# 克莱布希－高登系数

（重定向自克里布希－戈登係數

## 角动量耦合的一般理论

${\displaystyle |jm\rangle ,\quad m=-j,-j+1,\dots ,j-1,j}$

${\displaystyle V=V_{1}\otimes V_{2}=\operatorname {span} (\{|j_{1}m_{1}\rangle |m_{1}=-j_{1},-j_{1}+1,\dots ,j_{1}-1,j_{1}\})\otimes \operatorname {span} (\{|j_{2}m_{2}\rangle |m_{2}=-j_{2},-j_{2}+1,\dots ,j_{2}-1,j_{2}\})}$

${\displaystyle V=\operatorname {span} (\{|j_{1}m_{1}\rangle |\otimes |j_{2}m_{2}\rangle |m_{1}=-j_{1},-j_{1}+1,\dots ,j_{1}-1,j_{1};m_{2}=-j_{2},-j_{2}+1,\dots ,j_{2}-1,j_{2}\})}$

${\displaystyle |j_{1}m_{1}j_{2}m_{2}\rangle =|j_{1}m_{1}\rangle \otimes |j_{2}m_{2}\rangle }$

${\displaystyle f\otimes g:V_{1}\otimes V_{2}\rightarrow V_{1}\otimes V_{2},u\otimes v\rightarrow (fu)\otimes (gv)}$

${\displaystyle f\rightarrow f\otimes 1,g\rightarrow 1\otimes g}$

${\displaystyle j_{\alpha }=j_{1,\alpha }+j_{2,\alpha }=j_{1,\alpha }\otimes 1+1\otimes j_{2,\alpha },\quad \alpha \in \{x,y,z\}}$
${\displaystyle \mathbf {j} =\mathbf {j} _{1}+\mathbf {j} _{2}=\mathbf {j} _{1}\otimes 1+1\otimes \mathbf {j} _{2},\quad \alpha \in \{x,y,z\}}$

${\displaystyle |jm\rangle =\sum _{m_{1},m_{2}}\langle j_{1}m_{1}j_{2}m_{2}|jm\rangle |j_{1}m_{1}j_{2}m_{2}\rangle }$

${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|jm\rangle }$

## 耦合表象中量子数的取值

${\displaystyle j_{z}=j_{1,z}+j_{2,z}}$

${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|jm\rangle =\delta _{m_{1}+m_{2},m}\langle j_{1}m_{1}j_{2}m_{2}|jm_{1}+m_{2}\rangle }$

${\displaystyle j_{\max }=m_{\max }=m_{1,\max }+m_{2,\max }=j_{1}+j_{2}}$

j 最大的可能取值是 j1j2 的和，且它只出现一次。此时

${\displaystyle m=-j_{\max },-j_{\max }+1,\dots ,j_{\max }-1,j_{\max }}$

${\displaystyle m_{1}=j_{1}-1,m_{2}=j_{2}{\text{ or }}m_{1}=j_{1},m_{2}=j_{2}-1}$

${\displaystyle j_{\min },j_{\min }+1,\dots ,j_{\max }-1,j_{\max }}$

${\displaystyle j_{\max }-j_{\min }\in \mathbb {Z} }$

${\displaystyle \sum _{n=j_{\min }}^{j_{\max }}(2n+1)=(2j_{1}+1)(2j_{2}+1)}$

${\displaystyle j_{\min }=|j_{1}-j_{2}|}$

## 一个例子

${\displaystyle j_{1}=j_{2}={\frac {1}{2}}}$  为例[2]

${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|f|j_{1}m_{1}j_{2}m_{2}\rangle }$

${\displaystyle j_{z}={\frac {1}{2}}\left({\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}+{\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&-1\end{bmatrix}}\right)={\begin{bmatrix}1&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&-1\end{bmatrix}}}$
${\displaystyle j_{+}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&0&0\\1&0&0&0\\0&0&0&0\\0&0&1&0\end{bmatrix}}={\begin{bmatrix}0&0&0&0\\1&0&0&0\\1&0&0&0\\0&1&1&0\end{bmatrix}}=j_{-}^{\dagger }}$
${\displaystyle \mathbf {j} ^{2}={\frac {1}{2}}[j_{+},j_{-}]_{+}+j_{z}^{2}={\begin{bmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&2\end{bmatrix}}}$

