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Spherical and Hyperbolic Spin Networks: The q-extensions of Wigner-Racah 6j Coefficients and General Orthogonal Discrete Basis Sets in Applied Quantum Mechanics

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Computational Science and Its Applications – ICCSA 2017 (ICCSA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10408))

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Abstract

Discrete basis sets continue to have an important role in mathematics and quantum mechanics. Racah recoupling coefficients or their closely related Wigner 6j symbols form a remarkably rich source of such functions, and now their properties are well understood for Euclidean space where \(q = 1\). Here we report a unified treatment of their q-extensions to non-Euclidean spaces: hyperbolic, for real q different from 1, and spherical, for \(q = r^{th}\) root of unity. We calculate the non-Euclidean coefficients as the eigenvectors of a real symmetric tridiagonal matrix. The eigenvectors form a discrete ortho-normal basis set, and the eigenvalues can be interpreted as energy levels. We provide extensive numerical results, and also show the Neville volume formula for a tetrahedron appears to be valid for both hyperbolic and spherical cases in the semiclassical limit. This q-extended volume is used to scale the magnitude of the eigenvectors in a familiar fashion. We also determine for spherical space that the radius, r, has a sharp minimum value to support all x,y given by triangular relations for a quadrilateral (not necessarily planar) with side lengths a, b, c, d and forming a tetrahedron. The ranges of x and y are truncated for \(r < a+b+c+d+2\).

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Author information

Authors and Affiliations

  1. Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA, 95064, USA

    Roger W. Anderson

  2. Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, via Elce di Sotto, 8, 06183, Perugia, Italy

    Vincenzo Aquilanti

  3. Istituto di Struttura Della Materia, Consiglio Nazionale Delle Ricerche, Piazzale Aldo Moro, 7, 00185, Rome, Italy

    Vincenzo Aquilanti

Authors
  1. Roger W. Anderson
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  2. Vincenzo Aquilanti
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Correspondence to Roger W. Anderson .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Trieste, Trieste, Italy

    Giuseppe Borruso

  5. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

  6. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  7. Monash University, Clayton, Victoria, Australia

    David Taniar

  8. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  9. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  10. University of Trieste, Trieste, Italy

    Alfredo Cuzzocrea

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Anderson, R.W., Aquilanti, V. (2017). Spherical and Hyperbolic Spin Networks: The q-extensions of Wigner-Racah 6j Coefficients and General Orthogonal Discrete Basis Sets in Applied Quantum Mechanics. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10408. Springer, Cham. https://doi.org/10.1007/978-3-319-62404-4_25

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  • DOI: https://doi.org/10.1007/978-3-319-62404-4_25

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