Таблица коэффициентов Клебша – Гордана - Table of Clebsch–Gordan coefficients - Wikipedia

Это таблица Коэффициенты Клебша – Гордана используется для добавления угловой момент ценности в квантовая механика. Общий знак коэффициентов для каждого набора констант ${\displaystyle j_{1}}$, ${\displaystyle j_{2}}$, ${\displaystyle j}$ является до некоторой степени произвольным и фиксируется в соответствии с соглашением о знаках Кондона – Шортли и Вигнера, как обсуждалось Бэрдом и Биденхарн.^[1] Таблицы с одинаковым условным обозначением можно найти в Группа данных о частицах с Обзор свойств частиц^[2] и в онлайн-таблицах.^[3]

Формулировка

Коэффициенты Клебша – Гордана являются решениями

${\displaystyle |j_{1},j_{2};J,M\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1},m_{1};j_{2},m_{2}\rangle \langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle }$

Ясно:

${\displaystyle {\begin{aligned}&\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle \\[6pt]={}&\delta _{M,m_{1}+m_{2}}{\sqrt {\frac {(2J+1)(J+j_{1}-j_{2})!(J-j_{1}+j_{2})!(j_{1}+j_{2}-J)!}{(j_{1}+j_{2}+J+1)!}}}\ \times {}\\[6pt]&{\sqrt {(J+M)!(J-M)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times {}\\[6pt]&\sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-J-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(J-j_{2}+m_{1}+k)!(J-j_{1}-m_{2}+k)!}}.\end{aligned}}}$

Суммирование распространяется по всем целым k для которого аргумент каждого факториала неотрицателен.^[4]

Для краткости решения с M < 0 и j₁ < j₂ опущены. Их можно вычислить с помощью простых соотношений

${\displaystyle \langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{1},j_{2};-m_{1},-m_{2}\mid j_{1},j_{2};J,-M\rangle .}$

и

${\displaystyle \langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{2},j_{1};m_{2},m_{1}\mid j_{2},j_{1};J,M\rangle .}$

Конкретные значения

Коэффициенты Клебша – Гордана для j значения меньше или равные 5/2 приведены ниже.^[5]

 j₂ = 0

Когда j₂ = 0, коэффициенты Клебша – Гордана имеют вид ${\displaystyle \delta _{j,j_{1}}\delta _{m,m_{1}}}$.

 j₁ = 1/2,  j₂ = 1/2

м = 1

j м1, м2	1
1/2, 1/2	${\displaystyle 1}$

м = −1

j м1, м2	1
−1/2, −1/2	${\displaystyle 1}$

м = 0

j м1, м2	1	0
1/2, −1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

 j₁ = 1,  j₂ = 1/2

м = 3/2

j м1, м2	3/2
1, 1/2	${\displaystyle 1}$

м = 1/2

j м1, м2	3/2	1/2
1, −1/2	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle {\sqrt {\frac {2}{3}}}}$
0, 1/2	${\displaystyle {\sqrt {\frac {2}{3}}}}$	${\displaystyle -{\sqrt {\frac {1}{3}}}}$

 j₁ = 1,  j₂ = 1

м = 2

j м1, м2	2
1, 1	${\displaystyle 1}$

м = 1

j м1, м2	2	1
1, 0	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
0, 1	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

м = 0

j м1, м2	2	1	0
1, −1	${\displaystyle {\sqrt {\frac {1}{6}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$
0, 0	${\displaystyle {\sqrt {\frac {2}{3}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {1}{3}}}}$
−1, 1	${\displaystyle {\sqrt {\frac {1}{6}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$

 j₁ = 3/2,  j₂ = 1/2

м = 2

j м1, м2	2
3/2, 1/2	${\displaystyle 1}$

м = 1

j м1, м2	2	1
3/2, −1/2	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\sqrt {\frac {3}{4}}}}$
1/2, 1/2	${\displaystyle {\sqrt {\frac {3}{4}}}}$	${\displaystyle -{\frac {1}{2}}}$

