クレブシュ–ゴルダン係数
j ^ 2 | j m ⟩ = ħ 2 j ( j + 1 ) | j m ⟩ ( j = 0 , 1 2 , 1 , 3 2 , 2 , ... ) {\displaystyle \mathbf {\hat {j}} ^{2}|j\,m\rangle =\hbar ^{2}j(j+1)|j\,m\rangle \quad \quad (j=0,{\frac {1}{2}},1,{\frac {3}{2}},2,\ldots )}
j ^ z | j m ⟩ = ħ m | j m ⟩ ( m = − j , − j + 1 , ... , j . ) {\displaystyle {\hat {j}}_{z}|j\,m\rangle =\hbar m|j\,m\rangle \quad \quad \quad \quad \quad (m=-j,-j+1,\ldots ,j.)}
m {\displaystyle m\ } の値は昇降演算子で変化する。
j ^ ± | j m ⟩ = C ± ( j , m ) | j m ± 1 ⟩ {\displaystyle {\hat {j}}_{\pm }|j\,m\rangle =C_{\pm }(j,m)|j\,m\pm 1\rangle }
⟨ j 1 m 1 j 2 m 2 | J M ⟩ = ( − 1 ) j 1 + j 2 − J ⟨ j 1 − m 1 j 2 − m 2 | J − M ⟩ = ( − 1 ) j 1 + j 2 − J ⟨ j 2 m 2 j 1 m 1 | J M ⟩ = ( − 1 ) j 1 − m 1 2 J + 1 2 j 2 + 1 ⟨ j 1 m 1 J − M | j 2 − m 2 ⟩ = ( − 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 ⟨ J − M j 2 m 2 | j 1 − m 1 ⟩ = ( − 1 ) j 1 − m 1 2 J + 1 2 j 2 + 1 ⟨ J M j 1 − m 1 | j 2 m 2 ⟩ = ( − 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 ⟨ j 2 − m 2 J M | j 1 m 1 ⟩ {\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}}}