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6-j symbol

Wigner's 6 − j symbols were introduced by Eugene Paul Wigner in 1940, and published in 1965. They are related to Racah's W-coefficients by

They have higher symmetry than Racah's W-coefficients.

Contents 1 Symmetry relations 2 Special case 3 Orthogonality relation 4 See also 5 External links 6 References

Symmetry relations

The 6 − j symbol is invariant under the permutation of any two columns:

The 6 − j symbol is also invariant if upper and lower arguments are interchanged in any two columns:

The 6 − j symbol

is zero unless j₁, j₂, and j₃ satisfy triangle conditions, i.e.,

In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for (j₁,j₅,j₆), (j₄,j₂,j₆), and (j₄,j₅,j₃).

Special case

When j₆ = 0 the expression for the 6-j symbol is:

The function Δ(j₁,j₂,j₃) is equal to 1 when (j₁,j₂,j₃) satisfy the triangle conditions, and zero otherwise. The symmetry relations can be used to find the expression when another j is equal to zero.

Orthogonality relation

The 6-j symbols satisfy this orthogonality relation:

See also

• Clebsch-Gordan coefficient
• 3-jm symbol
• Racah W-coefficient
• 9-j symbol

External links

• Anthony Stone’s Wigner coefficient calculator (Gives exact answer)
• Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator
• 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science

References

• Biedenharn, L. C.; van Dam, H. (1965). Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers. New York: Academic Press. ISBN 0120960567. 

• Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9. 

• Condon, Edward U.; Shortley, G. H. (1970). "Chapter 3", The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4. 

• Messiah, Albert (1981). Quantum Mechanics (Volume II), 12th edition, New York: North Holland Publishing. ISBN 0-7204-0045-7. 

• Brink, D. M.; Satchler, G. R. (1993). "Chapter 2", Angular Momentum, 3rd edition, Oxford: Clarendon Press. ISBN 0-19-851759-9. 

• Zare, Richard N. (1988). "Chapter 2", Angular Momentum. New York: John Wiley. ISBN 0-471-85892-7. 

• Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0201135078.