Table of Clebsch–Gordan coefficients

Short description: Used for adding angular momentum values in quantum mechanics

This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant $j_{1}$ , $j_{2}$ , $j$ is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign convention as discussed by Baird and Biedenharn.^[1] Tables with the same sign convention may be found in the Particle Data Group's Review of Particle Properties^[2] and in online tables.^[3]

Formulation

The Clebsch–Gordan coefficients are the solutions to

| j_{1}, j_{2}; J, M ⟩ = \sum_{m_{1} = - j_{1}}^{j_{1}} \sum_{m_{2} = - j_{2}}^{j_{2}} | j_{1}, m_{1}; j_{2}, m_{2} ⟩ ⟨ j_{1}, j_{2}; m_{1}, m_{2} ∣ j_{1}, j_{2}; J, M ⟩

Explicitly:

\begin{aligned} ⟨ j_{1}, j_{2}; m_{1}, m_{2} ∣ j_{1}, j_{2}; J, M ⟩ \\ = & δ_{M, m_{1} + m_{2}} \sqrt{\frac{(2 J + 1) (J + j_{1} - j_{2})! (J - j_{1} + j_{2})! (j_{1} + j_{2} - J)!}{(j_{1} + j_{2} + J + 1)!}} \times \\ \sqrt{(J + M)! (J - M)! (j_{1} - m_{1})! (j_{1} + m_{1})! (j_{2} - m_{2})! (j_{2} + m_{2})!} \times \\ \sum_{k} \frac{(- 1)^{k}}{k! (j_{1} + j_{2} - J - k)! (j_{1} - m_{1} - k)! (j_{2} + m_{2} - k)! (J - j_{2} + m_{1} + k)! (J - j_{1} - m_{2} + k)!} . \end{aligned}

The summation is extended over all integer $k$ for which the argument of every factorial is nonnegative.^[4]

For brevity, solutions with $M < 0$ and $j 1 < j 2$ are omitted. They may be calculated using the simple relations

⟨ j_{1}, j_{2}; m_{1}, m_{2} ∣ j_{1}, j_{2}; J, M ⟩ = (- 1)^{J - j_{1} - j_{2}} ⟨ j_{1}, j_{2}; - m_{1}, - m_{2} ∣ j_{1}, j_{2}; J, - M ⟩ .

and

⟨ j_{1}, j_{2}; m_{1}, m_{2} ∣ j_{1}, j_{2}; J, M ⟩ = (- 1)^{J - j_{1} - j_{2}} ⟨ j_{2}, j_{1}; m_{2}, m_{1} ∣ j_{2}, j_{1}; J, M ⟩ .

Specific values

The Clebsch–Gordan coefficients for j values less than or equal to 5/2 are given below.^[5]

$j 2 = 0$

When $j 2 = 0$ , the Clebsch–Gordan coefficients are given by $δ_{j, j_{1}} δ_{m, m_{1}}$ .

$j 1 = 1 / 2, j 2 = 1 / 2$

$m = 1$
$j$ $m 1, m 2$	1
1/2, 1/2	$1$

$m = -1$
$j$ $m 1, m 2$	1
−1/2, −1/2	$1$

$m = 0$
$j$ $m 1, m 2$	1	0
1/2, −1/2	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
−1/2, 1/2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$j 1 = 1, j 2 = 1 / 2$

$m = 3 / 2$
$j$ $m 1, m 2$	3/2
1, 1/2	$1$

$m = 1 / 2$
$j$ $m 1, m 2$	3/2	1/2
1, −1/2	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{2}{3}}$
0, 1/2	$\sqrt{\frac{2}{3}}$	$- \sqrt{\frac{1}{3}}$

$j 1 = 1, j 2 = 1$

$m = 2$
$j$ $m 1, m 2$	2
1, 1	$1$

$m = 1$
$j$ $m 1, m 2$	2	1
1, 0	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
0, 1	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$m = 0$
$j$ $m 1, m 2$	2	1	0
1, −1	$\sqrt{\frac{1}{6}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{3}}$
0, 0	$\sqrt{\frac{2}{3}}$	$0$	$- \sqrt{\frac{1}{3}}$
−1, 1	$\sqrt{\frac{1}{6}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{3}}$

