# 3j-symbol

In physics and mathematics, Wigner **3 -jm symbols**, also called 3

*j*symbols, are related to the Clebsch-Gordan coefficients of the groups SU(2) and SO(3) through

The 3*j* symbols show more symmetry in permutation of the labels than the corresponding Clebsch-Gordan coefficients.
Exactly as is true for the Clebsch-Gordan coefficients, the *j*-values are positive and either integral: (0, 1, 2,..) or half-integral: (1/2, 3/2, 5/2, ...).

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## [edit] Note on phases

All 3*j* symbols are real, which means that the value of *n* in overall phase factors of the kind:

must be integral, otherwise exp[*iπn*] would not be on the real axis in the complex plane. For half-integral *n* the phase factor is purely imaginary and a 3*j* symbol containing the factor would be too, so that half-integral values do not appear as powers of −1. The overall powers of −1 are odd or even integral numbers. The following relation (*n* integral) seems not to be known to everyone contributing to the corresponding article on Wikipedia:

Note that the expression

is necessarily integral, since *m* runs in unit steps and *j*−*m*_{max} = *j*+*m*_{min} =0. Likewise *j*_{1}−*j*_{2}±*m*_{3} is integral [±*m*_{3} = ±(−*m*_{1}−*m*_{2})].

## [edit] Inverse relation

The inverse relation—the Clebsch-Gordan coefficient given by a 3*j* symbol—can be found by noting that *j*_{1} - *j*_{2} - *m*_{3} is an integral number and making the substitution

## [edit] Symmetry properties

The symmetry properties of 3*j* symbols are more convenient than those of
Clebsch-Gordan coefficients. A 3*j* symbol is invariant under an even
permutation of its columns:

An odd permutation of the columns gives a phase factor:

Changing the sign of the *m* quantum numbers also gives a phase:

## [edit] Selection rules

The Wigner 3j is zero unless
*m*_{1} + *m*_{2} + *m*_{3} = 0, *j*_{1} + *j*_{2} + *j*_{3} is integer, and .

## [edit] Scalar invariant

The contraction of the product of three rotational states with a 3*j* symbol,

is invariant under rotations.

## [edit] Orthogonality Relations

## [edit] References

- E. P. Wigner,
*On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups*, unpublished (1940). Reprinted in: L. C. Biedenharn and H. van Dam,*Quantum Theory of Angular Momentum*, Academic Press, New York (1965). - A. R. Edmonds,
*Angular Momentum in Quantum Mechanics*, 2nd edition, Princeton University Press, Pinceton, 1960. - D. M. Brink and G. R. Satchler,
*Angular Momentum*, 3rd edition, Clarendon, Oxford, 1993. - L. C. Biedenharn and J. D. Louck,
*Angular Momentum in Quantum Physics*, volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981. - D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii,
*Quantum Theory of Angular Momentum*, World Scientific Publishing Co., Singapore, 1988.