Four-dimensional Harmonic Oscillator Research Articles

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

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

Coherent States of Harmonic and Reversed Harmonic Oscillator

A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a time-dependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the Kepler-Coulomb problem when transformed to the model of a four-dimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically.

Fictitious-time wave-packet dynamics. I. Nondispersive wave packets in the quantum Coulomb problem

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of ``restricted Gaussian wave packets'' consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well-defined angular momentum quantum numbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \check{c}\else \v{c}\fi{}, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].

Efficient Orbit Integration by Linear Transformation for Kustaanheimo-Stiefel Regularization

We extended the quadruple scaling method of the Kustaanheimo-Stiefel (K-S) regularization by adding three independent components of the four-dimensional fictitious angular momentum tensor to the quasi-conserved quantities to be monitored during the numerical integration. By using a linear transformation in the four-dimensional fictitious space to make the newly introduced components and the full harmonic energies approximately consistent, we adjust the four amplitudes and the three phase differences for the four-dimensional harmonic oscillator associated with the K-S regularization at every integration step. The determination of transformation parameters is unique, simple, and universal. The new method is superior to the quadruple scaling method in the sense that the errors in all unperturbed orbital elements except the mean longitude at epoch are reduced to the machine epsilon level independently of the precision of the numerical integration. For perturbed orbits, the errors increase more slowly than the quadruple scaling method. Although the number of variables to be integrated is increased to 16 per celestial body, the new method provides the best performance among the manifold correction methods we developed for K-S-regularized orbital motions.

Efficient Integration of Highly Eccentric Orbits by Quadruple Scaling for Kustaanheimo-Stiefel Regularization

We have extended the single scaling method for Kustaanheimo-Stiefel (K-S) regularization by applying different scaling factors to each component of the four-dimensional harmonic oscillator associated with the regularization. This is achieved by monitoring the time development of the total energy of each component of the harmonic oscillator and rescaling the magnitude of the position and velocity of the corresponding component so as to satisfy the defining relation for the harmonic energy. To perform the monitoring increases the number of variables to be integrated per celestial body from 10 to 13; however, the extra cost of computation is still negligible compared with that of the perturbing acceleration. The resulting method of quadruple scaling significantly enhances the numerical stability of orbit integrations. In the case of unperturbed orbits, the new method when applied at every integration step reduces to the machine-epsilon level the errors in all the orbital elements except the mean longitude at epoch, if sufficiently high order integrators are used with sufficiently small step sizes. This remarkable feature is lost when the scaling is applied at every apocenter. In the case of perturbed orbits, we confirm the superiority of the quadruple scaling method applied at every integration step over the other scaling methods for K-S regularized orbital motions unless round-off plays the key role in the accumulation of integration error.

On the use of the group SO(4, 2) in atomic and molecular physics

The dynamical non-invariance group SO(4, 2) for a hydrogen-like atom is derived through two different approaches. The first one is by an established traditional ascent process starting from the symmetry group SO(3). This approach is presented in a mathematically oriented original way with a special emphasis on maximally superintegrable systems, N-dimensional extension and little groups. The second approach is by a new symmetry descent process starting from the non-invariance dynamical group Sp(8, R) for a four-dimensional harmonic oscillator. It is based on the little known concept of a Lie algebra under constraints and corresponds in some sense to a symmetry breaking mechanism. This paper ends with a brief discussion of the interest of SO(4, 2) for a new group-theoretical approach to the periodic table of chemical elements. In this connection, a general ongoing programme based on the use of a complete set of commuting operators is briefly described. It is believed that the present paper could be useful not only to the atomic and molecular community but also to people working in theoretical and mathematical physics.

The Kustaanheimo–Stiefel transformation in geometric algebra

The Kustaanheimo-Stiefel (KS) transformation maps the non-linear and singular equations of motion of the three-dimensional Kepler problem to the linear and regular equations of a four-dimensional harmonic oscillator. It is used extensively in studies of the perturbed Kepler problem in celestial mechanics and atomic physics. In contrast to the conventional matrix-based approach, the formulation of the KS transformation in the language of geometric Clifford algebra offers the advantages of a clearer geometrical interpretation and greater computational simplicity. It is demonstrated that the geometric algebra formalism can readily be used to derive a Lagrangian and Hamiltonian description of the KS dynamics in arbitrary static electromagnetic fields. For orbits starting at the Coulomb centre, initial conditions are derived and a framework is set up that allows a discussion of the stability of these orbits.

Nucleon spin structure functions in the resonance region with constituent quark models

Q 2 -evolution of the nucleon spin structure functions g 1 , g 2 and the generalized Gerasimov–Drell–Hearn sum rule in the small- Q 2 region are investigated by considering the resonance contributions based on nonrelativistic constituent quark model and on relativistic four-dimensional harmonic oscillator model, respectively. Results of the two models are compared and the relativistic effects are discussed. Moreover, influences of the longitudinal transitions of the resonances on the sum rules of the nucleon spin structure functions are addressed.

A transformation of a Feynman–Kac formula for holomorphic families of type B

A transformation formula for resolvents of families of Schrödinger operators H(ξ)=−12Δ+ξQ, which are assumed to be holomorphic of type B, is proved. It can be used to derive the well-known correspondence between three-dimensional Coulomb problem and four-dimensional harmonic oscillator.

Kustaanheimo–Stiefel Transformation of the Monopolar Hydrogen Atom into the Four-Dimensional Oscillator

We show that the monopolar hydrogen atom is connected to a four-dimensionalharmonic oscillator with a monopole-dependent constraint by theKustaanheimo–Stiefel transformation.