$j 1 = 3 / 2, j 2 = 1 / 2$

$m = 2$
$j$ $m 1, m 2$	2
3/2, 1/2	$1$

$m = 1$
$j$ $m 1, m 2$	2	1
3/2, −1/2	$\frac{1}{2}$	$\sqrt{\frac{3}{4}}$
1/2, 1/2	$\sqrt{\frac{3}{4}}$	$- \frac{1}{2}$

$m = 0$
$j$ $m 1, m 2$	2	1
1/2, −1/2	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
−1/2, 1/2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$j 1 = 3 / 2, j 2 = 1$

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
3/2, 1	$1$

$m = 3 / 2$
$j$ $m 1, m 2$	5/2	3/2
3/2, 0	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{3}{5}}$
1/2, 1	$\sqrt{\frac{3}{5}}$	$- \sqrt{\frac{2}{5}}$

$m = 1 / 2$
$j$ $m 1, m 2$	5/2	3/2	1/2
3/2, −1	$\sqrt{\frac{1}{10}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{2}}$
1/2, 0	$\sqrt{\frac{3}{5}}$	$\sqrt{\frac{1}{15}}$	$- \sqrt{\frac{1}{3}}$
−1/2, 1	$\sqrt{\frac{3}{10}}$	$- \sqrt{\frac{8}{15}}$	$\sqrt{\frac{1}{6}}$

$j 1 = 3 / 2, j 2 = 3 / 2$

$m = 3$
$j$ $m 1, m 2$	3
3/2, 3/2	$1$

$m = 2$
$j$ $m 1, m 2$	3	2
3/2, 1/2	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
1/2, 3/2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$m = 1$
$j$ $m 1, m 2$	3	2	1
3/2, −1/2	$\sqrt{\frac{1}{5}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$
1/2, 1/2	$\sqrt{\frac{3}{5}}$	$0$	$- \sqrt{\frac{2}{5}}$
−1/2, 3/2	$\sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$

$m = 0$
$j$ $m 1, m 2$	3	2	1	0
3/2, −3/2	$\sqrt{\frac{1}{20}}$	$\frac{1}{2}$	$\sqrt{\frac{9}{20}}$	$\frac{1}{2}$
1/2, −1/2	$\sqrt{\frac{9}{20}}$	$\frac{1}{2}$	$- \sqrt{\frac{1}{20}}$	$- \frac{1}{2}$
−1/2, 1/2	$\sqrt{\frac{9}{20}}$	$- \frac{1}{2}$	$- \sqrt{\frac{1}{20}}$	$\frac{1}{2}$
−3/2, 3/2	$\sqrt{\frac{1}{20}}$	$- \frac{1}{2}$	$\sqrt{\frac{9}{20}}$	$- \frac{1}{2}$

$j 1 = 2, j 2 = 1 / 2$

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
2, 1/2	$1$

$m = 3 / 2$
$j$ $m 1, m 2$	5/2	3/2
2, −1/2	$\sqrt{\frac{1}{5}}$	$\sqrt{\frac{4}{5}}$
1, 1/2	$\sqrt{\frac{4}{5}}$	$- \sqrt{\frac{1}{5}}$

$m = 1 / 2$
$j$ $m 1, m 2$	5/2	3/2
1, −1/2	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{3}{5}}$
0, 1/2	$\sqrt{\frac{3}{5}}$	$- \sqrt{\frac{2}{5}}$

$j 1 = 2, j 2 = 1$

$m = 3$
$j$ $m 1, m 2$	3
2, 1	$1$

$m = 2$
$j$ $m 1, m 2$	3	2
2, 0	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{2}{3}}$
1, 1	$\sqrt{\frac{2}{3}}$	$- \sqrt{\frac{1}{3}}$

$m = 1$
$j$ $m 1, m 2$	3	2	1
2, −1	$\sqrt{\frac{1}{15}}$	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{3}{5}}$
1, 0	$\sqrt{\frac{8}{15}}$	$\sqrt{\frac{1}{6}}$	$- \sqrt{\frac{3}{10}}$
0, 1	$\sqrt{\frac{2}{5}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{10}}$

