Momentum distribution in heavy deformed nuclei: role of effective mass

The impact of nuclear deformation on the momentum distributions (MD) of occupied proton states in $^{238}$U is studied with a phenomenological Woods-Saxon (WS) shell model and the self-consistent Skyrme-Hartree-Fock (SHF) scheme. Four Skyrme parameterizations (SkT6, SkM*, SLy6, SkI3) with different effective masses are used. The calculations reveal significant deformation effects in the low-momentum domain of $K^{\pi}=1/2^{\pm}$ states, mainly of those lying near the Fermi surface. For other states, the deformation effect on MD is rather small and may be neglected. The most remarkable result is that the very different Skyrme parameterizations and the WS potential give about identical MD. This means that the value of effective mass, being crucial for the description of the spectra, is not important for the spatial shape of the wave functions and thus for the MD. In general, it seems that, for the description of MD at $0\le k \le 300$ MeV/c, one may use any single-particle scheme (phenomenological or self-consistent) fitted properly to the global ground state properties.Comment: 14 pages, 6 figure

Connection Between Type A and E Factorizations and Construction of Satellite Algebras

Recently, we introduced a new class of symmetry algebras, called satellite algebras, which connect with one another wavefunctions belonging to different potentials of a given family, and corresponding to different energy eigenvalues. Here the role of the factorization method in the construction of such algebras is investigated. A general procedure for determining an so(2,2) or so(2,1) satellite algebra for all the Hamiltonians that admit a type E factorization is proposed. Such a procedure is based on the known relationship between type A and E factorizations, combined with an algebraization similar to that used in the construction of potential algebras. It is illustrated with the examples of the generalized Morse potential, the Rosen-Morse potential, the Kepler problem in a space of constant negative curvature, and, in each case, the conserved quantity is identified. It should be stressed that the method proposed is fairly general since the other factorization types may be considered as limiting cases of type A or E factorizations.Comment: 20 pages, LaTeX, no figure, to be published in J. Phys.

Generalized Morse Potential: Symmetry and Satellite Potentials

We study in detail the bound state spectrum of the generalized Morse potential~(GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schr\"odinger equation with the Laplace equation on the hyperboloid and the Schr\"odinger equation for the P\"oschl-Teller potential, we explain the exact solvability of the problem by an $so(2,2)$ symmetry algebra, and obtain an explicit realization of the latter as $su(1,1) \oplus su(1,1)$. We prove that some of the $so(2,2)$ generators connect among themselves wave functions belonging to different GMP's (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, $so(2,2)$ belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.Comment: 23 pages, LaTeX, 2 figures (on request). one LaTeX problem settle

Supersymmetry and superalgebra for the two-body system with a Dirac oscillator interaction

Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$\omega$. By considering the latter together with its partner corresponding to the replacement of~$\omega$ by~$-\omega$, we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.Comment: 21 pages, LaTeX, 1 figure (can be obtained from the authors), to appear in J. Phys.

Teacher Questioning in Problem Solving in Community College Algebra Classrooms

In this chapter, we focus on the ways two community college instructors worked with students to demonstrate the solution of contextualized algebra problems in their college algebra lessons. We use two classroom episodes to illustrate how they sought to elicit students' mathematical ideas of algebraic topics, attending primarily to teachers' questioning approaches. We found that the instructors mostly asked questions of lower cognitive demand and used a variety of approaches to elicit the mathematical ideas of the problems, such as using examples relevant to the students and dividing the problems into smaller tasks, that together help identify a solution. We conclude by offering considerations for instruction at community colleges and potential areas for professional development

Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid

A class of quantum superintegrable Hamiltonians defined on a two-dimensional hyperboloid is considered together with a set of intertwining operators connecting them. It is shown that such intertwining operators close a su(2,1) Lie algebra and determine the Hamiltonians through the Casimir operators. By means of discrete symmetries a broader set of operators is obtained closing a so(4,2) algebra. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of unitary representations of su(2,1) and so(4,2).Comment: 11 pages, 5 figure

Deformations of the Boson sp(4,R)sp(4,R) Representation and its Subalgebras

The boson representation of the sp(4,R) algebra and two distinct deformations of it, are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space. One of the deformed representation is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4,R) (compact u(2) and three representations of the noncompact u(1,1) are also deformed and are contained in this deformed algebra. They are reducible in the action spaces of sp(4,R) and decompose into irreducible representations. The other deformed representation, is realized by means of a transformation of the q-deformed bosons into q-tensors (spinor-like) with respect to the standard deformed su(2). All of its generators are deformed and have expressions in terms of tensor products of spinor-like operators. In this case, an other deformation of su(2) appears in a natural way as a subalgebra and can be interpreted as a deformation of the angular momentum algebra so(3). Its representation is reducible and decomposes into irreducible ones that yields a complete description of the same