Tabulka Clebsch-Gordanových koeficientů - Table of Clebsch–Gordan coefficients - Wikipedia

Tohle je tabulka Clebsch – Gordanovy koeficienty slouží k přidání moment hybnosti hodnoty v kvantová mechanika. Celkové znaménko koeficientů pro každou sadu konstanty ${ displaystyle j_ {1}}$ , ${ displaystyle j_ {2}}$ , ${ displaystyle j}$ je do jisté míry libovolný a byl opraven podle konvence Condon – Shortley a Wigner, jak je popsáno v Baird a Biedenharn.^[1] Tabulky se stejnou znakovou konvencí najdete v Skupina dat o částicích je Kontrola vlastností částic^[2] a v online tabulkách.^[3]

Formulace

Clebsch – Gordanovy koeficienty jsou řešením

{ 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}

Výslovně:

{ 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)!}}} krát {} [6pt] & { sqrt {(J + M)! (JM)! (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} -Jk)! (j_ {1} -m_ {1} -k)! (j_ {2} + m_ {2} -k)! (J -j_ {2} + m_ {1} + k)! (J-j_ {1} -m_ {2} + k)!}}. end {zarovnáno}}}

Součet je rozšířen na celé celé číslo $k$ pro které argument každého faktoriálu není negativní.^[4]

Pro stručnost, řešení s $M < 0$ a $j 1 < j 2$ jsou vynechány. Mohou být vypočítány pomocí jednoduchých vztahů

{ 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 zvonit.}

a

{ 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.}

Specifické hodnoty

Clebsch – Gordanovy koeficienty pro j hodnoty menší nebo rovné 5/2 jsou uvedeny níže.^[5]

$j 2 = 0$

Když $j 2 = 0$ , Clebsch – Gordan koeficienty jsou dány ${ displaystyle delta _ {j, j_ {1}} delta _ {m, m_ {1}}}$ .

$j 1 = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} 1 / 2, j 2 = 1 / 2$

$m = 1$
$j$ $m 1, m 2$	1
1/2, 1/2	${ displaystyle 1}$

$m = -1$
$j$ $m 1, m 2$	1
−1/2, −1/2	${ displaystyle 1}$

$m = 0$
$j$ $m 1, m 2$	1	0
1/2, −1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
−1/2, 1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

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

$m = 3 / 2$
$j$ $m 1, m 2$	3/2
1, 1/2	${ displaystyle 1}$

$m = 1 / 2$
$j$ $m 1, m 2$	3/2	1/2
1, −1/2	${ displaystyle { sqrt { frac {1} {3}}}}$	${ displaystyle { sqrt { frac {2} {3}}}}$
0, 1/2	${ displaystyle { sqrt { frac {2} {3}}}}$	${ displaystyle - { sqrt { frac {1} {3}}}}$

$j 1 = 1, j 2 = 1$

$m = 2$
$j$ $m 1, m 2$	2
1, 1	${ displaystyle 1}$

$m = 1$
$j$ $m 1, m 2$	2	1
1, 0	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
0, 1	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$m = 0$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {6}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {3}}}}$

$j 1 = 3 / 2, j 2 = 1 / 2$

$m = 2$
$j$ $m 1, m 2$	2
3/2, 1/2	${ displaystyle 1}$

$m = 1$
$j$ $m 1, m 2$	2	1
3/2, −1/2	${ displaystyle { frac {1} {2}}}$	${ displaystyle { sqrt { frac {3} {4}}}}$
1/2, 1/2	${ displaystyle { sqrt { frac {3} {4}}}}$	${ displaystyle - { frac {1} {2}}}$

$m = 0$
$j$ $m 1, m 2$	2	1
1/2, −1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
−1/2, 1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$j 1 = 3 / 2, j 2 = 1$

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
3/2, 1	${ displaystyle 1}$

$m = 3 / 2$
$j$ $m 1, m 2$	5/2	3/2
3/2, 0	${ displaystyle { sqrt { frac {2} {5}}}}$	${ displaystyle { sqrt { frac {3} {5}}}}$
1/2, 1	${ displaystyle { sqrt { frac {3} {5}}}}$	${ displaystyle - { sqrt { frac {2} {5}}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {3} {10}}}}$	${ displaystyle - { sqrt { frac {8} {15}}}}$	${ displaystyle { sqrt { frac {1} {6}}}}$

