角運動量演算子 $ \hat{\boldsymbol { L }} = \hat{\boldsymbol { x }} \times \hat{\boldsymbol { p }} ~,~(\hat{ L }_i= \epsilon _ {ijk} \hat{x} _ j \hat{p} _ k ~(i,j,k=1,2,3))$ とその大きさの2乗 $ \hat{\boldsymbol{ L }} ^ { 2 } $ は次の交換関係を満たす．$ \epsilon _ {ijk} $ はレビチビタ記号．アインシュタインの総和記法を用いる．

\begin{align} [ \hat{L} _ { i } , \hat{x} _ { j } ] &= i \hbar \epsilon _ { i j k } \hat{x} _ { k } \\ [ \hat{L} _ { i } , \hat{p} _ { j } ] &= i \hbar \epsilon _ { i j k } \hat{p} _ { k } \\ [ \hat{L} _ { i } , \hat{L} _ { j } ] &= i \hbar \epsilon _ { i j k } \hat{L} _ { k } \\ [ \hat{L} _ { i } , \hat{\boldsymbol { x } }^ { 2 } ] &= 0 \\ [ \hat{L} _ { i } , \hat{\boldsymbol { p }} ^ { 2 } ] &= 0 \\ [ \hat{L} _ { i } , \hat{\boldsymbol{ L }} ^ { 2 } ] &= 0 \\ \hat{\boldsymbol{ L }} ^ { 2 } &= \hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} \\ [\hat{ \boldsymbol{ L }} ^ { 2 } , \hat{x} _ { i } ] &= -i\hbar (2 \hat{\boldsymbol { x } }^ { 2 } \hat{p}_i - \hat{x}_i(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})- (\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})\hat{x}_i + i\hbar \hat{x}_i)\\ [ \hat{\boldsymbol{ L }} ^ { 2 } , \hat{p} _ { i } ] &= i\hbar (2 \hat{x}_i\hat{\boldsymbol { p } }^ { 2 } - \hat{p}_i(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})- (\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})\hat{p}_i + i\hbar \hat{p}_i) \\ [ \hat{H} , \hat{L} _ { i } ] &= 0 \end{align}

ただし，$ \hat{H} = \hat{\boldsymbol { p }} ^ { 2 }/2m + V(\hat{\boldsymbol { x }} ^ { 2 }) $．

