The Future of Quantum Theory of Angular Momentum

Discrete Mathematics and Combinatorics
  • James D. Louck


THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [1] and Landé [2] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [3], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [4] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [5]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [6], Pauli [7], and Güttinger and Pauli [8]. These works, and the books by Born and Jordan [9] and Condon and Shortley [10], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.


Angular Momentum Quantum Theory Binary Tree Binary Sequence Kronecker Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Sommerfeld, Zur Quantentheorie der Spektrallinien, Ann. Physik 51 (1916), 1–94, 125-167.MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    E. Back and A. Landé, Zeemaneffekt und Multiplettstruktur der Spektrallinien, Springer-Verlag, Berlin, 1924.Google Scholar
  3. [3]
    M. Born, W. Heisenberg, P. Jordan, Zur Quantenmechanik II, Z. Physik 35 (1926), 557–615.ADSMATHCrossRefGoogle Scholar
  4. [4]
    P. A. M. Dirac, The elimination of the nodes in quantum mechanics, Proc. Roy. Soc. A111 (1926), 281–305.ADSGoogle Scholar
  5. [5]
    P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford Univ. Press, London, 1st. edition, 1930.MATHGoogle Scholar
  6. [6]
    W. Heisenberg and P. Jordan, Anwendung der Quantenmechanik auf das Problem der anomalen Zeemaneffekte, Z. Physik 37 (1926), 263–277.ADSMATHCrossRefGoogle Scholar
  7. [7]
    W. Pauli, Zur Quantenmechanik des magnetischen Elektrons, Z. Physik 43 (1927), 601–623.ADSMATHCrossRefGoogle Scholar
  8. [8]
    P. Güttinger and W. Pauli, Zur Hyperfeinstruktur von Li+, Teil II. Mathematisher Anhang, Z. Physik 67 (1931), 754–765.Google Scholar
  9. [9]
    M. Born and P. Jordan, Elementare Quantenmechanic, J. Springer, 1930.Google Scholar
  10. [10]
    E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge Univ. Press, London, 1935.Google Scholar
  11. [11]
    H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen, London, 1931 (Dover, New York, 1949). Translation by H. P. Robertson of the 1928 German edition.MATHGoogle Scholar
  12. [12]
    J. von Neumann and E. P. Wigner, Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons, Z. Physik 47 (1928), 203–220; 49 (1928), 73-94; 51 (1928), 844-858.ADSCrossRefGoogle Scholar
  13. [13]
    E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. Translation by J. J. Griffin of the 1931 German edition.MATHGoogle Scholar
  14. [14]
    C. Eckart, The application of group theory to the quantum dynamics of monatomic systems, Rev. Mod. Phys. 2 (1930), 305–380.ADSMATHCrossRefGoogle Scholar
  15. [15]
    J. C. Slater, Solid State and Molecular Theory: A Scientific Biography, Wiley-Interscience, NY, 1975.Google Scholar
  16. [16]
    G. Racah, Theory of complex spectra. I. Phys. Rev. 61 (1942), 186–197; II. 62 (1942), 438-462; HI. 63 (1943), 368-382; IV. 76 (1949), 1352-1365.ADSGoogle Scholar
  17. [17]
    R. M. F. Houtappel, H. van Dam, and E. P. Wigner, The conceptual basis and use of the geometrical invariance principles, Rev. Mod. Phys. 37 (1965), 595–632.ADSCrossRefGoogle Scholar
  18. [18]
    S. Lie, Gesammelte Abhandlungen V (eds., F. Engel and P. Heegaard, Leipzig, Teubner, 1924).Google Scholar
  19. [19]
    E. Cartan, Sur la Structure des Groupes Finis and Continus, Thesis, Paris, Nony, 1894 (Oeuvres Complète, Part I, pp. 137–287, Gauthier Villars, Paris, 1952).Google Scholar