м = 0

j м1, м2	2	1
1/2, −1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

 j₁ = 3/2,  j₂ = 1

м = 5/2

j м1, м2	5/2
3/2, 1	${\displaystyle 1}$

м = 3/2

j м1, м2	5/2	3/2
3/2, 0	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {3}{5}}}}$
1/2, 1	${\displaystyle {\sqrt {\frac {3}{5}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$

м = 1/2

j м1, м2	5/2	3/2	1/2
3/2, −1	${\displaystyle {\sqrt {\frac {1}{10}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
1/2, 0	${\displaystyle {\sqrt {\frac {3}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{15}}}}$	${\displaystyle -{\sqrt {\frac {1}{3}}}}$
−1/2, 1	${\displaystyle {\sqrt {\frac {3}{10}}}}$	${\displaystyle -{\sqrt {\frac {8}{15}}}}$	${\displaystyle {\sqrt {\frac {1}{6}}}}$

 j₁ = 3/2,  j₂ = 3/2

м = 3

j м1, м2	3
3/2, 3/2	${\displaystyle 1}$

м = 2

j м1, м2	3	2
3/2, 1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
1/2, 3/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

м = 1

j м1, м2	3	2	1
3/2, −1/2	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$
1/2, 1/2	${\displaystyle {\sqrt {\frac {3}{5}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$
−1/2, 3/2	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$

м = 0

j м1, м2	3	2	1	0
3/2, −3/2	${\displaystyle {\sqrt {\frac {1}{20}}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\sqrt {\frac {9}{20}}}}$	${\displaystyle {\frac {1}{2}}}$
1/2, −1/2	${\displaystyle {\sqrt {\frac {9}{20}}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle -{\sqrt {\frac {1}{20}}}}$	${\displaystyle -{\frac {1}{2}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {9}{20}}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle -{\sqrt {\frac {1}{20}}}}$	${\displaystyle {\frac {1}{2}}}$
−3/2, 3/2	${\displaystyle {\sqrt {\frac {1}{20}}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\sqrt {\frac {9}{20}}}}$	${\displaystyle -{\frac {1}{2}}}$

 j₁ = 2,  j₂ = 1/2

м = 5/2

j м1, м2	5/2
2, 1/2	${\displaystyle 1}$

м = 3/2

j м1, м2	5/2	3/2
2, −1/2	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle {\sqrt {\frac {4}{5}}}}$
1, 1/2	${\displaystyle {\sqrt {\frac {4}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$

м = 1/2

j м1, м2	5/2	3/2
1, −1/2	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {3}{5}}}}$
0, 1/2	${\displaystyle {\sqrt {\frac {3}{5}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$

 j₁ = 2,  j₂ = 1

м = 3

j м1, м2	3
2, 1	${\displaystyle 1}$

м = 2

j м1, м2	3	2
2, 0	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle {\sqrt {\frac {2}{3}}}}$
1, 1	${\displaystyle {\sqrt {\frac {2}{3}}}}$	${\displaystyle -{\sqrt {\frac {1}{3}}}}$

м = 1

j м1, м2	3	2	1
2, −1	${\displaystyle {\sqrt {\frac {1}{15}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle {\sqrt {\frac {3}{5}}}}$
1, 0	${\displaystyle {\sqrt {\frac {8}{15}}}}$	${\displaystyle {\sqrt {\frac {1}{6}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$
0, 1	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{10}}}}$

м = 0

j м1, м2	3	2	1
1, −1	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$
0, 0	${\displaystyle {\sqrt {\frac {3}{5}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$
−1, 1	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$

 j₁ = 2,  j₂ = 3/2

м = 7/2

j м1, м2	7/2
2, 3/2	${\displaystyle 1}$

м = 5/2

j м1, м2	7/2	5/2
2, 1/2	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {4}{7}}}}$
1, 3/2	${\displaystyle {\sqrt {\frac {4}{7}}}}$	${\displaystyle -{\sqrt {\frac {3}{7}}}}$

