamics, some applications are done to the non-relativistic isotropic three-dimensional harmonic oscillator, Rovelli's model, and SO(3, 1) BF theories.

Two-dimensional isotropic harmonic oscillator approach to classical and quantum Stokes parameters

We show that the well-known Stokes operators, defined as elements of the Jordan–Schwinger map with the Pauli matrices of two independent bosons, are equal to the constants of motion of the two-dimensional isotropic harmonic oscillator. Taking the expectation value of the Stokes operators in a two-mode coherent state, we obtain the corresponding classical Stokes parameters. We show that this classical limit of the Stokes operators, the 2 × 2 unit matrix and the Pauli matrices may be used to expand the polarization matrix. Finally, by means of the constants of motion of the classical two-dimensional isotropic harmonic oscillator, we describe the geometric properties of the polarization ellipse. Our study is restricted to the case of a monochromatic quantized-plane electromagnetic wave that propagates along the z axis.PACS Nos.: 42.50.–p, 42.25.Ja, 11.30.–j

Nonlinear Dynamical Symmetries of Some Two-Dimensional Quantum Systems**The project supported by National Natural Science Foundation of China under Grant No. 10275038, partly by the State Key Basic Research Development Programs under Grant Nos. G2000077400 and G2000077604, and Tsinghua Natural Science Foundation

In this paper nonlinear dynamical symmetries of three quantum systems are studied in detail, such as the Kepler–Coulomb system and the isotropic harmonic oscillator in a two-dimensional curved space, and the generalized pseudo-oscillators in the two-dimensional flat space. Their nonlinear spectrum generating algebras are shown to be relevant to polynomial angular momentum algebras.

Symplectic quantization, inequivalent quantum theories, and Heisenberg’s principle of uncertainty

We analyze the quantum dynamics of the non-relativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence does not arise from ambiguities in the ordering of operators but from the fact of having several symplectic structures defined with respect to the same set of coordinates. It is also shown that the uncertainty relations between the fundamental observables depend on the particular quantum theory chosen. It is important to emphasize that these (somehow paradoxical) results emerge from the combination of two paradigms: Dirac's quantization rule and the usual Copenhagen interpretation of quantum mechanics.

Reduction of the classical MICZ-Kepler problem to a two-dimensional linear isotropic harmonic oscillator

The classical MICZ-Kepler problem is shown to be reducible to an isotropic two-dimensional system of linear harmonic oscillators and a conservation law in terms of new variables related to the Ermanno–Bernoulli constants and the components of the Poincaré vector. An algorithmic route to linearization is shown based on Lie symmetry analysis and the reduction method [ Nucci, J. Math. Phys. 37, 1772 (1996) ]. First integrals are also obtained by symmetry analysis and the reduction method [ Marcelli and Nucci,J. Math. Phys. 44, 2111 (2002) ].

Quantum shifts for qubits in heterostructures

The effects of the solid-state heterostructure environment on the properties of embedded qubits are investigated. An electric dipole emitter in the form of an $F$ center has recently been used as the qubit for quantum information processing and, possibly, quantum computing. Modeled here in terms of an isotropic three-dimensional harmonic oscillator localized in a dielectric, we show that it experiences level shifts due to its proximity to a metallic contact plane. Appropriate parameters are used to model a nitrogen $F$ center localized in diamond within nanometer distances from a metallic surface.

Exact solutions of the spherically symmetric multidimensional isotropic harmonic oscillator

The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic harmonic oscillator, Hyperspherical harmonics, Multidimensional space. Global Jnl Pure & Applied Sciences Vol.10(2) 2004: 343-349

Jordan–Schwinger map, 3D harmonic oscillator constants of motion, and classical and quantum parameters characterizing electromagnetic wave polarization

In this work we introduce a generalization of the Jauch and Rohrlich quantum Stokes operators when the arrival direction from the source is unknown a priori. We define the generalized Stokes operators as the Jordan–Schwinger map of a triplet of harmonic oscillators with the Gell–Mann and Ne'eman matrices of the SU(3) symmetry group. We show that the elements of the Jordan–Schwinger map are the constants of motion of the three-dimensional isotropic harmonic oscillator. Also, we show that the generalized Stokes operators together with the Gell–Mann and Ne'eman matrices may be used to expand the polarization matrix. By taking the expectation value of the Stokes operators in a three-mode coherent state of the electromagnetic field, we obtain the corresponding generalized classical Stokes parameters. Finally, by means of the constants of motion of the classical 3D isotropic harmonic oscillator we describe the geometrical properties of the polarization ellipse.

Four kinds of raising and lowering operators of three-dimensional isotropic harmonic oscillators with spin-orbit coupling

The method of raising and lowering operator is used to solve the problem of threedimensional isotropic harmonic oscillator with Spinorbit coupling. The energy level splitting of the harmonic oscillator is analyzed and discussed.