$m = 0$
$j$ $m 1, m 2$	3	2	1
1, −1	$\sqrt{\frac{1}{5}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$
0, 0	$\sqrt{\frac{3}{5}}$	$0$	$- \sqrt{\frac{2}{5}}$
−1, 1	$\sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$

$j 1 = 2, j 2 = 3 / 2$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
2, 3/2	$1$

$m = 5 / 2$
$j$ $m 1, m 2$	7/2	5/2
2, 1/2	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{4}{7}}$
1, 3/2	$\sqrt{\frac{4}{7}}$	$- \sqrt{\frac{3}{7}}$

$m = 3 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
2, −1/2	$\sqrt{\frac{1}{7}}$	$\sqrt{\frac{16}{35}}$	$\sqrt{\frac{2}{5}}$
1, 1/2	$\sqrt{\frac{4}{7}}$	$\sqrt{\frac{1}{35}}$	$- \sqrt{\frac{2}{5}}$
0, 3/2	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{18}{35}}$	$\sqrt{\frac{1}{5}}$

$m = 1 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2	1/2
2, −3/2	$\sqrt{\frac{1}{35}}$	$\sqrt{\frac{6}{35}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{2}{5}}$
1, −1/2	$\sqrt{\frac{12}{35}}$	$\sqrt{\frac{5}{14}}$	$0$	$- \sqrt{\frac{3}{10}}$
0, 1/2	$\sqrt{\frac{18}{35}}$	$- \sqrt{\frac{3}{35}}$	$- \sqrt{\frac{1}{5}}$	$\sqrt{\frac{1}{5}}$
−1, 3/2	$\sqrt{\frac{4}{35}}$	$- \sqrt{\frac{27}{70}}$	$\sqrt{\frac{2}{5}}$	$- \sqrt{\frac{1}{10}}$

$j 1 = 2, j 2 = 2$

$m = 4$
$j$ $m 1, m 2$	4
2, 2	$1$

$m = 3$
$j$ $m 1, m 2$	4	3
2, 1	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
1, 2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$m = 2$
$j$ $m 1, m 2$	4	3	2
2, 0	$\sqrt{\frac{3}{14}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{2}{7}}$
1, 1	$\sqrt{\frac{4}{7}}$	$0$	$- \sqrt{\frac{3}{7}}$
0, 2	$\sqrt{\frac{3}{14}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{2}{7}}$

$m = 1$
$j$ $m 1, m 2$	4	3	2	1
2, −1	$\sqrt{\frac{1}{14}}$	$\sqrt{\frac{3}{10}}$	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{1}{5}}$
1, 0	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{14}}$	$- \sqrt{\frac{3}{10}}$
0, 1	$\sqrt{\frac{3}{7}}$	$- \sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{14}}$	$\sqrt{\frac{3}{10}}$
−1, 2	$\sqrt{\frac{1}{14}}$	$- \sqrt{\frac{3}{10}}$	$\sqrt{\frac{3}{7}}$	$- \sqrt{\frac{1}{5}}$

$m = 0$
$j$ $m 1, m 2$	4	3	2	1	0
2, −2	$\sqrt{\frac{1}{70}}$	$\sqrt{\frac{1}{10}}$	$\sqrt{\frac{2}{7}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{5}}$
1, −1	$\sqrt{\frac{8}{35}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{14}}$	$- \sqrt{\frac{1}{10}}$	$- \sqrt{\frac{1}{5}}$
0, 0	$\sqrt{\frac{18}{35}}$	$0$	$- \sqrt{\frac{2}{7}}$	$0$	$\sqrt{\frac{1}{5}}$
−1, 1	$\sqrt{\frac{8}{35}}$	$- \sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{14}}$	$\sqrt{\frac{1}{10}}$	$- \sqrt{\frac{1}{5}}$
−2, 2	$\sqrt{\frac{1}{70}}$	$- \sqrt{\frac{1}{10}}$	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{5}}$

$j 1 = 5 / 2, j 2 = 1 / 2$

$m = 3$
$j$ $m 1, m 2$	3
5/2, 1/2	$1$

$m = 2$
$j$ $m 1, m 2$	3	2
5/2, −1/2	$\sqrt{\frac{1}{6}}$	$\sqrt{\frac{5}{6}}$
3/2, 1/2	$\sqrt{\frac{5}{6}}$	$- \sqrt{\frac{1}{6}}$