${\displaystyle {\begin{bmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&2\end{bmatrix}}{\begin{bmatrix}0&0&1&0\\-2^{-1/2}&2^{-1/2}&0&0\\2^{-1/2}&2^{-1/2}&0&0\\0&0&0&1\end{bmatrix}}={\begin{bmatrix}0&0&1&0\\-2^{-1/2}&2^{-1/2}&0&0\\2^{-1/2}&2^{-1/2}&0&0\\0&0&0&1\end{bmatrix}}\operatorname {diag} \{0,2,2,2\}}$

m=1 j=

${\displaystyle m_{1},m_{2}=}$
 1 1/2, 1/2 ${\displaystyle 1\!\,}$
m=0 j=

m1, m2=
 1 0 1/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ -1/2, 1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$
m=-1 j=

m1, m2=
 1 -1/2, -1/2 ${\displaystyle 1\!\,}$

## Racah 表达式

Racah 用代数方法得出了克莱布希－高登系数的有限级数表达式[3]

${\displaystyle {\begin{array}{cl}&\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle \\=&\delta _{m_{3},m_{1}+m_{2}}\left[(2j_{3}+1){\frac {(j_{1}+j_{2}-j_{3})!(j_{2}+j_{3}-j_{1})!(j_{3}+j_{1}-j_{2})!}{(j_{1}+j_{2}+j_{3}+1)!}}\times \prod _{i=1,2,3}(j_{i}+m_{i})!(j_{i}-m_{i})!\right]^{1/2}\\\times &\sum _{\nu }[(-1)^{\nu }\nu !(j_{1}+j_{2}-j_{3}-\nu )!(j_{1}-m_{1}-\nu )!(j_{2}+m_{2}-\nu )!(j_{3}-j_{1}-m_{2}+\nu )!(j_{3}-j_{2}+m_{1}+\nu )!]^{-1}\end{array}}}$

## 对称性

{\displaystyle {\begin{aligned}\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle &=(-1)^{j_{1}+j_{2}-J}\langle j_{1}\,{-m_{1}}j_{2}\,{-m_{2}}|J\,{-M}\rangle \\&=(-1)^{j_{1}+j_{2}-J}\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle \\&=(-1)^{j_{1}-m_{1}}{\sqrt {\frac {2J+1}{2j_{2}+1}}}\langle j_{1}m_{1}J\,{-M}|j_{2}\,{-m_{2}}\rangle \\&=(-1)^{j_{2}+m_{2}}{\sqrt {\frac {2J+1}{2j_{1}+1}}}\langle J\,{-M}j_{2}m_{2}|j_{1}\,{-m_{1}}\rangle \\&=(-1)^{j_{1}-m_{1}}{\sqrt {\frac {2J+1}{2j_{2}+1}}}\langle JMj_{1}\,{-m_{1}}|j_{2}m_{2}\rangle \\&=(-1)^{j_{2}+m_{2}}{\sqrt {\frac {2J+1}{2j_{1}+1}}}\langle j_{2}\,{-m_{2}}JM|j_{1}m_{1}\rangle \end{aligned}}}

## 与维格纳 3-j 符号的关系

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}m_{1}j_{2}m_{2}|j_{3}\,{-m_{3}}\rangle .}$

{\displaystyle {\begin{aligned}&{}\quad \int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\[8pt]0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}}

## 参考

1. 曾谨言. 10. 量子力学卷 I （第四版）. 科学出版社. [2011]. ISBN 9787030181398.
2. ^ William O. Straub. EFFICIENT COMPUTATION OF CLEBSCH-GORDAN COEFFICIENTS (PDF). [2014-09-09]. （原始内容存档 (PDF)于2019-08-19）.
3. ^ Giulio Racah. Theory of Complex Spectra. II. Phys. Rev.: 438. doi:10.1103/PhysRev.62.438.
4. Maximon, Leonard C., 3j,6j,9j Symbols, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248