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Chinese Physics C> 2015, Vol. 39> Issue(6) : 063104 DOI: 10.1088/1674-1137/39/6/063104

Rotational invariants constructed by the products of three spherical harmonic polynomials

  • The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors in the compact form, where the coefficients are calculated explicitly in this paper.
      PCAS:
    • 02.30.Gp(Special functions)
    • 03.65.Fd(Algebraic methods)
    • 02.20.Qs(General properties, structure, and representation of Lie groups)

  • References

    [1] Weyl H. The Classical Groups: Their Invariants and Representations. Princeton, New Jersey: Princeton University Press, 1946[2] Biedenharn L C, Louck J D. The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Application. Vol. 9. Ed. Rota G C. Massachusetts: Addison-Wesley, 1981[3] Fromm D M, Hill R N. Phys. Rev. A, 1987, 36: 1013[4] Harris F E. Integrals for Exponentially Correlated Four-body Systems of General Angular Symmetry. Vol. III, Ed. Brndas E J, Kryachko E S. Fundamental World of Quantum Chemistry, Dordrecht, Kluwer, 2004. 115[5] Manakov N L, Marmo S I, Meremianin A V. J. Phys. B, 1996, 29: 2711[6] Manakov N L, Meremianin A V, Starace A F. Phys. Rev. A, 1998, 57: 3233[7] Meremianin A V, Briggs J S. Phys. Rep., 2003, 384: 121[8] Borisenko O, Kushnir V. Ukr. J. Phys., 2006, 51: N 1, 90[9] Joshi N, Jhingan S, Souradeep T, Hajian A. Phys. Rev. D, 2010, 81: 083012[10] Rocha G, Hobson M P, Smith S, Ferreira P, Challinor A. Mon. Not. R. Astron. Soc., 2005, 357: 1[11] Pápai P, Szapudi I. Mon. Not. R. Astron. Soc., 2008, 389: 292[12] Raccanelli A, Samushia L, Percival W J. Mon. Not. R. Astron. Soc., 2010, 409: 1525[13] GU X Y, DUAN B, MA Z Q. Phys. Rev. A, 2001, 64: 042108[14] MA Z Q. Group Theory for Physicists. Singapore: World Scientific, 2007[15] Gradshteyn I S, Ryzhik I M. Table of Integrals, Series, and Products. Ed. Jeffrey A, Zwillinger D. 7th Ed. Academic Press, 2007

  • References

    [1] Weyl H. The Classical Groups: Their Invariants and Representations. Princeton, New Jersey: Princeton University Press, 1946[2] Biedenharn L C, Louck J D. The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Application. Vol. 9. Ed. Rota G C. Massachusetts: Addison-Wesley, 1981[3] Fromm D M, Hill R N. Phys. Rev. A, 1987, 36: 1013[4] Harris F E. Integrals for Exponentially Correlated Four-body Systems of General Angular Symmetry. Vol. III, Ed. Brndas E J, Kryachko E S. Fundamental World of Quantum Chemistry, Dordrecht, Kluwer, 2004. 115[5] Manakov N L, Marmo S I, Meremianin A V. J. Phys. B, 1996, 29: 2711[6] Manakov N L, Meremianin A V, Starace A F. Phys. Rev. A, 1998, 57: 3233[7] Meremianin A V, Briggs J S. Phys. Rep., 2003, 384: 121[8] Borisenko O, Kushnir V. Ukr. J. Phys., 2006, 51: N 1, 90[9] Joshi N, Jhingan S, Souradeep T, Hajian A. Phys. Rev. D, 2010, 81: 083012[10] Rocha G, Hobson M P, Smith S, Ferreira P, Challinor A. Mon. Not. R. Astron. Soc., 2005, 357: 1[11] Pápai P, Szapudi I. Mon. Not. R. Astron. Soc., 2008, 389: 292[12] Raccanelli A, Samushia L, Percival W J. Mon. Not. R. Astron. Soc., 2010, 409: 1525[13] GU X Y, DUAN B, MA Z Q. Phys. Rev. A, 2001, 64: 042108[14] MA Z Q. Group Theory for Physicists. Singapore: World Scientific, 2007[15] Gradshteyn I S, Ryzhik I M. Table of Integrals, Series, and Products. Ed. Jeffrey A, Zwillinger D. 7th Ed. Academic Press, 2007

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1. Ma, Z.-Q.. Group Theory for Physicists, Second Edition[J]. Group Theory for Physicists, Second Edition, 2019. doi: 10.1142/11187
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MA Zhong-Qi and YAN Zong-Chao. Rotational invariants constructed by the products of three spherical harmonic polynomials[J]. Chinese Physics C, 2015, 39(6): 063104. doi: 10.1088/1674-1137/39/6/063104
MA Zhong-Qi and YAN Zong-Chao. Rotational invariants constructed by the products of three spherical harmonic polynomials[J]. Chinese Physics C, 2015, 39(6): 063104.  doi: 10.1088/1674-1137/39/6/063104 shu
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Received: 2014-09-09
Revised: 2014-11-28
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Rotational invariants constructed by the products of three spherical harmonic polynomials

    Corresponding author: MA Zhong-Qi,
    Corresponding author: YAN Zong-Chao,
  • Received Date: 2014-09-09
  • Accepted Date: 2014-11-28
  • Available Online: 2015-06-05

Abstract: The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors in the compact form, where the coefficients are calculated explicitly in this paper.

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