м = 3/2

j м1, м2	7/2	5/2	3/2
2, −1/2	${\displaystyle {\sqrt {\frac {1}{7}}}}$	${\displaystyle {\sqrt {\frac {16}{35}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$
1, 1/2	${\displaystyle {\sqrt {\frac {4}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{35}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$
0, 3/2	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle -{\sqrt {\frac {18}{35}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$

м = 1/2

j м1, м2	7/2	5/2	3/2	1/2
2, −3/2	${\displaystyle {\sqrt {\frac {1}{35}}}}$	${\displaystyle {\sqrt {\frac {6}{35}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$
1, −1/2	${\displaystyle {\sqrt {\frac {12}{35}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$
0, 1/2	${\displaystyle {\sqrt {\frac {18}{35}}}}$	${\displaystyle -{\sqrt {\frac {3}{35}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
−1, 3/2	${\displaystyle {\sqrt {\frac {4}{35}}}}$	${\displaystyle -{\sqrt {\frac {27}{70}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{10}}}}$

 j₁ = 2,  j₂ = 2

м = 4

j м1, м2	4
2, 2	${\displaystyle 1}$

м = 3

j м1, м2	4	3
2, 1	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
1, 2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

м = 2

j м1, м2	4	3	2
2, 0	${\displaystyle {\sqrt {\frac {3}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$
1, 1	${\displaystyle {\sqrt {\frac {4}{7}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {3}{7}}}}$
0, 2	${\displaystyle {\sqrt {\frac {3}{14}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$

м = 1

j м1, м2	4	3	2	1
2, −1	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
1, 0	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{14}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$
0, 1	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{14}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$
−1, 2	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$

м = 0

j м1, м2	4	3	2	1	0
2, −2	${\displaystyle {\sqrt {\frac {1}{70}}}}$	${\displaystyle {\sqrt {\frac {1}{10}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
1, −1	${\displaystyle {\sqrt {\frac {8}{35}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle -{\sqrt {\frac {1}{10}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$
0, 0	${\displaystyle {\sqrt {\frac {18}{35}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {2}{7}}}}$	${\displaystyle 0}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
−1, 1	${\displaystyle {\sqrt {\frac {8}{35}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{10}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$
−2, 2	${\displaystyle {\sqrt {\frac {1}{70}}}}$	${\displaystyle -{\sqrt {\frac {1}{10}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$

 j₁ = 5/2,  j₂ = 1/2

м = 3

j м1, м2	3
5/2, 1/2	${\displaystyle 1}$

м = 2

j м1, м2	3	2
5/2, −1/2	${\displaystyle {\sqrt {\frac {1}{6}}}}$	${\displaystyle {\sqrt {\frac {5}{6}}}}$
3/2, 1/2	${\displaystyle {\sqrt {\frac {5}{6}}}}$	${\displaystyle -{\sqrt {\frac {1}{6}}}}$

м = 1

j м1, м2	3	2
3/2, −1/2	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle {\sqrt {\frac {2}{3}}}}$
1/2, 1/2	${\displaystyle {\sqrt {\frac {2}{3}}}}$	${\displaystyle -{\sqrt {\frac {1}{3}}}}$

м = 0

j м1, м2	3	2
1/2, −1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

 j₁ = 5/2,  j₂ = 1

м = 7/2

j м1, м2	7/2
5/2, 1	${\displaystyle 1}$

м = 5/2

j м1, м2	7/2	5/2
5/2, 0	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle {\sqrt {\frac {5}{7}}}}$
3/2, 1	${\displaystyle {\sqrt {\frac {5}{7}}}}$	${\displaystyle -{\sqrt {\frac {2}{7}}}}$

