The Screen representation of spin networks. Images of 6j symbols and semiclassical features

This article presents and discusses in detail the results of extensive exact calculations of the most basic ingredients of spin networks, the Racah coefficients (or Wigner 6j symbols), exhibiting their salient features when considered as a function of two variables - a natural choice due to their origin as elements of a square orthogonal matrix - and illustrated by use of a projection on a square "screen" introduced recently. On these screens, shown are images which provide a systematic classification of features previously introduced to represent the caustic and ridge curves (which delimit the boundaries between oscillatory and evanescent behaviour according to the asymptotic analysis of semiclassical approaches). Particular relevance is given to the surprising role of the intriguing symmetries discovered long ago by Regge and recently revisited; from their use, together with other newly discovered properties and in conjunction with the traditional combinatorial ones, a picture emerges of the amplitudes and phases of these discrete wavefunctions, of interest in wide areas as building blocks of basic and applied quantum mechanics.Comment: 16 pages, 13 figures, presented at ICCSA 2013 13th International Conference on Computational Science and Applicatio

The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics

This paper treats 6j symbols or their orthonormal forms as a function of two variables spanning a square manifold which we call the "screen". We show that this approach gives important and interesting insight. This two dimensional perspective provides the most natural extension to exhibit the role of these discrete functions as matrix elements that appear at the very foundation of the modern theory of classical discrete orthogonal polynomials. Here we present 2D and 1D recursion relations that are useful for the direct computation of the orthonormal 6j, which we name U. We present a convention for the order of the arguments of the 6j that is based on their classical and Regge symmetries, and a detailed investigation of new geometrical aspects of the 6j symbols. Specifically we compare the geometric recursion analysis of Schulten and Gordon with the methods of this paper. The 1D recursion relation, written as a matrix diagonalization problem, permits an interpretation as a discrete Schr\"odinger-like equations and an asymptotic analysis illustrates semiclassical and classical limits in terms of Hamiltonian evolution.Comment: 14 pages,9 figures, presented at ICCSA 2013 13th International Conference on Computational Science and Applicatio

The screen representation of vector coupling coefficients or Wigner 3j symbols: exact computation and illustration of the asymptotic behavior

The Wigner $3j$ symbols of the quantum angular momentum theory are related to the vector coupling or Clebsch-Gordan coefficients and to the Hahn and dual Hahn polynomials of the discrete orthogonal hyperspherical family, of use in discretization approximations. We point out the important role of the Regge symmetries for defining the screen where images of the coefficients are projected, and for discussing their asymptotic properties and semiclassical behavior. Recursion relationships are formulated as eigenvalue equations, and exploited both for computational purposes and for physical interpretations.Comment: 14 pages, 8 figures, presented at ICCSA 2014, 14th International Conference on Computational Science and Application

Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective

A unified vision of the symmetric coupling of angular momenta and of the quantum mechanical volume operator is illustrated. The focus is on the quantum mechanical angular momentum theory of Wigner's 6j symbols and on the volume operator of the symmetric coupling in spin network approaches: here, crucial to our presentation are an appreciation of the role of the Racah sum rule and the simplification arising from the use of Regge symmetry. The projective geometry approach permits the introduction of a symmetric representation of a network of seven spins or angular momenta. Results of extensive computational investigations are summarized, presented and briefly discussed.Comment: 15 pages, 10 figures, presented at ICCSA 2014, 14th International Conference on Computational Science and Application