$j 1 = 3 / 2, j 2 = 3 / 2$

$m = 3$
$j$ $m 1, m 2$	3
3/2, 3/2	${ displaystyle 1}$

$m = 2$
$j$ $m 1, m 2$	3	2
3/2, 1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
1/2, 3/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$m = 1$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {5}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {3} {10}}}}$

$m = 0$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {20}}}}$	${ displaystyle - { frac {1} {2}}}$	${ displaystyle { sqrt { frac {9} {20}}}}$	${ displaystyle - { frac {1} {2}}}$

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

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
2, 1/2	${ displaystyle 1}$

$m = 3 / 2$
$j$ $m 1, m 2$	5/2	3/2
2, −1/2	${ displaystyle { sqrt { frac {1} {5}}}}$	${ displaystyle { sqrt { frac {4} {5}}}}$
1, 1/2	${ displaystyle { sqrt { frac {4} {5}}}}$	${ displaystyle - { sqrt { frac {1} {5}}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	5/2	3/2
1, −1/2	${ displaystyle { sqrt { frac {2} {5}}}}$	${ displaystyle { sqrt { frac {3} {5}}}}$
0, 1/2	${ displaystyle { sqrt { frac {3} {5}}}}$	${ displaystyle - { sqrt { frac {2} {5}}}}$

$j 1 = 2, j 2 = 1$

$m = 3$
$j$ $m 1, m 2$	3
2, 1	${ displaystyle 1}$

$m = 2$
$j$ $m 1, m 2$	3	2
2, 0	${ displaystyle { sqrt { frac {1} {3}}}}$	${ displaystyle { sqrt { frac {2} {3}}}}$
1, 1	${ displaystyle { sqrt { frac {2} {3}}}}$	${ displaystyle - { sqrt { frac {1} {3}}}}$

$m = 1$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {2} {5}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {10}}}}$

$m = 0$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {5}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {3} {10}}}}$

$j 1 = 2, j 2 = 3 / 2$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
2, 3/2	${ displaystyle 1}$

$m = 5 / 2$
$j$ $m 1, m 2$	7/2	5/2
2, 1/2	${ displaystyle { sqrt { frac {3} {7}}}}$	${ displaystyle { sqrt { frac {4} {7}}}}$
1, 3/2	${ displaystyle { sqrt { frac {4} {7}}}}$	${ displaystyle - { sqrt { frac {3} {7}}}}$

$m = 3 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {2} {7}}}}$	${ displaystyle - { sqrt { frac {18} {35}}}}$	${ displaystyle { sqrt { frac {1} {5}}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {4} {35}}}}$	${ displaystyle - { sqrt { frac {27} {70}}}}$	${ displaystyle { sqrt { frac {2} {5}}}}$	${ displaystyle - { sqrt { frac {1} {10}}}}$

$j 1 = 2, j 2 = 2$

$m = 4$
$j$ $m 1, m 2$	4
2, 2	${ displaystyle 1}$

$m = 3$
$j$ $m 1, m 2$	4	3
2, 1	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
1, 2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$m = 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {3} {14}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {2} {7}}}}$

$m = 1$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {14}}}}$	${ displaystyle - { sqrt { frac {3} {10}}}}$	${ displaystyle { sqrt { frac {3} {7}}}}$	${ displaystyle - { sqrt { frac {1} {5}}}}$

$m = 0$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {70}}}}$	${ displaystyle - { sqrt { frac {1} {10}}}}$	${ displaystyle { sqrt { frac {2} {7}}}}$	${ displaystyle - { sqrt { frac {2} {5}}}}$	${ displaystyle { sqrt { frac {1} {5}}}}$

$j 1 = 5 / 2, j 2 = 1 / 2$

$m = 3$
$j$ $m 1, m 2$	3
5/2, 1/2	${ displaystyle 1}$

$m = 2$
$j$ $m 1, m 2$	3	2
5/2, −1/2	${ displaystyle { sqrt { frac {1} {6}}}}$	${ displaystyle { sqrt { frac {5} {6}}}}$
3/2, 1/2	${ displaystyle { sqrt { frac {5} {6}}}}$	${ displaystyle - { sqrt { frac {1} {6}}}}$

$m = 1$
$j$ $m 1, m 2$	3	2
3/2, −1/2	${ displaystyle { sqrt { frac {1} {3}}}}$	${ displaystyle { sqrt { frac {2} {3}}}}$
1/2, 1/2	${ displaystyle { sqrt { frac {2} {3}}}}$	${ displaystyle - { sqrt { frac {1} {3}}}}$