  20. [20]
    W. Miller, Jr., Symmetry andSeparation ofVariables, Vol. 4, Encycl. of Math. and Its Appl. (ed., G. C. Rota, Addison-Wesley, Reading, MA, 1977).Google Scholar
  21. [21]
    W. Pauli, Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Phys. 36 (1926), 336–363.ADSMATHCrossRefGoogle Scholar
  22. [22]
    G. Racah, Group Theory and Spectroscopy, Princeton Lectures, 1951. Published in Ergeb Exakt. Naturw. 37 (1965), 28–84.MathSciNetGoogle Scholar
  23. [23]
    Quantum Theory of Angular Momentum, L. C. Biedenharn and H.van Dam, eds., Academic Press, NY, 1965.Google Scholar
  24. [24]
    L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8; The Racah-Wigner Algebra in Quantum Theory, Vol 9., Encycl. of Math, and Its Appl. (ed., G. C. Rota, Cambridge Univ. Press, Cambridge, 1981, originally published by Addison-Wesley).Google Scholar
  25. [25]
    M. Rotenberg, N. Metropolis, R. Bivins, and J. K. Wooten, Jr., The 3 — J and 6-j Symbols, M.I.T. Press, Cambridge, MA, 1959.Google Scholar
  26. [26]
    E. Cartan, Les groupes projectif’s qui ne laisse invariants aucune multiplicité plane, Bull. Soc. Math. France 41 (1913), 53–96.MathSciNetMATHGoogle Scholar
  27. [27]
    E. P. Wigner, On the Matrices Which Reduce the Kronecker Product of S.R. Groups. Princeton Lectures, 1940. Published in Quantum Theory of Angular Momentum, L. C. Biedenharn and H. van Dam, eds., Academic Press, NY, 1965.Google Scholar
  28. [28]
    A. Clebsch, Theorie der binären algebraischen Formen, Teubner, Leipzig (1872).Google Scholar
  29. [29]
    P. Gordan, Über das Formensystem binärer Formen, Teubner, Leipzig (1875).Google Scholar
  30. [30]
    J. D. Louck, Angular momentum theory, in: Atomic, Molecular, and Optical Physics Reference Book, G. W. F. Drake and N. Hedgecock, eds., Am. Inst. Phys., NY, 1996, 6–55.Google Scholar
  31. [31]
    E. P. Wigner, Berichtigung zu der Arbeit: Einige Folgerungen aus der Schrö-dingerschen Theorie für die Termstrukturen, Z. Physik 45 (1927), 601–602.ADSMATHCrossRefGoogle Scholar
  32. [32]
    E. P. Wigner, Application of Group Theory to Special Functions of Mathematical Physics, (Unpublished lecture notes, Princeton Univ., 1955).Google Scholar
  33. [33]
    J. Talman, Special Function: A Group Theoretical Approach, Benjamin, NY, 1968.Google Scholar
  34. [34]
    N. Vilenkin, Special functions and the theory of group representations, Transi. Amer. Math. Soc. 22 (1968).Google Scholar
  35. [35]
    J. Schwinger, On Angular Momentum. U.S. Atomic Energy Commission Report NYO-3071, 1952 (unpublished). Published in: Quantum Theory of Angular Momentum, L. C. Biedenharn and H. van Dam, eds., Academic Press, NY, 1965, 229–279.Google Scholar
  36. [36]
    T. Regge, Symmetry properties of Clebsch-Gordan coefficients, Nuovo Cimento 10 (1958), 544–545.MATHCrossRefGoogle Scholar
  37. [37]
    V. Bargmann, On the representations of the rotation group, Rev. Mod. Phys. 34 (1962), 829–845.MathSciNetADSMATHCrossRefGoogle Scholar
  38. [38]
    J. D. Louck, MacMahon’s master theorem, double tableau polynomials, and representations of groups, Adv. Appl. Math. 17 (1996), 143–168.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    J. D. Louck and P. Stein, Weight-2 zeros of 3; coefficients and the Pell equation, J. Math. Phys. 28 (1987), 2812–2823.MathSciNetADSMATHCrossRefGoogle Scholar
  40. [40]
