جدول معاملات كلبسش-غوردان

الصيغة

الصيغة:Formulation

معاملات كلبسش-غوردان هي حلول ل:

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

بشكل مباشر (صريح) إلى:

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

The summation is extended over all integer k for which the argument of every factorial is nonnegative.^[1]

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

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

و

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

القائمة الكاملة ^[2]

 j₂=0

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

 j₁=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

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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