The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these `curved' harmonic oscillators simultaneously on any such configuration space, using a Cayley-Klein (CK) type approach, with two free parameters \ki, \kii which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere ${\bf S}^2$, hyperbolic plane ${\bf H}^2$, AntiDeSitter sphere {\bf AdS}^{\unomasuno} and DeSitter sphere {\bf dS}^{\unomasuno}) appear in this family, with the Euclidean and Minkowski spaces as flat limits. We solve the equations of motion for the `curved' harmonic oscillator and obtain explicit expressions for the orbits by using three different methods: first by direct integration, second by obtaining the general CK version of the Binet's equation and third, as a consequence of its superintegrable character. The orbits are conics with centre at the potential origin in any CK space, thereby extending this well known Euclidean property to any constant curvature configuration space. The final part of the article, that has a more geometric character, presents those results of the theory of conics on spaces of constant curvature which are pertinent.Comment: 29 pages, 6 figure

Some integrals ocurring in a topology change problem

In a paper presented a few years ago, De Lorenci et al. showed, in the context of canonical quantum cosmology, a model which allowed space topology changes (Phys. Rev. D 56, 3329 (1997)). The purpose of this present work is to go a step further in that model, by performing some calculations only estimated there for several compact manifolds of constant negative curvature, such as the Weeks and Thurston spaces and the icosahedral hyperbolic space (Best space).Comment: RevTeX article, 4 pages, 1 figur

Weisskopf-Wigner model for wave packet excitation

We consider a laser induced molecular excitation process as a decay of a single energy state into a continuum. The analytic results based on Weisskopf-Wigner approach and perturbation calculations are compared with numerical wave packet results. We find that the decay model describes the excitation process well within the expected parameter region.Comment: 14 pages, Latex2.09, 9 Postscript figures embedded using psfig, see also http://www.physics.helsinki.fi/~kasuomin

Interacting Preformed Cooper Pairs in Resonant Fermi Gases

We consider the normal phase of a strongly interacting Fermi gas, which can have either an equal or an unequal number of atoms in its two accessible spin states. Due to the unitarity-limited attractive interaction between particles with different spin, noncondensed Cooper pairs are formed. The starting point in treating preformed pairs is the Nozi\`{e}res-Schmitt-Rink (NSR) theory, which approximates the pairs as being noninteracting. Here, we consider the effects of the interactions between the Cooper pairs in a Wilsonian renormalization-group scheme. Starting from the exact bosonic action for the pairs, we calculate the Cooper-pair self-energy by combining the NSR formalism with the Wilsonian approach. We compare our findings with the recent experiments by Harikoshi {\it et al.} [Science {\bf 327}, 442 (2010)] and Nascimb\`{e}ne {\it et al.} [Nature {\bf 463}, 1057 (2010)], and find very good agreement. We also make predictions for the population-imbalanced case, that can be tested in experiments.Comment: 10 pages, 6 figures, accepted version for PRA, discussion of the imbalanced Fermi gas added, new figure and references adde