$m = 1$
$j$ $m 1, m 2$	3	2
3/2, −1/2	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{2}{3}}$
1/2, 1/2	$\sqrt{\frac{2}{3}}$	$- \sqrt{\frac{1}{3}}$

$m = 0$
$j$ $m 1, m 2$	3	2
1/2, −1/2	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
−1/2, 1/2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$j 1 = 5 / 2, j 2 = 1$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
5/2, 1	$1$

$m = 5 / 2$
$j$ $m 1, m 2$	7/2	5/2
5/2, 0	$\sqrt{\frac{2}{7}}$	$\sqrt{\frac{5}{7}}$
3/2, 1	$\sqrt{\frac{5}{7}}$	$- \sqrt{\frac{2}{7}}$

$m = 3 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
5/2, −1	$\sqrt{\frac{1}{21}}$	$\sqrt{\frac{2}{7}}$	$\sqrt{\frac{2}{3}}$
3/2, 0	$\sqrt{\frac{10}{21}}$	$\sqrt{\frac{9}{35}}$	$- \sqrt{\frac{4}{15}}$
1/2, 1	$\sqrt{\frac{10}{21}}$	$- \sqrt{\frac{16}{35}}$	$\sqrt{\frac{1}{15}}$

$m = 1 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
3/2, −1	$\sqrt{\frac{1}{7}}$	$\sqrt{\frac{16}{35}}$	$\sqrt{\frac{2}{5}}$
1/2, 0	$\sqrt{\frac{4}{7}}$	$\sqrt{\frac{1}{35}}$	$- \sqrt{\frac{2}{5}}$
−1/2, 1	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{18}{35}}$	$\sqrt{\frac{1}{5}}$

$j 1 = 5 / 2, j 2 = 3 / 2$

$m = 4$
$j$ $m 1, m 2$	4
5/2, 3/2	$1$

$m = 3$
$j$ $m 1, m 2$	4	3
5/2, 1/2	$\sqrt{\frac{3}{8}}$	$\sqrt{\frac{5}{8}}$
3/2, 3/2	$\sqrt{\frac{5}{8}}$	$- \sqrt{\frac{3}{8}}$

$m = 2$
$j$ $m 1, m 2$	4	3	2
5/2, −1/2	$\sqrt{\frac{3}{28}}$	$\sqrt{\frac{5}{12}}$	$\sqrt{\frac{10}{21}}$
3/2, 1/2	$\sqrt{\frac{15}{28}}$	$\sqrt{\frac{1}{12}}$	$- \sqrt{\frac{8}{21}}$
1/2, 3/2	$\sqrt{\frac{5}{14}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{7}}$

$m = 1$
$j$ $m 1, m 2$	4	3	2	1
5/2, −3/2	$\sqrt{\frac{1}{56}}$	$\sqrt{\frac{1}{8}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{2}}$
3/2, −1/2	$\sqrt{\frac{15}{56}}$	$\sqrt{\frac{49}{120}}$	$\sqrt{\frac{1}{42}}$	$- \sqrt{\frac{3}{10}}$
1/2, 1/2	$\sqrt{\frac{15}{28}}$	$- \sqrt{\frac{1}{60}}$	$- \sqrt{\frac{25}{84}}$	$\sqrt{\frac{3}{20}}$
−1/2, 3/2	$\sqrt{\frac{5}{28}}$	$- \sqrt{\frac{9}{20}}$	$\sqrt{\frac{9}{28}}$	$- \sqrt{\frac{1}{20}}$

$m = 0$
$j$ $m 1, m 2$	4	3	2	1
3/2, −3/2	$\sqrt{\frac{1}{14}}$	$\sqrt{\frac{3}{10}}$	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{1}{5}}$
1/2, −1/2	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{14}}$	$- \sqrt{\frac{3}{10}}$
−1/2, 1/2	$\sqrt{\frac{3}{7}}$	$- \sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{14}}$	$\sqrt{\frac{3}{10}}$
−3/2, 3/2	$\sqrt{\frac{1}{14}}$	$- \sqrt{\frac{3}{10}}$	$\sqrt{\frac{3}{7}}$	$- \sqrt{\frac{1}{5}}$