Stability of Jahn-Teller distortion ordering in LaMn1-x ScxO3

We have investigated the role of Mn3+ Jahn-Teller distortion in driving ferromagnetism in the LaMn1-xScxO3 series. The replacement of Mn by Sc in LaMnO3 decreases the orthorhombic distortion of the Pbnm cell, but the unit cell remains distorted even in the LaScO3 sample. The analysis of the x-ray diffraction patterns indicates a continuous evolution from the typical Jahn-Teller distorted octahedron in LaMnO3 into a nearly regular one in LaScO3. Surprisingly, x-ray absorption spectroscopy measurements at the Mn and Sc K edges reveal the stability of both Jahn-Teller distorted MnO6 octahedron and nearly regular ScO6 octahedron along the whole Sc-substituted series. Moreover, the structure is described as a random distribution of Jahn-Teller distorted MnO6 octahedra spatially ordered as in LaMnO3 and nearly regular ScO6 octahedra. This result contrasts with the pseudocubic phase and the appearance of regular MnO6 octahedra in LaMn1-xGaxO3 for x>0.5. Thus the occurrence of Jahn-Teller distortion strongly depends on the distorted orthorhombic crystallographic structure of the matrix in which the Mn3+ atom is allocated. Besides, a ferromagnetic ground state is observed for x>0.5 in both series independently of the presence (or not) of Jahn-Teller distortions around Mn3+, which discards either the spin flipping or the vibronic superexchange models proposed for the ferromagnetism in LaMn1-xBxO3 (B=Sc or Ga).The authors thank ESRF, ALBA, and Elettra Synchrotrons for granting beam time. Financial support from the Spanish MINECO (Projects No. MAT2012-38213-C02-01 and No. MAT2011-23791) and the Diputacion General de Aragón (CAMRADS) are acknowledged.Peer Reviewe

The two-dimensional hydrogen atom revisited

The bound state energy eigenvalues for the two-dimensional Kepler problem are found to be degenerate. This "accidental" degeneracy is due to the existence of a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schroedinger equation. This equation is solved by projecting the two-dimensional momentum space onto the surface of a three-dimensional sphere. The eigenfunctions are then expanded in terms of spherical harmonics, and this leads to an integral relation in terms of special functions which has not previously been tabulated. The dynamical symmetry of the problem is also considered, and it is shown that the two components of the Runge-Lenz vector in real space correspond to the generators of infinitesimal rotations about the respective coordinate axes in momentum space.Comment: 10 pages, no figures, RevTex

Boundary Conditions on Internal Three-Body Wave Functions

For a three-body system, a quantum wave function $\Psi^\ell_m$ with definite $\ell$ and $m$ quantum numbers may be expressed in terms of an internal wave function $\chi^\ell_k$ which is a function of three internal coordinates. This article provides necessary and sufficient constraints on $\chi^\ell_k$ to ensure that the external wave function $\Psi^\ell_m$ is analytic. These constraints effectively amount to boundary conditions on $\chi^\ell_k$ and its derivatives at the boundary of the internal space. Such conditions find similarities in the (planar) two-body problem where the wave function (to lowest order) has the form $r^{|m|}$ at the origin. We expect the boundary conditions to prove useful for constructing singularity free three-body basis sets for the case of nonvanishing angular momentum.Comment: 41 pages, submitted to Phys. Rev.

Dependence of the Jahn-Teller distortion in LaMn1-xScxO3 on the isovalent Mn-site substitution

We investigated the relative importance of removing the Mn3+ Jahn-Teller distortion in driving ferromagnetism in LaMn1-xScxO3 combining x-ray powder diffraction and x-ray absorption spectroscopy at the Mn and Sc K-edges. By increasing the Sc content, the orthorhombic distortion of the Pbnm cell in LaMnO3 decreases but the unit-cell remains slightly distorted in LaScO3. Besides, the nearly tetragonal-distorted MO6 in LaMnO3 continuously evolves into a nearly regular one in LaScO3. On the other hand, x-ray absorption spectra show that the MnO6 octahedron remains Jahn-Teller distorted and the ScO6 octahedron is nearly regular along the whole series. Moreover, the ordering of the Mn3+ Jahn-Teller distortion is not disrupted in the ab plane for any Sc concentration. This contrasts with the Gasubstituted compounds, where a regular MnO6 is found for x > 0.5. However, both LaMn0.5Sc0.5O3 and LaMn0.5Ga0.5O3 show ferromagnetic behavior independently of the presence (or not) of Jahn-Teller distorted Mn3+. Thus, our results point to the Mn-sublattice dilution as the main effect in driving ferromagnetism in these manganites over local structure effects previously proposed by the spin flipping or the vibronic superexchange models