I've been asked to prove $\left[ \vec{L}^2, x_i \right] = -2i\hbar \varepsilon_{ijk}L_j x_k -2\hbar^2 x_i $ and I don't seem to get it correctly.

I propose $\left[ \vec{L}^2, x_i \right] = \left[ L_l L_l, x_i \right] = L_l\left[ L_l, x_i \right] +\left[ L_l, x_i \right]L_l$ which leads me to $\left[ \vec{L}^2, x_i \right] = -i\hbar \varepsilon_{ilk}L_lx_k - i\hbar\varepsilon_{ilk}x_kL_l$ but I think I'm using some property not properly because I arrive to the following result $\left[ \vec{L}^2, x_i \right] = -i\hbar \varepsilon_{ilk}L_lx_k -i\hbar\varepsilon_{ilk}\left[L_lx_k -i\hbar\varepsilon_{lkm}x_m\right] =-2i\hbar \varepsilon_{ilk}L_lx_k +\hbar^2 \varepsilon_{ilk}\varepsilon_{lmk}x_m = -2i\hbar \varepsilon_{ilk}L_lx_k$

which is false. I used $\varepsilon_{ilk}\varepsilon_{lkm}=-\varepsilon_{lik}\varepsilon_{lkm} = -\left(\delta_{ik}\delta_{km}-\delta_{im}\delta_{kk}\right) = 0$ and $\left[L_l, x_k\right] = i\hbar \varepsilon_{lkm}x_m$.

Could you give me some hints? I'm starting to learn how to use Levi-Civita tensor

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    $\begingroup$ First problem: after the "which leads me to" replace $j\rightarrow l$. $\endgroup$ – Sean E. Lake Mar 17 '20 at 14:49
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    $\begingroup$ Next: What is $\epsilon_{ijk}\epsilon_{klm}$ in terms of Kronecker deltas? $\endgroup$ – Sean E. Lake Mar 17 '20 at 14:50
  • $\begingroup$ @SeanE.Lake I edited the question. I have corrected the fact with the $j$ and the $l$. I continue having the same problem :( $\endgroup$ – Patrick Mar 17 '20 at 15:19

Found my error. $\delta_{ik}\delta_{km}-\delta_{im}\delta_{kk} = (1-3)\delta_{im} \neq 0$.

Thank you all!


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