Table of Clebsch–Gordan coefficients

Template:SHORTDESC: 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 $k$ 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 $δ_{j, j_{1}} δ_{m, m_{1}}$ .

$j1 = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/2⁠, j2 = ⁠1/2⁠$

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

^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)..
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Table 1.4 resumes the most common.
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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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[3] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[4] (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)..

[5] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Table 1.4 resumes the most common.

[6] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[1]

[2]

[3]

[4]

[5]

[6]

Table of Clebsch–Gordan coefficients

Contents

Formulation

Specific values

$j 2 = 0$

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

$j 1 = 1, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = 2, j 2 = 1$

$j 1 = 2, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 5 / 2 ⁠$

SU(N) Clebsch–Gordan coefficients

References

External links

Navigation menu

Table of Clebsch–Gordan coefficients

Formulation

Specific values

j2 = 0

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

References

External links

Navigation menu

Search

$j 2 = 0$

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

$j 1 = 1, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = 2, j 2 = 1$

$j 1 = 2, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 5 / 2 ⁠$