$m = 0$
$j$ $m 1, m 2$	3	2
1/2, −1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
−1/2, 1/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$j 1 = 5 / 2, j 2 = 1$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
5/2, 1	${ displaystyle 1}$

$m = 5 / 2$
$j$ $m 1, m 2$	7/2	5/2
5/2, 0	${ displaystyle { sqrt { frac {2} {7}}}}$	${ displaystyle { sqrt { frac {5} {7}}}}$
3/2, 1	${ displaystyle { sqrt { frac {5} {7}}}}$	${ displaystyle - { sqrt { frac {2} {7}}}}$

$m = 3 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {10} {21}}}}$	${ displaystyle - { sqrt { frac {16} {35}}}}$	${ displaystyle { sqrt { frac {1} {15}}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {2} {7}}}}$	${ displaystyle - { sqrt { frac {18} {35}}}}$	${ displaystyle { sqrt { frac {1} {5}}}}$

$j 1 = 5 / 2, j 2 = 3 / 2$

$m = 4$
$j$ $m 1, m 2$	4
5/2, 3/2	${ displaystyle 1}$

$m = 3$
$j$ $m 1, m 2$	4	3
5/2, 1/2	${ displaystyle { sqrt { frac {3} {8}}}}$	${ displaystyle { sqrt { frac {5} {8}}}}$
3/2, 3/2	${ displaystyle { sqrt { frac {5} {8}}}}$	${ displaystyle - { sqrt { frac {3} {8}}}}$

$m = 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {5} {14}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {7}}}}$

$m = 1$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {5} {28}}}}$	${ displaystyle - { sqrt { frac {9} {20}}}}$	${ displaystyle { sqrt { frac {9} {28}}}}$	${ displaystyle - { sqrt { frac {1} {20}}}}$

$m = 0$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {14}}}}$	${ displaystyle - { sqrt { frac {3} {10}}}}$	${ displaystyle { sqrt { frac {3} {7}}}}$	${ displaystyle - { sqrt { frac {1} {5}}}}$

$j 1 = 5 / 2, j 2 = 2$

$m = 9 / 2$
$j$ $m 1, m 2$	9/2
5/2, 2	${ displaystyle 1}$

$m = 7 / 2$
$j$ $m 1, m 2$	9/2	7/2
5/2, 1	${ displaystyle { frac {2} {3}}}$	${ displaystyle { sqrt { frac {5} {9}}}}$
3/2, 2	${ displaystyle { sqrt { frac {5} {9}}}}$	${ displaystyle - { frac {2} {3}}}$

$m = 5 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {5} {18}}}}$	${ displaystyle - { sqrt { frac {32} {63}}}}$	${ displaystyle { sqrt { frac {3} {14}}}}$

$m = 3 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {5} {42}}}}$	${ displaystyle - { sqrt { frac {8} {21}}}}$	${ displaystyle { sqrt { frac {27} {70}}}}$	${ displaystyle - { sqrt { frac {4} {35}}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {5} {126}}}}$	${ displaystyle - { sqrt { frac {64} {315}}}}$	${ displaystyle { sqrt { frac {27} {70}}}}$	${ displaystyle - { sqrt { frac {32} {105}}}}$	${ displaystyle { sqrt { frac {1} {15}}}}$

$j 1 = 5 / 2, j 2 = 5 / 2$

$m = 5$
$j$ $m 1, m 2$	5
5/2, 5/2	${ displaystyle 1}$

$m = 4$
$j$ $m 1, m 2$	5	4
5/2, 3/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {1} {2}}}}$
3/2, 5/2	${ displaystyle { sqrt { frac {1} {2}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$

$m = 3$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {2} {9}}}}$	${ displaystyle - { sqrt { frac {1} {2}}}}$	${ displaystyle { sqrt { frac {5} {18}}}}$

$m = 2$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {12}}}}$	${ displaystyle - { sqrt { frac {9} {28}}}}$	${ displaystyle { sqrt { frac {5} {12}}}}$	${ displaystyle - { sqrt { frac {5} {28}}}}$