м = 3/2

j м1, м2	7/2	5/2	3/2
5/2, −1	${\displaystyle {\sqrt {\frac {1}{21}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle {\sqrt {\frac {2}{3}}}}$
3/2, 0	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle {\sqrt {\frac {9}{35}}}}$	${\displaystyle -{\sqrt {\frac {4}{15}}}}$
1/2, 1	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle -{\sqrt {\frac {16}{35}}}}$	${\displaystyle {\sqrt {\frac {1}{15}}}}$

м = 1/2

j м1, м2	7/2	5/2	3/2
3/2, −1	${\displaystyle {\sqrt {\frac {1}{7}}}}$	${\displaystyle {\sqrt {\frac {16}{35}}}}$	${\displaystyle {\sqrt {\frac {2}{5}}}}$
1/2, 0	${\displaystyle {\sqrt {\frac {4}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{35}}}}$	${\displaystyle -{\sqrt {\frac {2}{5}}}}$
−1/2, 1	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle -{\sqrt {\frac {18}{35}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$

 j₁ = 5/2,  j₂ = 3/2

м = 4

j м1, м2	4
5/2, 3/2	${\displaystyle 1}$

м = 3

j м1, м2	4	3
5/2, 1/2	${\displaystyle {\sqrt {\frac {3}{8}}}}$	${\displaystyle {\sqrt {\frac {5}{8}}}}$
3/2, 3/2	${\displaystyle {\sqrt {\frac {5}{8}}}}$	${\displaystyle -{\sqrt {\frac {3}{8}}}}$

м = 2

j м1, м2	4	3	2
5/2, −1/2	${\displaystyle {\sqrt {\frac {3}{28}}}}$	${\displaystyle {\sqrt {\frac {5}{12}}}}$	${\displaystyle {\sqrt {\frac {10}{21}}}}$
3/2, 1/2	${\displaystyle {\sqrt {\frac {15}{28}}}}$	${\displaystyle {\sqrt {\frac {1}{12}}}}$	${\displaystyle -{\sqrt {\frac {8}{21}}}}$
1/2, 3/2	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$

м = 1

j м1, м2	4	3	2	1
5/2, −3/2	${\displaystyle {\sqrt {\frac {1}{56}}}}$	${\displaystyle {\sqrt {\frac {1}{8}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
3/2, −1/2	${\displaystyle {\sqrt {\frac {15}{56}}}}$	${\displaystyle {\sqrt {\frac {49}{120}}}}$	${\displaystyle {\sqrt {\frac {1}{42}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$
1/2, 1/2	${\displaystyle {\sqrt {\frac {15}{28}}}}$	${\displaystyle -{\sqrt {\frac {1}{60}}}}$	${\displaystyle -{\sqrt {\frac {25}{84}}}}$	${\displaystyle {\sqrt {\frac {3}{20}}}}$
−1/2, 3/2	${\displaystyle {\sqrt {\frac {5}{28}}}}$	${\displaystyle -{\sqrt {\frac {9}{20}}}}$	${\displaystyle {\sqrt {\frac {9}{28}}}}$	${\displaystyle -{\sqrt {\frac {1}{20}}}}$

м = 0

j м1, м2	4	3	2	1
3/2, −3/2	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
1/2, −1/2	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{14}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$	${\displaystyle -{\sqrt {\frac {1}{14}}}}$	${\displaystyle {\sqrt {\frac {3}{10}}}}$
−3/2, 3/2	${\displaystyle {\sqrt {\frac {1}{14}}}}$	${\displaystyle -{\sqrt {\frac {3}{10}}}}$	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle -{\sqrt {\frac {1}{5}}}}$

 j₁ = 5/2,  j₂ = 2

м = 9/2

j м1, м2	9/2
5/2, 2	${\displaystyle 1}$

м = 7/2

j м1, м2	9/2	7/2
5/2, 1	${\displaystyle {\frac {2}{3}}}$	${\displaystyle {\sqrt {\frac {5}{9}}}}$
3/2, 2	${\displaystyle {\sqrt {\frac {5}{9}}}}$	${\displaystyle -{\frac {2}{3}}}$

