核磁気モーメント
s → ⋅ j → = 1 2 ( j → ⋅ j → − l → ⋅ l → + s → ⋅ s → ) {\displaystyle {\overrightarrow {s}}\cdot {\overrightarrow {j}}={1 \over 2}\left({\overrightarrow {j}}\cdot {\overrightarrow {j}}-{\overrightarrow {l}}\cdot {\overrightarrow {l}}+{\overrightarrow {s}}\cdot {\overrightarrow {s}}\right)}
と書き直すと μ = 1 ( j + 1 ) ⟨ ( l , s ) , j , m j = j | ( g ( l ) 1 2 ( j → ⋅ j → + l → ⋅ l → − s → ⋅ s → ) + g ( s ) 1 2 ( j → ⋅ j → − l → ⋅ l → + s → ⋅ s → ) | ( l , s ) , j , m j = j ⟩ {\displaystyle \mu ={1 \over (j+1)}\langle (l,s),j,m_{j}=j|(g^{(l)}{1 \over 2}\left({\overrightarrow {j}}\cdot {\overrightarrow {j}}+{\overrightarrow {l}}\cdot {\overrightarrow {l}}-{\overrightarrow {s}}\cdot {\overrightarrow {s}}\right)+g^{(s)}{1 \over 2}\left({\overrightarrow {j}}\cdot {\overrightarrow {j}}-{\overrightarrow {l}}\cdot {\overrightarrow {l}}+{\overrightarrow {s}}\cdot {\overrightarrow {s}}\right)|(l,s),j,m_{j}=j\rangle } = 1 ( j + 1 ) ( g ( l ) 1 2 ( j ( j + 1 ) + l ( l + 1 ) − s ( s + 1 ) ) + g ( s ) 1 2 ( j ( j + 1 ) − l ( l + 1 ) + s ( s + 1 ) ) ) {\displaystyle ={1 \over (j+1)}\left(g^{(l)}{1 \over 2}\left(j(j+1)+l(l+1)-s(s+1)\right)+g^{(s)}{1 \over 2}\left(j(j+1)-l(l+1)+s(s+1)\right)\right)} となる。