    J. D. Louck, Survey of zeros of 3j and 6J coefficients by Diophantine equation methods, in: International Symposium on Group Theory and Special Symmetries in Nuclear Physics, J. P. Draayer and J. Janecke, eds., World Scientific, Singapore, 1992.Google Scholar
  41. [41]
    J. P. S. Kung and G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. 10 (1984), 27–85.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    L. C. Biedenharn, An identity satisfied by the Racah coefficients, J. Math. Phys. 31 (1953), 287–293.MathSciNetMATHGoogle Scholar
  43. [43]
    J. P. Elliott, Theoretical studies in nuclear spectroscopy V. The matrix elements of non-central forces with applications to the 2p-shell, Proc. Roy. Soc. A218 (1953), 345–370.ADSGoogle Scholar
  44. [44]
    V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, Studies in Mathematics 19, de Gruyter, NY, 1994.Google Scholar
  45. [45]
    R. Askey and J. A. Wilson, A set of hypergeometric polynomials that generalize the Racah or 6-j symbols, SIAMJ. Math. Anal. 10 (1979), 1008–1016.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    R. Askey and J. A. Wilson, A set of hypergeometric orthogonal polynomials, SIAMJ. Math. Anal. 13 (1982), 651–655.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    A. P. Yutsis (Jucys), I. B. Levinson, and V. V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum, Russian edition, 1960. Translation by A. Sen and A. R. Sen, Jerusalem, 1962. Translated and published by Gordon and Breach, 1964.Google Scholar
  48. [48]
    A. P. Yutsis (Jucys) and A. A. Bandzaitis, Angular Momentum Theory in Quantum Mechanics, (Molslas, Vilnius, 1977).Google Scholar
  49. [49]
    L. C. Biedenharn, W. Y. C. Chen, M. A. Lohe, and J. D. Louck, The role of 3nj coefficients in SU(3), in: Symmetry and Structural Properties of Condensed Matter, T. Lulek, W. Florek, and S. Walcerz, eds., Proc. Third SSCPM, World Scientific, Singapore, 1995, 150–182.Google Scholar
  50. [50]
    D. E. Knuth, Permutations, matrices and generalized Young tableaux, Pac. J. Math. 34 (1970), 709.MathSciNetMATHCrossRefGoogle Scholar
  51. [51]
    G. E. Baird and L. C. Biedenharn, On the representations of the semisimple Lie groups. IV. A canonical classification for tensor operators in SU3, J. Math. Phys. 5 (1965), 1730–1747.MathSciNetCrossRefGoogle Scholar
  52. [52]
    B. Kostant, A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73.MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    J. D. Louck and L. C. Biedenharn, Some properties of the intertwining number of the general linear group, Science and Computers, Adv. Math. Suppl. Studies 10(2) (1986), 265–311.Google Scholar
  54. [54]
    K. Baclawski, A new rule for computing Clebsch-Gordan series, Adv. Appl. Math. 5 (1984), 416–432.MathSciNetMATHCrossRefGoogle Scholar
  55. [55]
    L. C. Biedenharn, A. M. Bincer, M. A. Lohe, and J. D. Louck, New relations and identities for generalized hypergeometric coefficients, Adv. Appl Math. 13 (1992), 62–121.MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    J. D. Louck and L. C. Biedenharn, Special functions associated with SU(3) Wigner-Clebsch-Gordan coefficients, in: Symmetry and Structural Properties of Condensed Matter, T. Lulek, W. Florek, and M. Mucha, eds., Proc. First SSCP, World Scientific, Singapore, 1990, 207–229.Google Scholar
  57. [57]
