جدول معاملات كلبسش-غوردان

الصيغة

الصيغة:Formulation

معاملات كلبسش-غوردان هي حلول ل:

$|(j_{1}j_{2})jm\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle$

بشكل مباشر (صريح) إلى:

${\begin{aligned}\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =\ &\delta _{m,m_{1}+m_{2}}{\sqrt {\frac {(2j+1)(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!(j_{1}+j_{2}-j)!}{(j_{1}+j_{2}+j+1)!}}}\ \times \\&{\sqrt {(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times \\&\sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j-j_{2}+m_{1}+k)!(j-j_{1}-m_{2}+k)!}}.\end{aligned}}$

The summation is extended over all integer k for which the argument of every factorial is nonnegative.

For brevity, solutions with m < 0 and j₁ < j₂ are omitted. They may be calculated using the simple relations

$\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{1}j_{2};-m_{1},-m_{2}|j_{1}j_{2};j,-m\rangle$ .

و

$\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{2}j_{1};m_{2}m_{1}|j_{2}j_{1};jm\rangle$ .

القائمة الكاملة

 j₂=0

عندما تكون j₂=0، فإن معاملات كلبسش-غوردان تعطي $\delta _{j,j_{1}}\delta _{m,m_{1}}$ .

 j₁=1/2, j₂=1/2

m=1	j
m1, m2	1 1/2, 1/2 $1\!\,$

m=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 ${\sqrt {\frac {5}{126}}}\!\,$ $-{\sqrt {\frac {64}{315}}}\!\,$ ${\sqrt {\frac {27}{70}}}\!\,$ $-{\sqrt {\frac {32}{105}}}\!\,$ ${\sqrt {\frac {1}{15}}}\!\,$

وصلات خارجية

• Online, جاوة-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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