軌道角運動量
極座標で書き表すと:
L 2 ^ = − ħ 2 ( 1 sin  θ ∂ ∂ θ ( sin  θ ∂ ∂ θ ) + 1 sin 2  θ ∂ 2 ∂ φ 2 ) {\displaystyle {\hat {{\boldsymbol {L}}^{2}}}=-\hbar ^{2}\left({\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)}
である。
軌道角運動量演算子は以下のように定義される:
L ^ = ( L ^ x , L ^ y , L ^ z ) = ( − i ħ ( y ∂ ∂ z − z ∂ ∂ y ) , − i ħ ( z ∂ ∂ x − x ∂ ∂ z ) , − i ħ ( x ∂ ∂ y − y ∂ ∂ x ) ) {\displaystyle {\hat {\boldsymbol {L}}}=({\hat {L}}_{x},{\hat {L}}_{y},{\hat {L}}_{z})=\left(-i\hbar \left(y{\frac {\partial }{\partial z}}-z{\frac {\partial }{\partial y}}\right),-i\hbar \left(z{\frac {\partial }{\partial x}}-x{\frac {\partial }{\partial z}}\right),-i\hbar \left(x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}\right)\right)}