$j 1 = 5 / 2, j 2 = 2$

$m = 9 / 2$
$j$ $m 1, m 2$	9/2
5/2, 2	$1$

$m = 7 / 2$
$j$ $m 1, m 2$	9/2	7/2
5/2, 1	$\frac{2}{3}$	$\sqrt{\frac{5}{9}}$
3/2, 2	$\sqrt{\frac{5}{9}}$	$- \frac{2}{3}$

$m = 5 / 2$
$j$ $m 1, m 2$	9/2	7/2	5/2
5/2, 0	$\sqrt{\frac{1}{6}}$	$\sqrt{\frac{10}{21}}$	$\sqrt{\frac{5}{14}}$
3/2, 1	$\sqrt{\frac{5}{9}}$	$\sqrt{\frac{1}{63}}$	$- \sqrt{\frac{3}{7}}$
1/2, 2	$\sqrt{\frac{5}{18}}$	$- \sqrt{\frac{32}{63}}$	$\sqrt{\frac{3}{14}}$

$m = 3 / 2$
$j$ $m 1, m 2$	9/2	7/2	5/2	3/2
5/2, −1	$\sqrt{\frac{1}{21}}$	$\sqrt{\frac{5}{21}}$	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{2}{7}}$
3/2, 0	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{1}{70}}$	$- \sqrt{\frac{12}{35}}$
1/2, 1	$\sqrt{\frac{10}{21}}$	$- \sqrt{\frac{2}{21}}$	$- \sqrt{\frac{6}{35}}$	$\sqrt{\frac{9}{35}}$
−1/2, 2	$\sqrt{\frac{5}{42}}$	$- \sqrt{\frac{8}{21}}$	$\sqrt{\frac{27}{70}}$	$- \sqrt{\frac{4}{35}}$

$m = 1 / 2$
$j$ $m 1, m 2$	9/2	7/2	5/2	3/2	1/2
5/2, −2	$\sqrt{\frac{1}{126}}$	$\sqrt{\frac{4}{63}}$	$\sqrt{\frac{3}{14}}$	$\sqrt{\frac{8}{21}}$	$\sqrt{\frac{1}{3}}$
3/2, −1	$\sqrt{\frac{10}{63}}$	$\sqrt{\frac{121}{315}}$	$\sqrt{\frac{6}{35}}$	$- \sqrt{\frac{2}{105}}$	$- \sqrt{\frac{4}{15}}$
1/2, 0	$\sqrt{\frac{10}{21}}$	$\sqrt{\frac{4}{105}}$	$- \sqrt{\frac{8}{35}}$	$- \sqrt{\frac{2}{35}}$	$\sqrt{\frac{1}{5}}$
−1/2, 1	$\sqrt{\frac{20}{63}}$	$- \sqrt{\frac{14}{45}}$	$0$	$\sqrt{\frac{5}{21}}$	$- \sqrt{\frac{2}{15}}$
−3/2, 2	$\sqrt{\frac{5}{126}}$	$- \sqrt{\frac{64}{315}}$	$\sqrt{\frac{27}{70}}$	$- \sqrt{\frac{32}{105}}$	$\sqrt{\frac{1}{15}}$

$j 1 = 5 / 2, j 2 = 5 / 2$

$m = 5$
$j$ $m 1, m 2$	5
5/2, 5/2	$1$

$m = 4$
$j$ $m 1, m 2$	5	4
5/2, 3/2	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
3/2, 5/2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

$m = 3$
$j$ $m 1, m 2$	5	4	3
5/2, 1/2	$\sqrt{\frac{2}{9}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{5}{18}}$
3/2, 3/2	$\sqrt{\frac{5}{9}}$	$0$	$- \frac{2}{3}$
1/2, 5/2	$\sqrt{\frac{2}{9}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{5}{18}}$

