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Register a new userMaple procedures for the coupling of angular momenta. VI. LS-jj transformations
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Gaigalas Gediminas Habil. dr.
Publication date
2004
fields
Physics
and research's language is
English
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Transformation matrices between different coupling schemes are required, if a reliable classification of the level structure is to be obtained for open-shell atoms and ions. While, for instance, relativistic computations are traditionally carried out in jj-coupling, a LSJ coupling notation often occurs much more appropriate for classifying the valence-shell structure of atoms. Apart from the (known) transformation of single open shells, however, further demand on proper transformation coefficients has recently arose from the study of open d- and f-shell elements, the analysis of multiple--excited levels, or the investigation on inner-shell phenomena. Therefore, in order to facilitate a simple access to LS <-> jj transformation matrices, here we present an extension to the Racah program for the set-up and the transformation of symmetry-adapted functions. A flexible notation is introduced for defining and for manipulating open-shell configurations at different level of complexity which can be extended also to other coupling schemes and, hence, may help determine an optimum classification of atomic levels and processes in the future.
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We present a mathematical proof of the algorithm allowing to generate all - symmetric and non-symmetric - total angular momentum eigenstates in remote matter qubits by projective measurements, proposed in Maser et al. [Phys. Rev. A 79, 033833 (2009)]. By deriving a recursion formula for the algorithm we show that the generated states are equal to the total angular momentum eigenstates obtained via the usual quantum mechanical coupling of angular momenta. In this way we demonstrate that the algorithm is able to simulate the coupling of N spin-1/2 systems, and to implement the required Clebsch-Gordan coefficients, even though the particles never directly interact with each other.
Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical Hamiltonian picture of the evolution of a `quantum of space, as shown by the authors in a recent paper. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuous) variable. Key tool for this comparative analysis is the duality operation defined on the generators of the quadratic algebra and suitably extended to the various families of overlap functions (generalized recoupling coefficients). These families, recognized as lying at the top level of the Askey scheme, are classified and a few limiting cases are addressed.
We report a complete calculation of the quark and glue momenta and angular momenta in the proton. These include the quark contributions from both the connected and disconnected insertions. The quark disconnected insertion loops are computed with $Z_4$ noise, and the signal-to-noise is improved with unbiased subtractions. The glue operator is comprised of gauge-field tensors constructed from the overlap operator. The calculation is carried out on a $16^3 times 24$ quenched lattice at $beta = 6.0$ for Wilson fermions with $kappa=0.154, 0.155$, and $0.1555$ which correspond to pion masses at $650, 538$, and $478$~MeV, respectively. The chirally extrapolated $u$ and $d$ quark momentum/angular momentum fraction is found to be $0.64(5)/0.70(5)$, the strange momentum/angular momentum fraction is $0.024(6)/0.023(7)$, and that of the glue is $0.33(6)/0.28(8)$. The previous study of quark spin on the same lattice revealed that it carries a fraction of $0.25(12)$ of proton spin. The orbital angular momenta of the quarks are then obtained from subtracting the spin from their corresponding angular momentum components. We find that the quark orbital angular momentum constitutes $0.47(13)$ of the proton spin with almost all of it coming from the disconnected insertions.
We report a complete calculation of the quark and glue momenta and angular momenta in the proton. These include the quark contributions from both the connected and disconnected insertions. The calculation is carried out on a $16^3 times 24$ quenched lattice at $beta = 6.0$ and for Wilson fermions with $kappa = 0.154, 0.155,$ and 0.1555 which correspond to pion masses at 650, 538, and 478 MeV. The quark loops are calculated with $Z_4$ noise and signal-to-noise is improved further with unbiased subtractions. The glue operator is comprised of gauge-field tensors constructed from the overlap operator. The $u$ and $d$ quark momentum/angular momentum fraction is 0.66(5)/0.72(5), the strange momentum/angular momentum fraction is 0.024(6)/0.023(7), and that of the glue is 0.31(6)/0.25(8). The orbital angular momenta of the quarks are obtained from subtracting the angular momentum component from its corresponding spin. As a result, the quark orbital angular momentum constitutes 0.50(2) of the proton spin, with almost all it coming from the disconnected insertion. The quark spin carries a fraction 0.25(12) and glue carries a fraction 0.25(8) of the total proton spin.
We present the results of a suite of numerical simulations designed to explore the origin of the angular momenta of protostellar cores. Using the hydrodynamic grid code emph{Athena} with a sink implementation, we follow the formation of protostellar cores and protostars (sinks) from the subvirial collapse of molecular clouds on larger scales to investigate the range and relative distribution of core properties. We find that the core angular momenta are relatively unaffected by large-scale rotation of the parent cloud; instead, we infer that angular momenta are mainly imparted by torques between neighboring mass concentrations and exhibit a log-normal distribution. Our current simulation results are limited to size scales $sim 0.05$~pc ($sim 10^4 rm AU$), but serve as first steps toward the ultimate goal of providing initial conditions for higher-resolution studies of core collapse to form protoplanetary disks.
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