Table of Clebsch–Gordan coefficients

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_1j_2)jm\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} |j_1m_1j_2m_2\rangle \langle j_1j_2;m_1m_2|j_1j_2;jm\rangle

Explicitly:

\begin{align} \langle j_1j_2;m_1m_2|j_1j_2;jm\rangle = \ &\delta_{m,m_1+m_2} \sqrt{\frac{(2j+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{align}

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

\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2}\langle j_1j_2;-m_1,-m_2|j_1j_2;j,-m\rangle

and

\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2} \langle j_2j_1;m_2m_1|j_2j_1;jm\rangle

A complete list ^[5]

$j 2 =0$

When $j 2 = 0$ , the Clebsch–Gordan coefficients are given by $\delta_{j,j_1}\delta_{m,m_1}$ .

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

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	$\sqrt{\frac{5}{126}}\!\,$	$-\sqrt{\frac{64}{315}}\!\,$	$\sqrt{\frac{27}{70}}\!\,$	$-\sqrt{\frac{32}{105}}\!\,$	$\sqrt{\frac{1}{15}}\!\,$

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: 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095. Retrieved 2007-12-20.
↑ Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF). Phys. Rev. D. 66: 010001. Bibcode:2002PhRvD..66a0001H. doi:10.1103/PhysRevD.66.010001. Retrieved 2007-12-20.
↑ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). 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.
↑ Weisbluth, Michael (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. 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. arXiv:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13.

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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Table of Clebsch–Gordan coefficients

Formulation

A complete list [5]

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