導出

\begin{align} [ \hat{L} _ { i } , \hat{x} _ { j } ] &= [\epsilon _ {ikl} \hat{x} _ k \hat{p} _ l , \hat{x} _ { j }] = \epsilon _ {ikl} \hat{x} _ k\underbrace{[ \hat{p} _ l , \hat{x} _ { j }]}_{=-i\hbar \delta_{lj}} + \epsilon _ {ikl} \underbrace{[ \hat{x} _ k , \hat{x} _ { j }]}_{=0} \hat{p} _ l = i \hbar \epsilon _ { i j k } \hat{x} _ { k } \\ [ \hat{L} _ { i } , \hat{p} _ { j } ] &= [\epsilon _ {ikl} \hat{x} _ k \hat{p} _ l , \hat{p} _ { j }] = \epsilon _ {ikl} \hat{x} _ k\underbrace{[ \hat{p} _ l , \hat{p} _ { j }]}_{=0} + \epsilon _ {ikl} \underbrace{[ \hat{x} _ k , \hat{p} _ { j }]}_{=i\hbar \delta_{kj}} \hat{p} _ l = i \hbar \epsilon _ { i j k } \hat{p} _ { k } \\ [ \hat{L} _ { i } , \hat{L} _ { j } ] &= [ \hat{L} _ { i }, \epsilon _ {jkl} \hat{x} _ k \hat{p} _ l ] = \epsilon _ {jkl} \hat{x} _ k \underbrace{[ \hat{L} _ { i }, \hat{p} _ l ]}_{\rlap{=i \hbar \epsilon _ { i lm } \hat{p} _ { m }}} + \epsilon _ {jkl}\underbrace{[ \hat{L} _ { i }, \hat{x} _ k ]}_{\rlap{=i \hbar \epsilon _ { i k m } \hat{x} _ { m }}} \hat{p} _ l = i\hbar (\underbrace{\epsilon _ {jkl} \epsilon _ { i lm } \hat{x} _ k \hat{p} _ { m } }_{\substack{添え字の付けなおし\\ = \epsilon _ {jmk} \epsilon _ { i kl } \hat{x} _ m \hat{p} _ { l } }} + \epsilon _ {jkl} \epsilon _ { i km } \hat{x} _ m \hat{p} _ { l } ) \\ &= i\hbar (\epsilon _ {jmk} \underbrace{\epsilon _ { i kl }}_{\rlap{=-\epsilon _ { i lk }}} + \epsilon _ {jkl} \epsilon _ { i km } ) \hat{x} _ m \hat{p} _ { l } = i\hbar (-(\cancel{\delta_{ji}\delta_{ml}}-\delta_{jl}\delta_{mi}) + \cancel{\delta_{ji}\delta_{lm}}-\delta_{jm}\delta_{li} ) \hat{x} _ m \hat{p} _ { l } \\ &= i \hbar \epsilon _ { i j k } \epsilon _ { mlk } \hat{x} _ m \hat{p} _ { l } =i \hbar \epsilon _ { i j k } \hat{L} _ { k } \\ [ \hat{L} _ { i } , \hat{\boldsymbol { x } }^ { 2 } ] &= [ \hat{L} _ { i } , {\hat{ x}_j}^ { ~2 } ] = \underbrace{\hat{ x}_j \underbrace{[ \hat{L} _ { i } , \hat{ x}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{x} _ { k }}}_{=0} + \underbrace{\underbrace{[ \hat{L} _ { i } , \hat{ x}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{x} _ { k }}\hat{ x}_j}_{=0} = 0 \\ &対称な添え字と反対称な添え字の縮約は0になる．\\ [ \hat{L} _ { i } , \hat{\boldsymbol { p }} ^ { 2 } ] &= [ \hat{L} _ { i } , {\hat{ p}_j}^ { ~2 } ] = \underbrace{\hat{ p}_j \underbrace{[ \hat{L} _ { i } , \hat{ p}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{p} _ { k }}}_{=0} + \underbrace{\underbrace{[ \hat{L} _ { i } , \hat{ p}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{p} _ { k }}\hat{ p}_j}_{=0} = 0 \\ [ \hat{L} _ { i } , \hat{\boldsymbol{ L }} ^ { 2 } ] &= [ \hat{L} _ { i } , {\hat{ L}_j}^ { ~2 } ] = \hat{ L}_j \underbrace{[ \hat{L} _ { i } , \hat{ L}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{L} _ { k }} + \underbrace{[ \hat{L} _ { i } , \hat{ L}_j ]}_{=i \hbar \epsilon _ { i j k } \hat{L} _ { k }}\hat{ L}_j = i\hbar \epsilon _ { i j k } \{ \hat{L} _ { j } , \hat{L} _ { k } \} = 0 \\ \hat{\boldsymbol{ L }} ^ { 2 } &= \epsilon _ {ijk} \hat{x} _ j \hat{p} _ k\epsilon _ {ilm} \hat{x} _ l \hat{p} _ m =(\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl})\hat{x} _ j \hat{p} _ k \hat{x} _ l\hat{p} _ m \\ &= \delta_{jl}\delta_{km} \hat{x} _ j\underbrace{\hat{p} _ k \hat{x} _ l}_{\rlap{= \hat{x} _ l\hat{p} _ k - [\hat{x} _ l , \hat{p} _ k]}} \hat{p} _ m -\delta_{jm}\delta_{kl} \hat{x} _ j \hat{p} _ k \underbrace{\hat{x} _ l\hat{p} _ m}_{\rlap{= \hat{p} _ m\hat{x} _ l - [\hat{p} _ m , \hat{x} _ l]}} \\ &= \delta_{jl}\delta_{km} \hat{x} _ j \hat{x} _ l\hat{p} _ k \hat{p} _ m - i\hbar \underbrace{\delta_{jl}\delta_{km}\delta_{lk} \hat{x} _ j \hat{p} _ m}_{=\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }}} -\delta_{jm}\delta_{kl}\hat{x} _ j \underbrace{\hat{p} _ k \hat{p} _ m}_{=\hat{p} _ m \hat{p} _ k}\hat{x} _ l -i\hbar \underbrace{\delta_{jm}\delta_{kl}\delta_{ml}\hat{x} _ j \hat{p} _ k}_{=\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }}}\\ &=\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } - 2i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} - \delta_{jm}\delta_{kl}\hat{x} _ j \hat{p} _ m (\hat{x} _ l \hat{p} _ k - \underbrace{[\hat{x} _ l , \hat{p} _ k]}_{=i\hbar \delta_{lk}})\\ &= \hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } - 2i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} - (\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\delta_{jm}\underbrace{\delta_{kl}\delta_{lk}}_{=3}\hat{x} _ j \hat{p} _ m\\ &=\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} \\ [\hat{ \boldsymbol{ L }} ^ { 2 } , \hat{x} _ { i } ] &= [\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} , \hat{x} _ { i } ] = -i\hbar \frac{\partial}{\partial \hat{p}_i}(\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} ) \\ &=-i\hbar (2 \hat{\boldsymbol { x } }^ { 2 } \hat{p}_i - \hat{x}_i(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})- (\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})\hat{x}_i + i\hbar \hat{x}_i)\\ & \hat{x} と \hat{p} の関数との交換関係の公式を用いた．\\ [ \hat{\boldsymbol{ L }} ^ { 2 } , \hat{p} _ { i } ] &= [\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} , \hat{p} _ { i } ] = i\hbar \frac{\partial}{\partial \hat{x}_i}(\hat{\boldsymbol { x } }^ { 2 } \hat{\boldsymbol { p }} ^ { 2 } -(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} )^2 + i\hbar \hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }} ) \\ &=i\hbar (2 \hat{x}_i\hat{\boldsymbol { p } }^ { 2 } - \hat{p}_i(\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})- (\hat{\boldsymbol { x } } \cdot \hat{\boldsymbol { p }})\hat{p}_i + i\hbar \hat{p}_i) \end{align}