м = 5/2

j м1, м2	9/2	7/2	5/2
5/2, 0	${\displaystyle {\sqrt {\frac {1}{6}}}}$	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$
3/2, 1	${\displaystyle {\sqrt {\frac {5}{9}}}}$	${\displaystyle {\sqrt {\frac {1}{63}}}}$	${\displaystyle -{\sqrt {\frac {3}{7}}}}$
1/2, 2	${\displaystyle {\sqrt {\frac {5}{18}}}}$	${\displaystyle -{\sqrt {\frac {32}{63}}}}$	${\displaystyle {\sqrt {\frac {3}{14}}}}$

м = 3/2

j м1, м2	9/2	7/2	5/2	3/2
5/2, −1	${\displaystyle {\sqrt {\frac {1}{21}}}}$	${\displaystyle {\sqrt {\frac {5}{21}}}}$	${\displaystyle {\sqrt {\frac {3}{7}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$
3/2, 0	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {2}{7}}}}$	${\displaystyle -{\sqrt {\frac {1}{70}}}}$	${\displaystyle -{\sqrt {\frac {12}{35}}}}$
1/2, 1	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle -{\sqrt {\frac {2}{21}}}}$	${\displaystyle -{\sqrt {\frac {6}{35}}}}$	${\displaystyle {\sqrt {\frac {9}{35}}}}$
−1/2, 2	${\displaystyle {\sqrt {\frac {5}{42}}}}$	${\displaystyle -{\sqrt {\frac {8}{21}}}}$	${\displaystyle {\sqrt {\frac {27}{70}}}}$	${\displaystyle -{\sqrt {\frac {4}{35}}}}$

м = 1/2

j м1, м2	9/2	7/2	5/2	3/2	1/2
5/2, −2	${\displaystyle {\sqrt {\frac {1}{126}}}}$	${\displaystyle {\sqrt {\frac {4}{63}}}}$	${\displaystyle {\sqrt {\frac {3}{14}}}}$	${\displaystyle {\sqrt {\frac {8}{21}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$
3/2, −1	${\displaystyle {\sqrt {\frac {10}{63}}}}$	${\displaystyle {\sqrt {\frac {121}{315}}}}$	${\displaystyle {\sqrt {\frac {6}{35}}}}$	${\displaystyle -{\sqrt {\frac {2}{105}}}}$	${\displaystyle -{\sqrt {\frac {4}{15}}}}$
1/2, 0	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle {\sqrt {\frac {4}{105}}}}$	${\displaystyle -{\sqrt {\frac {8}{35}}}}$	${\displaystyle -{\sqrt {\frac {2}{35}}}}$	${\displaystyle {\sqrt {\frac {1}{5}}}}$
−1/2, 1	${\displaystyle {\sqrt {\frac {20}{63}}}}$	${\displaystyle -{\sqrt {\frac {14}{45}}}}$	${\displaystyle 0}$	${\displaystyle {\sqrt {\frac {5}{21}}}}$	${\displaystyle -{\sqrt {\frac {2}{15}}}}$
−3/2, 2	${\displaystyle {\sqrt {\frac {5}{126}}}}$	${\displaystyle -{\sqrt {\frac {64}{315}}}}$	${\displaystyle {\sqrt {\frac {27}{70}}}}$	${\displaystyle -{\sqrt {\frac {32}{105}}}}$	${\displaystyle {\sqrt {\frac {1}{15}}}}$

 j₁ = 5/2,  j₂ = 5/2

м = 5

j м1, м2	5
5/2, 5/2	${\displaystyle 1}$

м = 4

j м1, м2	5	4
5/2, 3/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$
3/2, 5/2	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$