GUI Matlab para o cálculo de funções de Bessel usando frações continuadas

[EN] Higher order Bessel functions are prevalent in physics and engineering and there exist different methods to evaluate them quickly and efficiently. Two of these methods are Miller's algorithm and the continued fractions algorithm. Miller's algorithm uses arbitrary starting values and normalization constants to evaluate Bessel functions. The continued fractions algorithm directly computes each value, keeping the error as small as possible. Both methods respect the stability of the Bessel function recurrence relations. Here we outline both methods and explain why the continued fractions algorithm is more efficient. The goal of this paper is both (1) to introduce the continued fractions algorithm to physics and engineering students and (2) to present a MATLAB GUI (Graphic User Interface) where this method has been used for computing the Semi-integer Bessel Functions and their zeros.[PT] Funções de Bessel de ordem mais alta são recorrentes em física e nas engenharias, sendo que há diferentes métodos para calculá-las de maneira rápida e eficiente. Dois destes métodos são o algoritmo de Miller e o algoritmo de frações continuadas. O primeiro faz uso de valores iniciais e constantes de normalização arbitrários, enquanto o segundo o faz calculando cada valor diretamente, minimizando tanto quanto possível o erro. Ambos respeitam a estabilidade das relações de recorrência das funções de Bessel. Neste trabalho descrevemos ambos os métodos e explicamos a razão pela qual o algoritmo das frações continuadas é mais eficiente. O objetivo do artigo é (1) introduzir o algoritmo de frações continuadas para estudantes de física e das engenharias e (2) apresentar um GUI (Graphic User Interface) em Matlab no qual este método foi utilizado para calcular funções de Bessel semi-inteiras e seus zeros.The authors wish to thank the financial support received from the Universidad Politécnica de Valencia under grant PAID-06-09-2734, from the Ministerio de Ciencia e Innovación through grant ENE2008-00599 and specially from the Generalitat Valenciana under grant reference 3012/2009.Hernandez Vargas, E.; Commeford, K.; Pérez Quiles, MJ. (2011). MATLAB GUI for computing Bessel functions using continued fractions algorithm. Revista Brasileira de Ensino de Física. 33(1):1303-1311. https://doi.org/10.1590/S1806-11172011000100003S13031311331Giladi, E. (2007). Asymptotically derived boundary elements for the Helmholtz equation in high frequencies. Journal of Computational and Applied Mathematics, 198(1), 52-74. doi:10.1016/j.cam.2005.11.024Havemann, S., & Baran, A. J. (2004). Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method. Journal of Quantitative Spectroscopy and Radiative Transfer, 89(1-4), 87-96. doi:10.1016/j.jqsrt.2004.05.014Segura, J., Fernández de Córdoba, P., & Ratis, Y. L. (1997). A code to evaluate modified bessel functions based on thecontinued fraction method. Computer Physics Communications, 105(2-3), 263-272. doi:10.1016/s0010-4655(97)00069-6Bastardo, J. L., Abraham Ibrahim, S., Fernández de Córdoba, P., Urchueguía Schölzel, J. F., & Ratis, Y. L. (2005). Evaluation of Fresnel integrals based on the continued fractions method. Applied Mathematics Letters, 18(1), 23-28. doi:10.1016/j.aml.2003.12.009Barnett, A. R., Feng, D. H., Steed, J. W., & Goldfarb, L. J. B. (1974). Coulomb wave functions for all real η and ϱ. Computer Physics Communications, 8(5), 377-395. doi:10.1016/0010-4655(74)90013-

Thinking about going to the dentist: a Contemplation Ladder to assess dentally-avoidant individuals' readiness to go to a dentist

<p>Abstract</p> <p>Background</p> <p>The Transtheoretical Model suggests that individuals vary according to their readiness to change behavior. Previous work in smoking cessation and other health areas suggests that interventions are more successful when they are tailored to an individual's stage of change with regards to the specific behavior. We report on the performance of a single-item measure ("Ladder") to assess the readiness to change dental-avoidant behavior.</p> <p>Methods</p> <p>An existing Contemplation Ladder for assessing stage of change in smoking cessation was modified to assess readiness to go to a dentist. The resulting Ladder was administered to samples of English-speaking adolescents (USA), Spanish-speaking adults (USA), and Norwegian military recruits (Norway) in order to assess construct validity. The Ladder was also administered to a sample of English-speaking avoidant adolescents and young adults who were enrolled in an intervention study (USA) in order to assess criterion validity. All participants also had dental examinations, and completed other questionnaires. Correlations, chi square, t tests and one-way ANOVAs were used to assess relationships between variables.</p> <p>Results</p> <p>In two samples, participants who do not go to the dentist had significantly more teeth with caries; in a third sample, participants who do not go to the dentist had significantly worse caries. Ladder scores were not significantly related to age, gender, caries, or dental fear. However, Ladder scores were significantly related to statements of intention to visit a dentist in the future and the importance of oral health. In a preliminary finding, Ladder scores at baseline also predicted whether or not the participants decided to go to a dentist in the intervention sample.</p> <p>Conclusions</p> <p>The data provide support for the convergent and divergent construct validity of the Ladder, and preliminary support for its criterion validity. The lack of relationship between dental fear and Ladder scores suggests that avoidant individuals may be helped to decide to go to a dentist using interventions which do not explicitly target their fear.</p

Vector Continued Fractions using a Generalised Inverse

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.Comment: Published form - minor change

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

Geometric descriptions of entangled states by auxiliaries varieties

The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as $2\times 2\times(n+1)$, for $n\geq 1$, quantum systems and a new example with the $2\times 3\times 3$ quantum system. Our description completes the approach of Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.Comment: 32 pages, 15 Tables, 5 Figures. References and remarks adde