    L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined by tableaux, Adv. Appl. Math. 10 (1989), 396–438.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    L. C. Biedenharn and J. D. Louck, Inhomogeneous basis set of symmetric polynomials defined by tableaux, Proc. Natl. Acad. USA 87 (1990), 1441–1445.MathSciNetADSMATHCrossRefGoogle Scholar
  59. [59]
    W. Y. C. Chen and J. D. Louck, The factorial Schur function, J. Math. Phys. 34 (1993), 4144–4160.MathSciNetADSMATHCrossRefGoogle Scholar
  60. [60]
    I. G. Macdonald, Schur functions: Themes and variations. Publ. IRMA Strasbourg, 1992, 498/S-27. Actes 28e Séminare Lotharingien, pp. 5-39.Google Scholar
  61. [61]
    I. Goulden and C. Greene, A new tableau representation for supersymmetric Schur functions, J. Algebra 170 (1994), 686–703.MathSciNetCrossRefGoogle Scholar
  62. [62]
    I. P. Goulden and A. M. Hamel, Shift operators and factorial symmetric functions, J. Combin. Theory Ser. A69 (1995), 51–60.MathSciNetCrossRefGoogle Scholar
  63. [63]
    A. Okounkov and G. Olshanski, Shifted Schur functions (preprint, February, 1996).Google Scholar
  64. [64]
    L. C. Biedenharn and M. A. Lohe, Quantum Group Symmetry and q-Tensor Algebras, World Scientific, Singapore, 1995.MATHCrossRefGoogle Scholar
  65. [65]
    J. D. Louck, Unitary symmetry, combinatorics and generating functions, in: Proc. of International Symposium on Combinatorics and Applications, Nankai Inst. Math., Tianjin, China, June, 1996), (to appear in Discrete Mathematics)Google Scholar
  66. [66]
    J. D. Louck, Unitary symmetry, combinatorics, and special functions, in: Group 21:Physical Applications and Mathematical Aspects of Geometry, Groups, andAlgebras, Vol. I, H.-D. Doebner, P. Mattermann, and W. Scherer, eds., Proc. XXI International Symposium on Group Theoretical Methods in Physics, Goslar, Germany, 1996, World Scientific, Singapore, 1997, 50–70.Google Scholar
  67. [67]
    J. D. Louck, Combinatorial aspects of representations of the unitary group, in: Symmetry and Structural Properties of Condensed Matter, T. Lulek, W. Florek, and B. Lulek, eds., Proc. Fourth SSCPM, World Scientific, Singapore, 1997, 231–252.Google Scholar
  68. [68]
    J. D. Louck, Power of a determinant with two physical applications, Int. J. Math. and Math. Sci., 22 (1999), 745–759.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    J. D. Louck, W. Y. C. Chen, and H. W. Galbraith, Combinatorics of 3n — j coefficients, in: Proc. VIII International Conf. on Symmetry Methods in Physics, July, 1997, Dubna, Russia, Physics of Atomic Nuclei 61 (1998), 1868–1873.MathSciNetGoogle Scholar
  70. [70]
    J. D. Louck, W. Y. C. Chen, and H. W. Galbraith, Generating functions for SU(2) binary recoupling coefficients, in: Symmetry and Structural Properties of Condensed Matter, T. Lulek, A. Wal, and B. Lulek, eds., Proc. Fifth SSCPM, World Scientific, Singapore, 1999, 112–137.Google Scholar
  71. [71]
    J. D. Louck, W. Y C. Chen, H. W. Galbraith, and J. D. Louck, Angular momentum theory and the umbral calculus, in: Conference on the Umbral Calculus, Cortona, Italy, June, 1998, to appear in Int. J. Computers and Math. Appl.Google Scholar
  72. [72]
    W. Y. C. Chen and J. D. Louck, The combinatorics of representation functions, Adv. Math. 140 (1998), 207–236.MathSciNetMATHCrossRefGoogle Scholar
  73. [73]
    J. D. Louck, From the rotation group to the Poincaré group, Symmetries in Science VI, B. Gruber, ed., Plenum Press, N.Y., 1993, 455–468.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • James D. Louck
    • 1
  1. 1.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations