Table of Clebsch–Gordan coefficients

Template:Short description 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{matrix} ⟨ 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{matrix}

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

For brevity, solutions with Template:Math and Template:Math 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]

Template:Diagonal split header	1	0
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
1, −Template:Sfrac	$\sqrt{\frac{1}{3}}$	$\sqrt{\frac{2}{3}}$
0, Template:Sfrac	$\sqrt{\frac{2}{3}}$	$- \sqrt{\frac{1}{3}}$

Template:Diagonal split header	2	1
1, 0	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
0, 1	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

Template:Diagonal split header	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	$\sqrt{\frac{1}{6}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{3}}$

Template:Diagonal split header	2	1
Template:Sfrac, −Template:Sfrac	$\frac{1}{2}$	$\sqrt{\frac{3}{4}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{3}{4}}$	$- \frac{1}{2}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
Template:Sfrac, 0	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{3}{5}}$
Template:Sfrac, 1	$\sqrt{\frac{3}{5}}$	$- \sqrt{\frac{2}{5}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac
Template:Sfrac, −1	$\sqrt{\frac{1}{10}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{2}}$
Template:Sfrac, 0	$\sqrt{\frac{3}{5}}$	$\sqrt{\frac{1}{15}}$	$- \sqrt{\frac{1}{3}}$
−Template:Sfrac, 1	$\sqrt{\frac{3}{10}}$	$- \sqrt{\frac{8}{15}}$	$\sqrt{\frac{1}{6}}$

Template:Diagonal split header	3	2	1
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{5}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{3}{5}}$	$0$	$- \sqrt{\frac{2}{5}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{1}{5}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{3}{10}}$

Template:Diagonal split header	3	2	1	0
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{20}}$	$\frac{1}{2}$	$\sqrt{\frac{9}{20}}$	$\frac{1}{2}$
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{9}{20}}$	$\frac{1}{2}$	$- \sqrt{\frac{1}{20}}$	$- \frac{1}{2}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{9}{20}}$	$- \frac{1}{2}$	$- \sqrt{\frac{1}{20}}$	$\frac{1}{2}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{1}{20}}$	$- \frac{1}{2}$	$\sqrt{\frac{9}{20}}$	$- \frac{1}{2}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
2, −Template:Sfrac	$\sqrt{\frac{1}{5}}$	$\sqrt{\frac{4}{5}}$
1, Template:Sfrac	$\sqrt{\frac{4}{5}}$	$- \sqrt{\frac{1}{5}}$

Template:Math
Template:Diagonal split header	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	$\sqrt{\frac{3}{14}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{2}{7}}$

Template:Diagonal split header	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	$\sqrt{\frac{2}{5}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{10}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
2, Template:Sfrac	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{4}{7}}$
1, Template:Sfrac	$\sqrt{\frac{4}{7}}$	$- \sqrt{\frac{3}{7}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac
2, −Template:Sfrac	$\sqrt{\frac{1}{7}}$	$\sqrt{\frac{16}{35}}$	$\sqrt{\frac{2}{5}}$
1, Template:Sfrac	$\sqrt{\frac{4}{7}}$	$\sqrt{\frac{1}{35}}$	$- \sqrt{\frac{2}{5}}$
0, Template:Sfrac	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{18}{35}}$	$\sqrt{\frac{1}{5}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac	Template:Sfrac
2, −Template:Sfrac	$\sqrt{\frac{1}{35}}$	$\sqrt{\frac{6}{35}}$	$\sqrt{\frac{2}{5}}$	$\sqrt{\frac{2}{5}}$
1, −Template:Sfrac	$\sqrt{\frac{12}{35}}$	$\sqrt{\frac{5}{14}}$	$0$	$- \sqrt{\frac{3}{10}}$
0, Template:Sfrac	$\sqrt{\frac{18}{35}}$	$- \sqrt{\frac{3}{35}}$	$- \sqrt{\frac{1}{5}}$	$\sqrt{\frac{1}{5}}$
−1, Template:Sfrac	$\sqrt{\frac{4}{35}}$	$- \sqrt{\frac{27}{70}}$	$\sqrt{\frac{2}{5}}$	$- \sqrt{\frac{1}{10}}$