$m = 2$
$j$ $m 1, m 2$	5	4	3	2
5/2, −1/2	$\sqrt{\frac{1}{12}}$	$\sqrt{\frac{9}{28}}$	$\sqrt{\frac{5}{12}}$	$\sqrt{\frac{5}{28}}$
3/2, 1/2	$\sqrt{\frac{5}{12}}$	$\sqrt{\frac{5}{28}}$	$- \sqrt{\frac{1}{12}}$	$- \sqrt{\frac{9}{28}}$
1/2, 3/2	$\sqrt{\frac{5}{12}}$	$- \sqrt{\frac{5}{28}}$	$- \sqrt{\frac{1}{12}}$	$\sqrt{\frac{9}{28}}$
−1/2, 5/2	$\sqrt{\frac{1}{12}}$	$- \sqrt{\frac{9}{28}}$	$\sqrt{\frac{5}{12}}$	$- \sqrt{\frac{5}{28}}$

$m = 1$
$j$ $m 1, m 2$	5	4	3	2	1
5/2, −3/2	$\sqrt{\frac{1}{42}}$	$\sqrt{\frac{1}{7}}$	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{7}}$
3/2, −1/2	$\sqrt{\frac{5}{21}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{30}}$	$- \sqrt{\frac{1}{7}}$	$- \sqrt{\frac{8}{35}}$
1/2, 1/2	$\sqrt{\frac{10}{21}}$	$0$	$- \sqrt{\frac{4}{15}}$	$0$	$\sqrt{\frac{9}{35}}$
−1/2, 3/2	$\sqrt{\frac{5}{21}}$	$- \sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{30}}$	$\sqrt{\frac{1}{7}}$	$- \sqrt{\frac{8}{35}}$
−3/2, 5/2	$\sqrt{\frac{1}{42}}$	$- \sqrt{\frac{1}{7}}$	$\sqrt{\frac{1}{3}}$	$- \sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{7}}$

$m = 0$
$j$ $m 1, m 2$	5	4	3	2	1	0
5/2, −5/2	$\sqrt{\frac{1}{252}}$	$\sqrt{\frac{1}{28}}$	$\sqrt{\frac{5}{36}}$	$\sqrt{\frac{25}{84}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{6}}$
3/2, −3/2	$\sqrt{\frac{25}{252}}$	$\sqrt{\frac{9}{28}}$	$\sqrt{\frac{49}{180}}$	$\sqrt{\frac{1}{84}}$	$- \sqrt{\frac{9}{70}}$	$- \sqrt{\frac{1}{6}}$
1/2, −1/2	$\sqrt{\frac{25}{63}}$	$\sqrt{\frac{1}{7}}$	$- \sqrt{\frac{4}{45}}$	$- \sqrt{\frac{4}{21}}$	$\sqrt{\frac{1}{70}}$	$\sqrt{\frac{1}{6}}$
−1/2, 1/2	$\sqrt{\frac{25}{63}}$	$- \sqrt{\frac{1}{7}}$	$- \sqrt{\frac{4}{45}}$	$\sqrt{\frac{4}{21}}$	$\sqrt{\frac{1}{70}}$	$- \sqrt{\frac{1}{6}}$
−3/2, 3/2	$\sqrt{\frac{25}{252}}$	$- \sqrt{\frac{9}{28}}$	$\sqrt{\frac{49}{180}}$	$- \sqrt{\frac{1}{84}}$	$- \sqrt{\frac{9}{70}}$	$\sqrt{\frac{1}{6}}$
−5/2, 5/2	$\sqrt{\frac{1}{252}}$	$- \sqrt{\frac{1}{28}}$	$\sqrt{\frac{5}{36}}$	$- \sqrt{\frac{25}{84}}$	$\sqrt{\frac{5}{14}}$	$- \sqrt{\frac{1}{6}}$

SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of $j_{1}$ and $j_{2}$ , or for the su(N) algebra instead of su(2), are known.^[6] A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.

References

↑ Baird, C.E.; L. C. Biedenharn (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_n". J. Math. Phys. 5 (12): 1723–1730. doi:10.1063/1.1704095. Bibcode: 1964JMP.....5.1723B.
↑ Hagiwara, K. (July 2002). "Review of Particle Properties". Phys. Rev. D 66 (1): 010001. doi:10.1103/PhysRevD.66.010001. Bibcode: 2002PhRvD..66a0001H. http://pdg.lbl.gov/2002/clebrpp.pdf. Retrieved 2007-12-20.
↑ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). http://www.mpia.de/~mathar/progs/CGord. Retrieved 2012-10-15.
↑ (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, ISBN:0-387-95330-2.
↑ Weissbluth, Mitchel (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. https://archive.org/details/atomsmolecules0000weis. Table 1.4 resumes the most common.
↑ Alex, A.; M. Kalus; A. Huckleberry; J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82: 023507. doi:10.1063/1.3521562. Bibcode: 2011JMP....52b3507A.

