## Anthony Stone’s Wigner coefficient calculator

### Wigner 3-j, 6-j and 9-j symbols

My root-rational-fraction program RRF was written to make it possible to calculate Wigner 3-j, 6-j and 9-j symbols exactly. These things appear in angular momentum theory, and can always be expressed as square roots of rational fractions. If you need to use them extensively, you might find the RRF program useful — you can download it and compile it on your own machine. However, if you only need occasional values, you can use this calculator, which uses the RRF program to evaluate individual coefficients. Just fill in the appropriate set of boxes below and push the button for the 3-j, 6-j or 9-j symbol that you need.

All entries take the form “n” or “n/2” where n is an integer.

The program has been tested extensively and used in a number of applications without any errors becoming apparent. However no guarantee can be given that the results are correct.

## 3j symbol

The 3j symbol is zero unless the sum of the m values is zero. The j values must all be positive or zero, and they must satisfy the triangle rule — i.e. their sum must be an integer, and the sum of any two must be not less than the third.

 ( j1 j2 j3 ) m1 m2 m3
 j1 = j2 = j3 = m1 = m2 = m3 =

## 6j symbol

The triples (j1, j2, j3), (j1, l2, l3), (l1, j2, l3) and (l1, l2, j3) must all satisfy the triangle rule.

 { j1 j2 j3 } l1 l2 l3
 j1 = j2 = j3 = l1 = l2 = l3 =

## 9j symbol

The three arguments in each row and in each column must satisfy the triangle rule.

 { a b c } d e f g h i
 a = b = c = d = e = f = g = h = i =