Advertisement

Symbolic Algorithm for Generating the Orthonormal Bargmann–Moshinsky Basis for \(\mathrm {SU(3)}\) Group

  • A. Deveikis
  • A. A. Gusev
  • V. P. Gerdt
  • S. I. VinitskyEmail author
  • A. Góźdź
  • A. Pȩdrak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)

Abstract

A symbolic algorithm which can be implemented in any computer algebra system for generating the Bargmann–Moshinsky (BM) basis with the highest weight vectors of \(\mathrm {SO(3)}\) irreducible representations is presented. The effective method resulting in analytical formula of overlap integrals in the case of the non-canonical BM basis [S. Alisauskas, P. Raychev, R. Roussev, J. Phys. G 7, 1213 (1981)] is used. A symbolic recursive algorithm for orthonormalisation of the obtained basis is developed. The effectiveness of the algorithms implemented in Mathematica 10.1 is investigated by calculation of the overlap integrals for up to \(\mu =5\) with \(\lambda > \mu \) and orthonormalization of the basis for up to \(\mu =4\) with \(\lambda > \mu \). The action of the zero component of the quadrupole operator onto the basis vectors with \(\mu =4\) is also obtained.

Keywords

SU(3) non-canonical basis Group theory Gram-Schmidt orthonormalization Symbolic algorithms 

Notes

Acknowledgment

The work was partially funded by the RFBR grant No. 16-01-00080, the Bogoliubov–Infeld program, and the RUDN University Program 5-100.

References

  1. 1.
    Afanasjev, G.N., Avramov, S.A., Raychev, P.P.: Realization of the physical basis for SU(3) and the probabilities of E2 transitions in the SU(3) formalism. Sov. J. Nucl. Phys. 16, 53–83 (1973)CrossRefGoogle Scholar
  2. 2.
    Alisauskas, S., Raychev, P., Roussev, R.: Analytical form of the orthonormal basis of the decomposition \(SU(3)\supset O(3)\supset O(2)\) for some \((\lambda,\mu )\) multiplets. J. Phys. G: Nucl. Phys. 7, 1213–1226 (1981)CrossRefGoogle Scholar
  3. 3.
    Bargmann, V., Moshinsky, M.: Group theory of harmonic oscillators (II). Nucl. Phys. 23, 177–199 (1961)CrossRefGoogle Scholar
  4. 4.
    Cseh, J.: Algebraic models for shell-like quarteting of nucleons. Phys. Lett. B 743, 213–217 (2015)CrossRefGoogle Scholar
  5. 5.
    Dudek, J., Goźdź, A., Schunck, N., Miśkiewicz, M.: Nuclear tetrahedral symmetry: possibly present throughout the periodic table. Phys. Rev. Lett. 88(25), 252502 (2002)CrossRefGoogle Scholar
  6. 6.
    Dytrych, T.: Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei. Comp. Phys. Comun. 207, 202–210 (2016)CrossRefGoogle Scholar
  7. 7.
    Elliott, J.P.: Collective motion in the nuclear shell model I. Proc. R. Soc. Lond. A 245, 128–145 (1958)CrossRefGoogle Scholar
  8. 8.
    Góźdź, A., Pȩdrak, A., Gusev, A.A., Vinitsky, S.I.: Point symmetries in the nuclear SU(3) partner groups model. Acta Phys. Pol. B Proc. Suppl. 11, 19–27 (2018)CrossRefGoogle Scholar
  9. 9.
    Harvey, M.: The nuclear \(SU_3\) model. In: Baranger, M., Vogt, E. (eds.) Advances in Nuclear Physics. Springer, Boston (1968).  https://doi.org/10.1007/978-1-4757-0103-6_2CrossRefGoogle Scholar
  10. 10.
    Moshinsky, M., Patera, J., Sharp, R.T., Winternitz, P.: Everything you always wanted to know about \(SU(3)\supset O(3)\). Ann. Phys. 95(N.Y.), 139–169 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pan, F., Yuan, S., Launey, K.D., Draayer, J.P.: A new procedure for constructing basis vectors of \(SU(3)\supset SO(3)\). Nucl. Phys. A 743, 70–99 (2016)CrossRefGoogle Scholar
  12. 12.
    Raychev, P., Roussev, R.: Matrix elements of the generators of SU(3) and of the basic O(3) scalars in the enveloping algebra of SU(3). J. Phys. G: Nucl. Phys. 7, 1227–1238 (1981)CrossRefGoogle Scholar
  13. 13.
    Varshalovitch, D.A., Moskalev, A.N., Hersonsky, V.K.: Quantum Theory of Angular Momentum. Nauka, Leningrad (1975). (Also World Scientific (1988))Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • A. Deveikis
    • 1
  • A. A. Gusev
    • 2
  • V. P. Gerdt
    • 2
    • 3
  • S. I. Vinitsky
    • 2
    • 3
    Email author
  • A. Góźdź
    • 4
  • A. Pȩdrak
    • 5
  1. 1.Department of Applied InformaticsVytautas Magnus UniversityKaunasLithuania
  2. 2.Joint Institute for Nuclear ResearchDubnaRussia
  3. 3.RUDN UniversityMoscowRussia
  4. 4.Institute of PhysicsMaria Curie-Skłodowska UniversityLublinPoland
  5. 5.National Centre for Nuclear ResearchWarsawPoland

Personalised recommendations