Comment on “Wigner phase-space distribution function for the hydrogen atom”

We object to the proposal that the mapping of the three-dimensional hydrogen atom into a four-dimensional harmonic oscillator can be readily used to determine the Wigner phase-space distribution function for the hydrogen atom.

The monopole-hydrogen atom system and its connection with a four-dimensional harmonic oscillator

The Hamiltonian of a hydrogen atom associated with a U(1) monopole is established in parallel to a usual hydrogen atom with conserved angular momentum and Pauli-Runge-Lenz vector . Both are associated with monopoles and form an algebra SO(4) for the bound motions. Due to the two corresponding Pauli relations and their bosonic realizations, the system can be connected to a four-dimensional harmonic oscillator with a monopole-dependent constraint. Furthermore, the monopole harmonics can be obtained by the operator method.

Wigner phase-space distribution function for the hydrogen atom

In this work we present a theoretical study of the hydrogen atom in quantum phase space. A coordinate transformation is used that maps the hydrogen atom into a four-dimensional harmonic oscillator and the Wigner distribution function for the hydrogen atom is then obtained. This exactly soluble model can shed some light on finite-size features of Wigner's distribution, which will be a vital experience for various dynamic problems.

ISOMORPHISM TRANSFORMATION BETWEEN THE HYDROGEN ATOM AND FOUR-DIMENSIONAL HARMONIC OSCILLATOR

By means of the SU(1,1) algebra we study therelationships between the coordinates, energy, angularmomentum, spherical harmonics, and radial function of ahydrogen atom and those of a four-dimensional harmonic oscillator. The energies and angularmomenta of the two quantum systems are found tocorrespond to one another.

Connection Between Angular Momenta of the Hydrogen Atom and a Four-Dimensional Harmonic Oscillator

After remodeling the form of the Kustaanheimo-Stiefel transformation, all the problems on the connection between angular momenta of the hydrogen atom and a four-dimensional harmonic oscillator are studied.

KUSTANNHEIMO-STIEFEL TRANSFORMATION FOR MESON STRUCTURE MODEL WITH A FOUR-DIMENSIONAL HARMONIC OSCILLATOR

By use of the Kustannheimo-Stiefel transformation, the problem of a four-dimensional covariant harmonic oscillator can be transformed to that of a three-dimensional hydrogen atom with constraint. On this basis, the difficulty of the excitation of the time degree of freedom can be avoided naturally and the mass-squared formula for mesons can be obtained.

Semiclassical mechanics of the quadratic Zeeman effect.

A comprehensive approach to semiclassical quantization of the quadratic Zeeman effect in the hydrogen atom (QZE) is presented which utilizes the connection between the Kepler system and the four-dimensional harmonic oscillator. Past attempts to quantize the QZE have encountered problems associated with singularities arising because of a classical separatrix. Here, a uniform semiclassical quantization scheme that passes smoothly through the separatrix is developed using classical perturbation theory. This method accounts for the topologically distinct kinds of dynamics that arise in the QZE, and provides good estimates of tunneling splittings. Extension to the quantization of arbitrarily-high-order classical perturbation expansions is straightforward.

Large-order perturbation expansion of three-dimensional Coulomb systems from four-dimensional anharmonic oscillators.

The three-dimensional Coulomb system can be mapped onto the four-dimensional harmonic oscillator. Additional perturbations of the type ${\mathit{gr}}^{\mathit{p}}$ [r=(${\mathit{x}}^{2}$+${\mathit{y}}^{2}$+${\mathit{z}}^{2}$${)}^{1/2}$] translate into anharmonic perturbations \ensuremath{\lambda}(${\mathbf{x}}^{2}$${)}^{\mathit{p}+1}$ (${\mathbf{x}}^{2}$=${\mathit{x}}_{1}^{2}$+${\mathit{x}}_{2}^{2}$+${\mathit{x}}_{3}^{2}$+${\mathit{x}}_{4}^{2}$) to the oscillator. We use this observation to relate the large-order behavior of perturbation series of the perturbed Coulomb systems to well-known large-order formulas for anharmonic oscillators. In this way, the leading behavior of Coulomb systems can be understood by simple scaling and symmetry arguments within the oscillator systems. Applications to more physical Stark- and Zeeman-type perturbations are briefly discussed.

On the spectrum of the hydrogen atom from the four-dimensional harmonic oscillator problem

A direct derivation is given of the classic hydrogen atom spectrum (including the degeneracy of the levels) from the four-dimensional harmonic oscillator problem by performing a generalised trace operation to the latter where the initial states are related to the final states by a phase transformation.

Hartree approximation to QCD in the Fock-Schwinger gauge.

The gluonic mean field in the Hartree approximation is obtained utilizing the Fock-Schwinger gauge. The potential in Euclidean space-time is color diagonal and local. The angular momentum decomposition of gluonic states can be done and the self-consistency equations then only depend on the (Euclidean) radial variable \ensuremath{\rho}=${x}_{\ensuremath{\mu}}$${x}_{\ensuremath{\mu}}$. Close to the gauge-fixing point we find that the potential resembles that of a four-dimensional harmonic oscillator without a mass term. The frequency \ensuremath{\omega}\ensuremath{\sim}(300 MeV${)}^{2}$ can be related to the gluon condensate and we estimate a lower bound for glueball masses ${m}_{G}$>1 GeV.