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 j₁ < j₂ 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]

 j₂=0

When j₂ = 0, the Clebsch–Gordan coefficients are given by $\delta_{j,j_1}\delta_{m,m_1}$ .

 j₁=1/2, j₂=1/2

m=1	j
m1, m2	1 1/2, 1/2 $1\!\,$

m=-1	j
m1, m2	1 -1/2, -1/2 $1\!\,$

m=0	j
m1, m2	1 0 1/2, -1/2 $\sqrt{\frac{1}{2}}\!\,$ $\sqrt{\frac{1}{2}}\!\,$ -1/2, 1/2 $\sqrt{\frac{1}{2}}\!\,$ $-\sqrt{\frac{1}{2}}\!\,$

 j₁=1, j₂=1/2

m=3/2	j
m1, m2	3/2 1, 1/2 $1\!\,$

m=1/2	j
m1, m2	3/2 1/2 1, -1/2 $\sqrt{\frac{1}{3}}\!\,$ $\sqrt{\frac{2}{3}}\!\,$ 0, 1/2 $\sqrt{\frac{2}{3}}\!\,$ $-\sqrt{\frac{1}{3}}\!\,$

 j₁=1, j₂=1

m=2	j
m1, m2	2 1, 1 $1\!\,$

m=1	j
m1, m2	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
m1, m2	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 $\sqrt{\frac{1}{6}}\!\,$ $-\sqrt{\frac{1}{2}}\!\,$ $\sqrt{\frac{1}{3}}\!\,$

 j₁=3/2, j₂=1/2

m=2	j
m1, m2	2 3/2, 1/2 $1\!\,$

m=1	j
m1, m2	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
m1, m2	2 1 1/2, -1/2 $\sqrt{\frac{1}{2}}\!\,$ $\sqrt{\frac{1}{2}}\!\,$ -1/2, 1/2 $\sqrt{\frac{1}{2}}\!\,$ $-\sqrt{\frac{1}{2}}\!\,$

 j₁=3/2, j₂=1

m=5/2	j
m1, m2	5/2 3/2, 1 $1\!\,$

m=3/2	j
m1, m2	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
m1, m2	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 $\sqrt{\frac{3}{10}}\!\,$ $-\sqrt{\frac{8}{15}}\!\,$ $\sqrt{\frac{1}{6}}\!\,$

 j₁=3/2, j₂=3/2

m=3	j
m1, m2	3 3/2, 3/2 $1\!\,$

m=2	j
m1, m2	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
m1, m2	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

m1, m2

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 $\sqrt{\frac{1}{20}}\!\,$ $-\frac{1}{2}\!\,$ $\sqrt{\frac{9}{20}}\!\,$ $-\frac{1}{2}\!\,$

 j₁=2, j₂=1/2

m=5/2	j
m1, m2	5/2 2, 1/2 $1\!\,$

m=3/2	j
m1, m2	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
m1, m2	5/2 3/2 1, -1/2 $\sqrt{\frac{2}{5}}\!\,$ $\sqrt{\frac{3}{5}}\!\,$ 0, 1/2 $\sqrt{\frac{3}{5}}\!\,$ $-\sqrt{\frac{2}{5}}\!\,$

 j₁=2, j₂=1

m=3	j
m1, m2	3 2, 1 $1\!\,$

m=2	j
m1, m2	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
m1, m2	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
m1, m2	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 $\sqrt{\frac{1}{5}}\!\,$ $-\sqrt{\frac{1}{2}}\!\,$ $\sqrt{\frac{3}{10}}\!\,$

 j₁=2, j₂=3/2

m=7/2	j
m1, m2	7/2 2, 3/2 $1\!\,$

m=5/2	j
m1, m2	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
m1, m2	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

m1, m2

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 $\sqrt{\frac{4}{35}}\!\,$ $-\sqrt{\frac{27}{70}}\!\,$ $\sqrt{\frac{2}{5}}\!\,$ $-\sqrt{\frac{1}{10}}\!\,$

 j₁=2, j₂=2

m=4	j
m1, m2	4 2, 2 $1\!\,$

m=3	j
m1, m2	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
m1, m2	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

m1, m2

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

m1, m2

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 $\sqrt{\frac{1}{70}}\!\,$ $-\sqrt{\frac{1}{10}}\!\,$ $\sqrt{\frac{2}{7}}\!\,$ $-\sqrt{\frac{2}{5}}\!\,$ $\sqrt{\frac{1}{5}}\!\,$

 j₁=5/2, j₂=1/2

m=3	j
m1, m2	3 5/2, 1/2 $1\!\,$

m=2	j
m1, m2	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
m1, m2	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
m1, m2	3 2 1/2, -1/2 $\sqrt{\frac{1}{2}}\!\,$ $\sqrt{\frac{1}{2}}\!\,$ -1/2, 1/2 $\sqrt{\frac{1}{2}}\!\,$ $-\sqrt{\frac{1}{2}}\!\,$

 j₁=5/2, j₂=1

m=7/2	j
m1, m2	7/2 5/2, 1 $1\!\,$

m=5/2	j
m1, m2	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
m1, m2	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
m1, m2	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 $\sqrt{\frac{2}{7}}\!\,$ $-\sqrt{\frac{18}{35}}\!\,$ $\sqrt{\frac{1}{5}}\!\,$

 j₁=5/2, j₂=3/2

m=4	j
m1, m2	4 5/2, 3/2 $1\!\,$

m=3	j
m1, m2	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
m1, m2	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

m1, m2

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

m1, m2

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 $\sqrt{\frac{1}{14}}\!\,$ $-\sqrt{\frac{3}{10}}\!\,$ $\sqrt{\frac{3}{7}}\!\,$ $-\sqrt{\frac{1}{5}}\!\,$

 j₁=5/2, j₂=2

m=9/2	j
m1, m2	9/2 5/2, 2 $1\!\,$

m=7/2	j
m1, m2	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
m1, m2	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

m1, m2

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

m1, m2

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

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: 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095. Retrieved 2007-12-20.
2. 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.
3. Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). 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. Weisbluth, Michael (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. 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. 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