LATIN‐AMERICAN SCHOOL OF PHYSICS—XL ELAF: Symmetries in Physics
1334(2011); http://dx.doi.org/10.1063/1.3555477View Description Hide Description
These summer school lectures cover the use of algebraic techniques in various subfields of nuclear physics. After a brief description of groups and algebras, concepts of dynamical symmetry, dynamical supersymmetry, and supersymmetric quantum mechanics are introduced. Appropriate tools such as quasiparticles, quasispin, and Bogoliubov transformations are discussed with an emphasis on group theoretical foundations of these tools. To illustrate these concepts three physics applications are worked out in some detail: i) Pairing in nuclear physics; ii) Subbarrier fusion and associated group transformations; and iii) Symmetries of neutrino mass and of a related neutrino many‐body problem.
1334(2011); http://dx.doi.org/10.1063/1.3555478View Description Hide Description
These lectures are devoted to the description of resonances, i.e. Gamow states, in an amenable mathematical formalism. We shall describe Gamow states in scattering theory (S‐matrix formalism), in extensions of the conventional Hilbert space (rigged Hilbert spaces and Gelfand triads), and in the Dirac formulation of quantum mechanics (analytical continuations, continuous linear functionals over analytical test functions). Since we aim at further applications in the domain of nuclear structure and nuclear many‐body problems, we shall address the issue in a physical oriented manner, restricting the discussion of mathematical concepts to the needed, unavoidable, background.
1334(2011); http://dx.doi.org/10.1063/1.3555480View Description Hide Description
The Cosmic Microwave Background is measured by satellite observation with great precision. It offers insight into its origin in early states of the universe. Unexpected low multipole amplitudes of the incoming CMB radiation may be due to a multiply connected topology of cosmic 3‐space. We present and analyze the geometry and homotopy for the family of Platonic spherical 3‐manifolds, provide their harmonic analysis, and formulate topological selection rules.
1334(2011); http://dx.doi.org/10.1063/1.3555481View Description Hide Description
Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.
1334(2011); http://dx.doi.org/10.1063/1.3555482View Description Hide Description
These lectures review some advances in the algebraic description of molecules from two point of views: structure and dynamics. We start by presenting the basic ideas involved in the traditional description of molecular structure in configuration space, where the Born‐Oppenheimer and rotor‐rigid approximations are assumed to be valid. We then focus on the vibrational degrees of freedom in order to introduce the traditional algebraic realization in terms of bosonic operators of harmonic oscillators. This analysis allows the algebraic methods based on dynamical unitary groups to be introduced as a anharmonization procedure where the local bosonic operators are translated into operators satisfying the su(2) commutation relations. Some examples of the vibrational spec‐troscopic description are presented. Concerned with the dynamical point of view an algebraic model to describe collinear collisions in the semiclassical approximation is presented.
1334(2011); http://dx.doi.org/10.1063/1.3555483View Description Hide Description
The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center‐of‐mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic symmetries like symplectic group is considered. Entropic uncertainty relations and inequalities for spin tomograms are reviewed.
1334(2011); http://dx.doi.org/10.1063/1.3555484View Description Hide Description
This work summarizes the most important developments in the construction and application of the Dirac‐Moshinsky oscillator (DMO). The literature on the subject is voluminous, mostly because of the avenues that exact solvability opens towards our understanding of relativistic quantum mechanics. Here we make an effort to present the subject in chronological order and also in increasing degree of complexity. We start our discussion with the seminal paper by Moshinsky and Szczepaniak and the immediate implications stemming from it. Then we analyze the extensions of this model to many particles. The one‐particle DMO is revisited in the light of the Jaynes‐Cummings model in quantum optics and exactly solvable extensions are presented. Applications and implementations in hexagonal lattices are given, with a particular emphasis in the emulation of graphene in electromagnetic billiards.
1334(2011); http://dx.doi.org/10.1063/1.3555485View Description Hide Description
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.
In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
1334(2011); http://dx.doi.org/10.1063/1.3555486View Description Hide Description
There has been tremendous progress in understanding the non‐perturbative phenomena in QCD. On the theoretical side it has been driven largely by lattice gauge simulations which when coupled with microscopic models can lead to a physical picture of the underlying microscopic dynamics. On the experimental side, new discoveries in the hadron spectrum are uncovering physics beyond the quark model and hopefully will resolve the role of gluons in confinement and mass generation. In these lectures I briefly review the main discussion points and give a list of relevant references.
1334(2011); http://dx.doi.org/10.1063/1.3555487View Description Hide Description
The use of dynamical symmetries or spectrum generating algebras for the solution of the nuclear many‐body problem is reviewed. General notions of symmetry and dynamical symmetry in quantum mechanics are introduced and illustrated with simple examples such as the SO(4) symmetry of the hydrogen atom and the isospin symmetry in nuclei. Two nuclear models, the shell model and the interacting boson model, are reviewed with particular emphasis on their use of group‐theoretical techniques.