Wigner-Eckart Theorem

  • C. A. Morrison
Part of the Lecture Notes in Chemistry book series (LNC, volume 47)


The Wigner-Eckart theorem states that if we have a spherical tensor Tkq in the space spanned by the wave functions |JM>, then the matrix elements are
$$<J'M'\left| {{T_{kq}}} \right|JM>=<J(M)k(q)\left| {J'(M')>\;<J'\left| {{T_k}} \right|J>\quad (Rose,\;1957)\;or\;<J'M'\left| {{T_{kq}}} \right|JM>= {{( - 1)}^{J' - M'}}(\begin{array}{*{20}{c}}{L'} & k & L \\{ - M'} & q & M \\\end{array} )(J'\left| {{T_k}} \right|J)\quad (Judd,\quad 1963;\;Wybourne,\;1965).} \right.$$


Matrix Element Angular Momentum Crystal Field Geometrical Factor Yttrium Aluminum Garnet 
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Annotated Bibliography and References

  1. Ballhausen, C. J. (1962), Introduction to Ligand Field Theory, McGraw-Hill, New York, NY.Google Scholar
  2. Brink, D. M., and G. R. Satchler (1962), Angular Momentum, Clarendon Press, Oxford, U.K.Google Scholar
  3. Hufner, S. (1978), Optical Spectra of Transparent Rare Earth Compounds, Academic Press, New York, NY.Google Scholar
  4. Judd, B. R. (1963), Operator Techniques in Atomic Spectroscopy, McGraw-Hill, New York, NY, chapters 1, 2, 3, and 4.Google Scholar
  5. Konig, E., and S. Kremer (1977), Ligand Field Energy Diagrams, Plenum Press, New York, NY. Pages 19 through 22 give relationships of the various notations used to describe the crystal field.Google Scholar
  6. Merzbacher, E. (1961), Quantum Mechanics, Wiley, New York, NY.Google Scholar
  7. Rose, M. E. (1957), Elementary Theory of Angular Momentum, Wiley, New York, NY.Google Scholar
  8. Rotenberg, M., R. Bevins, N. Metropolis, and J. K. Wooten, Jr. (1969), The 3-j and 6-j Symbols, MIT Press, Cambridge, MA.Google Scholar
  9. Schiff, L. I. (1968), Quantum Mechanics, 3rd ed., McGraw-Hill, New York, NY.Google Scholar
  10. Sobelman, I. I. (1979), Atomic Spectra and Radiative Transitions, Springer-Verlag, New York, NY.Google Scholar
  11. Watanabe, H. (1966), Operator Methods in Ligand Field Theory, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  12. Wybourne, B. G. (1965), Spectroscopic Properties of Rare Earths, Wiley, New York, NY.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • C. A. Morrison
    • 1
  1. 1.Harry Diamond LaboratoriesAdelphiUSA

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