м = 3

j м1, м2	5	4	3
5/2, 1/2	${\displaystyle {\sqrt {\frac {2}{9}}}}$	${\displaystyle {\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {5}{18}}}}$
3/2, 3/2	${\displaystyle {\sqrt {\frac {5}{9}}}}$	${\displaystyle 0}$	${\displaystyle -{\frac {2}{3}}}$
1/2, 5/2	${\displaystyle {\sqrt {\frac {2}{9}}}}$	${\displaystyle -{\sqrt {\frac {1}{2}}}}$	${\displaystyle {\sqrt {\frac {5}{18}}}}$

м = 2

j м1, м2	5	4	3	2
5/2, −1/2	${\displaystyle {\sqrt {\frac {1}{12}}}}$	${\displaystyle {\sqrt {\frac {9}{28}}}}$	${\displaystyle {\sqrt {\frac {5}{12}}}}$	${\displaystyle {\sqrt {\frac {5}{28}}}}$
3/2, 1/2	${\displaystyle {\sqrt {\frac {5}{12}}}}$	${\displaystyle {\sqrt {\frac {5}{28}}}}$	${\displaystyle -{\sqrt {\frac {1}{12}}}}$	${\displaystyle -{\sqrt {\frac {9}{28}}}}$
1/2, 3/2	${\displaystyle {\sqrt {\frac {5}{12}}}}$	${\displaystyle -{\sqrt {\frac {5}{28}}}}$	${\displaystyle -{\sqrt {\frac {1}{12}}}}$	${\displaystyle {\sqrt {\frac {9}{28}}}}$
−1/2, 5/2	${\displaystyle {\sqrt {\frac {1}{12}}}}$	${\displaystyle -{\sqrt {\frac {9}{28}}}}$	${\displaystyle {\sqrt {\frac {5}{12}}}}$	${\displaystyle -{\sqrt {\frac {5}{28}}}}$

м = 1

j м1, м2	5	4	3	2	1
5/2, −3/2	${\displaystyle {\sqrt {\frac {1}{42}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$
3/2, −1/2	${\displaystyle {\sqrt {\frac {5}{21}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{30}}}}$	${\displaystyle -{\sqrt {\frac {1}{7}}}}$	${\displaystyle -{\sqrt {\frac {8}{35}}}}$
1/2, 1/2	${\displaystyle {\sqrt {\frac {10}{21}}}}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {\frac {4}{15}}}}$	${\displaystyle 0}$	${\displaystyle {\sqrt {\frac {9}{35}}}}$
−1/2, 3/2	${\displaystyle {\sqrt {\frac {5}{21}}}}$	${\displaystyle -{\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{30}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$	${\displaystyle -{\sqrt {\frac {8}{35}}}}$
−3/2, 5/2	${\displaystyle {\sqrt {\frac {1}{42}}}}$	${\displaystyle -{\sqrt {\frac {1}{7}}}}$	${\displaystyle {\sqrt {\frac {1}{3}}}}$	${\displaystyle -{\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$