Template:Diagonal split header	4	3
2, 1	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{2}}$
1, 2	$\sqrt{\frac{1}{2}}$	$- \sqrt{\frac{1}{2}}$

Template:Diagonal split header	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	$\sqrt{\frac{1}{14}}$	$- \sqrt{\frac{3}{10}}$	$\sqrt{\frac{3}{7}}$	$- \sqrt{\frac{1}{5}}$

Template:Diagonal split header	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	$\sqrt{\frac{1}{70}}$	$- \sqrt{\frac{1}{10}}$	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{2}{5}}$	$\sqrt{\frac{1}{5}}$

Template:Diagonal split header	3	2
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{6}}$	$\sqrt{\frac{5}{6}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{5}{6}}$	$- \sqrt{\frac{1}{6}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
Template:Sfrac, 0	$\sqrt{\frac{2}{7}}$	$\sqrt{\frac{5}{7}}$
Template:Sfrac, 1	$\sqrt{\frac{5}{7}}$	$- \sqrt{\frac{2}{7}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac
Template:Sfrac, −1	$\sqrt{\frac{1}{21}}$	$\sqrt{\frac{2}{7}}$	$\sqrt{\frac{2}{3}}$
Template:Sfrac, 0	$\sqrt{\frac{10}{21}}$	$\sqrt{\frac{9}{35}}$	$- \sqrt{\frac{4}{15}}$
Template:Sfrac, 1	$\sqrt{\frac{10}{21}}$	$- \sqrt{\frac{16}{35}}$	$\sqrt{\frac{1}{15}}$

Table of Clebsch–Gordan coefficients

Formulation

Specific values

SU(N) Clebsch–Gordan coefficients

References

External links

Navigation menu

Search

Template:Diagonal split header	4	3
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{3}{8}}$	$\sqrt{\frac{5}{8}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{5}{8}}$	$- \sqrt{\frac{3}{8}}$

Template:Diagonal split header	4	3	2
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{3}{28}}$	$\sqrt{\frac{5}{12}}$	$\sqrt{\frac{10}{21}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{15}{28}}$	$\sqrt{\frac{1}{12}}$	$- \sqrt{\frac{8}{21}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{5}{14}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{1}{7}}$

Template:Diagonal split header	4	3	2	1
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{56}}$	$\sqrt{\frac{1}{8}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{2}}$
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{15}{56}}$	$\sqrt{\frac{49}{120}}$	$\sqrt{\frac{1}{42}}$	$- \sqrt{\frac{3}{10}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{15}{28}}$	$- \sqrt{\frac{1}{60}}$	$- \sqrt{\frac{25}{84}}$	$\sqrt{\frac{3}{20}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{5}{28}}$	$- \sqrt{\frac{9}{20}}$	$\sqrt{\frac{9}{28}}$	$- \sqrt{\frac{1}{20}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac
Template:Sfrac, 1	$\frac{2}{3}$	$\sqrt{\frac{5}{9}}$
Template:Sfrac, 2	$\sqrt{\frac{5}{9}}$	$- \frac{2}{3}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac	Template:Sfrac
Template:Sfrac, −1	$\sqrt{\frac{1}{21}}$	$\sqrt{\frac{5}{21}}$	$\sqrt{\frac{3}{7}}$	$\sqrt{\frac{2}{7}}$
Template:Sfrac, 0	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{2}{7}}$	$- \sqrt{\frac{1}{70}}$	$- \sqrt{\frac{12}{35}}$
Template:Sfrac, 1	$\sqrt{\frac{10}{21}}$	$- \sqrt{\frac{2}{21}}$	$- \sqrt{\frac{6}{35}}$	$\sqrt{\frac{9}{35}}$
−Template:Sfrac, 2	$\sqrt{\frac{5}{42}}$	$- \sqrt{\frac{8}{21}}$	$\sqrt{\frac{27}{70}}$	$- \sqrt{\frac{4}{35}}$