External links

Online, Java-based Clebsch–Gordan Coefficient Calculator by Paul Stevenson
Other formulae for Clebsch–Gordan coefficients.
Web interface for tabulating SU(N) Clebsch–Gordan coefficients

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[1] Baird, C.E.; L. C. Biedenharn (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_n". J. Math. Phys. 5 (12): 1723–1730. doi:10.1063/1.1704095. Bibcode: 1964JMP.....5.1723B.

[2] Hagiwara, K. (July 2002). "Review of Particle Properties". Phys. Rev. D 66 (1): 010001. doi:10.1103/PhysRevD.66.010001. Bibcode: 2002PhRvD..66a0001H. http://pdg.lbl.gov/2002/clebrpp.pdf. Retrieved 2007-12-20.

[3] Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). http://www.mpia.de/~mathar/progs/CGord. Retrieved 2012-10-15.

[4] (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, ISBN:0-387-95330-2.

[5] Weissbluth, Mitchel (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. https://archive.org/details/atomsmolecules0000weis. Table 1.4 resumes the most common.

[6] Alex, A.; M. Kalus; A. Huckleberry; J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82: 023507. doi:10.1063/1.3521562. Bibcode: 2011JMP....52b3507A.

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Formulation

Specific values

$j 2 = 0$

$j 1 = 1 / 2, j 2 = 1 / 2$

$j 1 = 1, j 2 = 1 / 2$

$j 1 = 1, j 2 = 1$

$j 1 = 3 / 2, j 2 = 1 / 2$

$j 1 = 3 / 2, j 2 = 1$

$j 1 = 3 / 2, j 2 = 3 / 2$

$j 1 = 2, j 2 = 1 / 2$

$j 1 = 2, j 2 = 1$

$j 1 = 2, j 2 = 3 / 2$

$j 1 = 2, j 2 = 2$

$j 1 = 5 / 2, j 2 = 1 / 2$

$j 1 = 5 / 2, j 2 = 1$

$j 1 = 5 / 2, j 2 = 3 / 2$

$j 1 = 5 / 2, j 2 = 2$

$j 1 = 5 / 2, j 2 = 5 / 2$

SU(N) Clebsch–Gordan coefficients

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Table of Clebsch–Gordan coefficients

Formulation

Specific values

j2 = 0

j1 = 1/2, j2 = 1/2

j1 = 1, j2 = 1/2

j1 = 1, j2 = 1

j1 = 3/2, j2 = 1/2

j1 = 3/2, j2 = 1

j1 = 3/2, j2 = 3/2

j1 = 2, j2 = 1/2

j1 = 2, j2 = 1

j1 = 2, j2 = 3/2

j1 = 2, j2 = 2

j1 = 5/2, j2 = 1/2

j1 = 5/2, j2 = 1

j1 = 5/2, j2 = 3/2

j1 = 5/2, j2 = 2

j1 = 5/2, j2 = 5/2

SU(N) Clebsch–Gordan coefficients

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$j 2 = 0$

$j 1 = 1 / 2, j 2 = 1 / 2$

$j 1 = 1, j 2 = 1 / 2$

$j 1 = 1, j 2 = 1$

$j 1 = 3 / 2, j 2 = 1 / 2$

$j 1 = 3 / 2, j 2 = 1$

$j 1 = 3 / 2, j 2 = 3 / 2$

$j 1 = 2, j 2 = 1 / 2$

$j 1 = 2, j 2 = 1$

$j 1 = 2, j 2 = 3 / 2$

$j 1 = 2, j 2 = 2$

$j 1 = 5 / 2, j 2 = 1 / 2$

$j 1 = 5 / 2, j 2 = 1$

$j 1 = 5 / 2, j 2 = 3 / 2$

$j 1 = 5 / 2, j 2 = 2$

$j 1 = 5 / 2, j 2 = 5 / 2$