I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##.(adsbygoogle = window.adsbygoogle || []).push({});

Something's gone wrong with my proof below. Can you point out the mistake?

##[J_{i},J_{j}]##

##=J_{i}J_{j}-J_{j}J_{i}##

##=(-i\epsilon_{ikl}x_{k}\nabla_{l})(-i\epsilon_{jmn}x_{m}\nabla_{n})-(-i\epsilon_{jmn}x_{m}\nabla_{n})(-i\epsilon_{ikl}x_{k}\nabla_{l})##

##=-\epsilon_{ikl}\epsilon_{jmn}[x_{k}\nabla_{l}(x_{m}\nabla_{n})-x_{m}\nabla_{n}(x_{k}\nabla_{l})##

##=-\epsilon_{ikl}\epsilon_{jmn}[x_{k}x_{m}\nabla_{l}\nabla_{n}+x_{k}\nabla_{n}\nabla_{l}x_{m}-x_{m}x_{k}\nabla_{n}\nabla_{l}-x_{m}\nabla_{l}\nabla_{n}x_{k}]##

##=-\epsilon_{ikl}\epsilon_{jmn}[x_{k}x_{m}\nabla_{l}\nabla_{n}+x_{k}\nabla_{n}\delta_{lm}-x_{m}x_{k}\nabla_{n}\nabla_{l}-x_{m}\nabla_{l}\delta_{nk}]##

##=-\epsilon_{ikl}\epsilon_{jmn}[x_{k}x_{m}\nabla_{l}\nabla_{n}+0-x_{m}x_{k}\nabla_{n}\nabla_{l}-0]##

##=0##

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# Commutation relations for angular momentum operator

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