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

Racah W-coefficient

Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory....

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

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 , , and 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 ,
, and .

Special case

When the expression for the 6-j symbol is:
The function is equal to 1 when satisfy the triangle conditions,
and zero otherwise. The symmetry relations can be used to find the expression when another is equal
to zero.

Orthogonality relation

The 6-j symbols satisfy this orthogonality relation:

See also

• Clebsch–Gordan coefficients
• 3-jm symbol
• Racah W-coefficient
  Racah W-coefficient
  Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory....
  
  
• 9-j symbol

External links

(Gives exact answer)

• Frederik J Simons: Matlab software archive, the code SIXJ.M Static table for j₁ <= 11/2.