Dynamics in discrete phase spaces and time interval operators

The von Neumann-Liouville time evolution equation is represented in a discrete quantum phase space. The mapped Liouville operator and the corresponding Wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found. The propagator is also discussed and a time interval operator, associated to a unitary operator which shifts the energy levels in the Zeeman spectrum, is introduced. This operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution. The pair of unitary operators which shifts the time and energy is shown to obey the Weyl-Schwinger algebra. (C) 1999 Elsevier B.V. B.V. All rights reserved.

Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials

The Dirac equation is exactly solved for a pseudoscalar linear plus Coulomb-like potential in a two-dimensional world. This sort of potential gives rise to an effective quadratic plus inversely quadratic potential in a Sturm-Liouville problem, regardless the sign of the parameter of the linear potential, in sharp contrast with the Schrodinger case. The generalized Dirac oscillator already analyzed in a previous work is obtained as a particular case. (C) 2004 Elsevier B.V. All rights reserved.

Thermally developing forced convection of non-Newtonian fluids inside elliptical ducts

Laminar-forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumptions used in this work are a non-Newtonian fluid, laminar flow, constant physical properties, and negligible axial heat diffusion (high Peclet number). Most of the previous research in elliptical ducts deal mainly with aspects of fully developed laminar flow forced convection, such as velocity profile, maximum velocity, pressure drop, and heat transfer quantities. In this work, we examine heat transfer in a hydrodynamically developed, thermally developing laminar forced convection flow of fluid inside an elliptical tube under a second kind of a boundary condition. To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm-Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform, where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number, and the average Nusselt number for various cross-section aspect ratios.

Estados ligados em mecânica quântica relativística

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Aceleração de Fermi em bilhares com fronteiras dependentes do tempo descritas por osciladores não lineares: caso conservativo e dissipativo

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Espalhamento e estados ligados de partículas de spin 1/2 em um potencial degrau suave com acoplamentos escalar e vetorial

Pós-graduação em Física - FEG

Soluções analíticas da equação de Duffin-Kemmer-Petiau

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Stationary states of fermions in a sign potential with a mixed vector-scalar coupling

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Effects of a mixed vector-scalar kink-like potential for spinless particles in two-dimensional space time

The intrinsically relativistic problem of spinless particles subject to a general mixing of vector and scalar kink- like potentials (similar to tanh gamma x) is investigated. The problem is mapped into the exactly solvable Sturm - Liouville problem with the Rosen - Morse potential and exact bounded solutions for particles and antiparticles are found. The behavior of the spectrum is discussed in some detail. An apparent paradox concerning the uncertainty principle is solved by recurring to the concept of effective Compton wavelength.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

An infinite level atom coupled to a heat bath

Wir untersuchen die Mathematik endlicher, an ein Wärmebad gekoppelter Teilchensysteme. Das Standard-Modell der Quantenelektrodynamik für Temperatur Null liefert einen Hamilton-Operator H, der die Energie von Teilchen beschreibt, welche mit Photonen wechselwirken. Im Heisenbergbild ist die Zeitevolution des physikalischen Systems durch die Wirkung einer Ein-Parameter-Gruppe auf eine Menge von Observablen A gegeben: Diese steht im Zusammenhang mit der Lösung der Schrödinger-Gleichung für H. Um Zustände von A, welche das physikalische System in der Nähe des thermischen Gleichgewichts zur Temperatur T darstellen, zu beschreiben, folgen wir dem Ansatz von Jaksic und Pillet, eine Darstellung von A zu konstruieren. Die Vektoren in dieser Darstellung definieren die Zustände, die Zeitentwicklung wird mit Hilfe des Standard Liouville-Operators L beschrieben. In dieser Doktorarbeit werden folgende Resultate bewiesen bzw. hergeleitet: - die Konstuktion einer Darstellung - die Selbstadjungiertheit des Standard Liouville-Operators - die Existenz eines Gleichgewichtszustandes in dieser Darstellung - der Limes des physikalischen Systems für große Zeiten.

A note on bifurcation from an interval

We study the global bifurcation of nonlinear Sturm-Liouville problems of the form -(pu')' + qu = lambda a(x)f(u), b(0)u(0) - c(0)u' (0) = 0, b(1)u(1) + c(1)u'(1) = 0 which are not linearizable in any neighborhood of the origin. (c) 2005 Published by Elsevier Ltd.

Analisi spettrale di operatori differenziali

L'elaborato è finalizzato a presentare l'analisi degli operatori differenziali agenti in meccanica quantistica e la teoria degli operatori di Sturm-Liouville. Nel primo capitolo vengono analizzati gli operatori differenziali e le relative proprietà. Viene studiata la loro autoaggiunzione su vari domini con diverse condizioni al contorno e vengono tratte delle conclusioni sul loro significato come osservabili. Nel secondo capitolo viene presentato il concetto di spettro e vengono studiate le sue proprietà.Vengono poi analizzati gli spettri degli operatori precedentemente introdotti. Nell'utimo capitolo vengono presentati gli operatori di Sturm-Liouville e alcune proprietà delle equazioni differenziali. Vengono imposte delle specifiche condizioni al contorno che determinano la realizzazione dei sistemi di Sturm-Liouville, di cui vengono studiati due esempi notevoli: le guide d'onda e la conduzione del calore.

On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

2000 Mathematics Subject Classification: 33D60, 26A33, 33C60

Liouville-space R-matrix-Floquet description of atomic radiative processes involving autoionizing states in the presence of intense electromagnetic fields

A reduced-density-operator description is developed for coherent optical phenomena in many-electron atomic systems, utilizing a Liouville-space, multiple-mode Floquet–Fourier representation. The Liouville-space formulation provides a natural generalization of the ordinary Hilbert-space (Hamiltonian) R-matrix-Floquet method, which has been developed for multi-photon transitions and laser-assisted electron–atom collision processes. In these applications, the R-matrix-Floquet method has been demonstrated to be capable of providing an accurate representation of the complex, multi-level structure of many-electron atomic systems in bound, continuum, and autoionizing states. The ordinary Hilbert-space (Hamiltonian) formulation of the R-matrix-Floquet method has been implemented in highly developed computer programs, which can provide a non-perturbative treatment of the interaction of a classical, multiple-mode electromagnetic field with a quantum system. This quantum system may correspond to a many-electron, bound atomic system and a single continuum electron. However, including pseudo-states in the expansion of the many-electron atomic wave function can provide a representation of multiple continuum electrons. The 'dressed' many-electron atomic states thereby obtained can be used in a realistic non-perturbative evaluation of the transition probabilities for an extensive class of atomic collision and radiation processes in the presence of intense electromagnetic fields. In order to incorporate environmental relaxation and decoherence phenomena, we propose to utilize the ordinary Hilbert-space (Hamiltonian) R-matrix-Floquet method as a starting-point for a Liouville-space (reduced-density-operator) formulation. To illustrate how the Liouville-space R-matrix-Floquet formulation can be implemented for coherent atomic radiative processes, we discuss applications to electromagnetically induced transparency, as well as to related pump–probe optical phenomena, and also to the unified description of radiative and dielectronic recombination in electron–ion beam interactions and high-temperature plasmas.