$m = 1$
$j$ $m 1, m 2$	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	${ displaystyle { sqrt { frac {1} {42}}}}$	${ displaystyle - { sqrt { frac {1} {7}}}}$	${ displaystyle { sqrt { frac {1} {3}}}}$	${ displaystyle - { sqrt { frac {5} {14}}}}$	${ displaystyle { sqrt { frac {1} {7}}}}$

$m = 0$
$j$ $m 1, m 2$	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) Clebsch – Gordanovy koeficienty

Algoritmy pro vytvoření Clebsch-Gordanových koeficientů pro vyšší hodnoty ${ displaystyle j_ {1}}$ a ${ displaystyle j_ {2}}$ , nebo pro su (N) algebru místo su (2), jsou známy.^[6]A webové rozhraní pro tabelaci SU (N) Clebsch – Gordanových koeficientů je snadno dostupný.

Reference

^ Baird, C.E .; L. C. Biedenharn (říjen 1964). „O reprezentacích polojednodušých lžových skupin. III. Explicitní konjugační operace pro SU_n". J. Math. Phys. 5 (12): 1723–1730. Bibcode:1964JMP ..... 5.1723B. doi:10.1063/1.1704095.
^ Hagiwara, K .; et al. (Červenec 2002). "Kontrola vlastností částic" (PDF). Phys. Rev. D. 66 (1): 010001. Bibcode:2002PhRvD..66a0001H. doi:10.1103 / PhysRevD.66.010001. Citováno 2007-12-20.
^ Mathar, Richard J. (2006-08-14). „SO (3) Clebsch Gordan coefficients“ (text). Citováno 2012-10-15.
^ (2,41), s. 172, Kvantová mechanika: základy a aplikaceArno Bohm, M. Loewe, New York: Springer-Verlag, 3. vydání, 1993, ISBN 0-387-95330-2.
^ Weissbluth, Mitchel (1978). Atomy a molekuly. AKADEMICKÝ TISK. str.28. ISBN 0-12-744450-5. Tabulka 1.4 uvádí nejběžnější.
^ Alex, A .; M. Kalus; A. Huckleberry; J. von Delft (únor 2011). "Numerický algoritmus pro explicitní výpočet SU (N) a SL (N, C) Clebsch-Gordanových koeficientů". J. Math. Phys. 82: 023507. arXiv:1009.0437. Bibcode:2011JMP .... 52b3507A. doi:10.1063/1.3521562.

externí odkazy

Online, Jáva -na základě Kalkulátor Clebsch – Gordanových koeficientů Paul Stevenson
Jiné vzorce pro Clebsch – Gordanovy koeficienty.
Webové rozhraní pro vytváření tabulek SU (N) Clebsch – Gordanových koeficientů

[1] Baird, C.E .; L. C. Biedenharn (říjen 1964). „O reprezentacích polojednodušých lžových skupin. III. Explicitní konjugační operace pro SU_n". J. Math. Phys. 5 (12): 1723–1730. Bibcode:1964JMP ..... 5.1723B. doi:10.1063/1.1704095.

[2] Hagiwara, K .; et al. (Červenec 2002). "Kontrola vlastností částic" (PDF). Phys. Rev. D. 66 (1): 010001. Bibcode:2002PhRvD..66a0001H. doi:10.1103 / PhysRevD.66.010001. Citováno 2007-12-20.

[3] Mathar, Richard J. (2006-08-14). „SO (3) Clebsch Gordan coefficients“ (text). Citováno 2012-10-15.

[4] (2,41), s. 172, Kvantová mechanika: základy a aplikaceArno Bohm, M. Loewe, New York: Springer-Verlag, 3. vydání, 1993, ISBN 0-387-95330-2.

[5] Weissbluth, Mitchel (1978). Atomy a molekuly. AKADEMICKÝ TISK. str.28. ISBN 0-12-744450-5. Tabulka 1.4 uvádí nejběžnější.

[6] Alex, A .; M. Kalus; A. Huckleberry; J. von Delft (únor 2011). "Numerický algoritmus pro explicitní výpočet SU (N) a SL (N, C) Clebsch-Gordanových koeficientů". J. Math. Phys. 82: 023507. arXiv:1009.0437. Bibcode:2011JMP .... 52b3507A. doi:10.1063/1.3521562.

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[6]

Tabulka Clebsch-Gordanových koeficientů - Table of Clebsch–Gordan coefficients - Wikipedia

Formulace

Specifické hodnoty

j2 = 0

j1 = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}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

j1 = 5/2, j2 = 5/2