м = 0

j м1, м2	5	4	3	2	1	0
5/2, −5/2	${\displaystyle {\sqrt {\frac {1}{252}}}}$	${\displaystyle {\sqrt {\frac {1}{28}}}}$	${\displaystyle {\sqrt {\frac {5}{36}}}}$	${\displaystyle {\sqrt {\frac {25}{84}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle {\sqrt {\frac {1}{6}}}}$
3/2, −3/2	${\displaystyle {\sqrt {\frac {25}{252}}}}$	${\displaystyle {\sqrt {\frac {9}{28}}}}$	${\displaystyle {\sqrt {\frac {49}{180}}}}$	${\displaystyle {\sqrt {\frac {1}{84}}}}$	${\displaystyle -{\sqrt {\frac {9}{70}}}}$	${\displaystyle -{\sqrt {\frac {1}{6}}}}$
1/2, −1/2	${\displaystyle {\sqrt {\frac {25}{63}}}}$	${\displaystyle {\sqrt {\frac {1}{7}}}}$	${\displaystyle -{\sqrt {\frac {4}{45}}}}$	${\displaystyle -{\sqrt {\frac {4}{21}}}}$	${\displaystyle {\sqrt {\frac {1}{70}}}}$	${\displaystyle {\sqrt {\frac {1}{6}}}}$
−1/2, 1/2	${\displaystyle {\sqrt {\frac {25}{63}}}}$	${\displaystyle -{\sqrt {\frac {1}{7}}}}$	${\displaystyle -{\sqrt {\frac {4}{45}}}}$	${\displaystyle {\sqrt {\frac {4}{21}}}}$	${\displaystyle {\sqrt {\frac {1}{70}}}}$	${\displaystyle -{\sqrt {\frac {1}{6}}}}$
−3/2, 3/2	${\displaystyle {\sqrt {\frac {25}{252}}}}$	${\displaystyle -{\sqrt {\frac {9}{28}}}}$	${\displaystyle {\sqrt {\frac {49}{180}}}}$	${\displaystyle -{\sqrt {\frac {1}{84}}}}$	${\displaystyle -{\sqrt {\frac {9}{70}}}}$	${\displaystyle {\sqrt {\frac {1}{6}}}}$
−5/2, 5/2	${\displaystyle {\sqrt {\frac {1}{252}}}}$	${\displaystyle -{\sqrt {\frac {1}{28}}}}$	${\displaystyle {\sqrt {\frac {5}{36}}}}$	${\displaystyle -{\sqrt {\frac {25}{84}}}}$	${\displaystyle {\sqrt {\frac {5}{14}}}}$	${\displaystyle -{\sqrt {\frac {1}{6}}}}$

SU (N) коэффициенты Клебша – Гордана

Алгоритмы для получения коэффициентов Клебша – Гордана для более высоких значений ${\displaystyle j_{1}}$ и ${\displaystyle j_{2}}$, или для алгебры su (N) вместо su (2).^[6]А веб-интерфейс для табулирования SU (N) коэффициентов Клебша – Гордана легко доступен.

Рекомендации

1. Baird, C.E .; Л. К. Биденхарн (октябрь 1964 г.). О представлениях полупростых групп Ли. III. Операция явного сопряжения для SU_п". J. Math. Phys. 5 (12): 1723–1730. Bibcode:1964JMP ..... 5.1723B. Дои:10.1063/1.1704095.
2. Hagiwara, K .; и другие. (Июль 2002 г.). «Обзор свойств частиц» (PDF). Phys. Ред. D. 66 (1): 010001. Bibcode:2002ПхРвД..66а0001Н. Дои:10.1103 / PhysRevD.66.010001. Получено 2007-12-20.
3. Матар, Ричард Дж. (14 августа 2006 г.). "SO (3) Коэффициенты Клебша Гордана" (текст). Получено 2012-10-15.
4. (2.41), стр. 172, Квантовая механика: основы и приложения, Арно Бом, М. Лоу, Нью-Йорк: Springer-Verlag, 3-е изд., 1993, ISBN  0-387-95330-2.
5. Вайсблут, Митчел (1978). Атомы и молекулы. АКАДЕМИЧЕСКАЯ ПРЕССА. п.28. ISBN  0-12-744450-5. В таблице 1.4 приведены самые распространенные.
6. Alex, A .; М. Калус; А. Гекльберри; Дж. Фон Делфт (февраль 2011 г.). «Численный алгоритм для явного вычисления SU (N) и SL (N, C) коэффициентов Клебша – Гордана». J. Math. Phys. 82: 023507. arXiv:1009.0437. Bibcode:2011JMP .... 52b3507A. Дои:10.1063/1.3521562.

внешняя ссылка

• В сети, Ява -основан Калькулятор коэффициентов Клебша – Гордана Пол Стивенсон
• Другие формулы для коэффициентов Клебша – Гордана.
• Веб-интерфейс для табулирования SU (N) коэффициентов Клебша – Гордана