# 3j-symbol

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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.

## Contents

## Inverse relation

The inverse relation can be found by noting that *j*_{1} - *j*_{2} - *m*_{3} is an integral number and making the substitution

## 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 quantum numbers also gives a phase:

## Selection rules

The Wigner 3j is zero unless , is integer, and .

## Scalar invariant

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

is invariant under rotations.

## Orthogonality Relations