Template:Diagonal split header	Template:Sfrac	Template:Sfrac	Template:Sfrac	Template:Sfrac	Template:Sfrac
Template:Sfrac, −2	$\sqrt{\frac{1}{126}}$	$\sqrt{\frac{4}{63}}$	$\sqrt{\frac{3}{14}}$	$\sqrt{\frac{8}{21}}$	$\sqrt{\frac{1}{3}}$
Template:Sfrac, −1	$\sqrt{\frac{10}{63}}$	$\sqrt{\frac{121}{315}}$	$\sqrt{\frac{6}{35}}$	$- \sqrt{\frac{2}{105}}$	$- \sqrt{\frac{4}{15}}$
Template:Sfrac, 0	$\sqrt{\frac{10}{21}}$	$\sqrt{\frac{4}{105}}$	$- \sqrt{\frac{8}{35}}$	$- \sqrt{\frac{2}{35}}$	$\sqrt{\frac{1}{5}}$
−Template:Sfrac, 1	$\sqrt{\frac{20}{63}}$	$- \sqrt{\frac{14}{45}}$	$0$	$\sqrt{\frac{5}{21}}$	$- \sqrt{\frac{2}{15}}$
−Template:Sfrac, 2	$\sqrt{\frac{5}{126}}$	$- \sqrt{\frac{64}{315}}$	$\sqrt{\frac{27}{70}}$	$- \sqrt{\frac{32}{105}}$	$\sqrt{\frac{1}{15}}$

Template:Diagonal split header	5	4	3
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{2}{9}}$	$\sqrt{\frac{1}{2}}$	$\sqrt{\frac{5}{18}}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{5}{9}}$	$0$	$- \frac{2}{3}$
Template:Sfrac, Template:Sfrac	$\sqrt{\frac{2}{9}}$	$- \sqrt{\frac{1}{2}}$	$\sqrt{\frac{5}{18}}$

Template:Diagonal split header	5	4	3	2	1	0
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{1}{252}}$	$\sqrt{\frac{1}{28}}$	$\sqrt{\frac{5}{36}}$	$\sqrt{\frac{25}{84}}$	$\sqrt{\frac{5}{14}}$	$\sqrt{\frac{1}{6}}$
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{25}{252}}$	$\sqrt{\frac{9}{28}}$	$\sqrt{\frac{49}{180}}$	$\sqrt{\frac{1}{84}}$	$- \sqrt{\frac{9}{70}}$	$- \sqrt{\frac{1}{6}}$
Template:Sfrac, −Template:Sfrac	$\sqrt{\frac{25}{63}}$	$\sqrt{\frac{1}{7}}$	$- \sqrt{\frac{4}{45}}$	$- \sqrt{\frac{4}{21}}$	$\sqrt{\frac{1}{70}}$	$\sqrt{\frac{1}{6}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{25}{63}}$	$- \sqrt{\frac{1}{7}}$	$- \sqrt{\frac{4}{45}}$	$\sqrt{\frac{4}{21}}$	$\sqrt{\frac{1}{70}}$	$- \sqrt{\frac{1}{6}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{25}{252}}$	$- \sqrt{\frac{9}{28}}$	$\sqrt{\frac{49}{180}}$	$- \sqrt{\frac{1}{84}}$	$- \sqrt{\frac{9}{70}}$	$\sqrt{\frac{1}{6}}$
−Template:Sfrac, Template:Sfrac	$\sqrt{\frac{1}{252}}$	$- \sqrt{\frac{1}{28}}$	$\sqrt{\frac{5}{36}}$	$- \sqrt{\frac{25}{84}}$	$\sqrt{\frac{5}{14}}$	$- \sqrt{\frac{1}{6}}$