I'm trying to show that $[L_i,x_k]=i\hbar \epsilon_{ikl}x_l$.

I seem to be off by a sign. Here's what I did:

$[L_i,x_k]=[\epsilon_{ikl}x_kp_l,x_k]=\epsilon_{ikl}(x_k[p_l,x_k]+[x_k,x_k]p_l) = \epsilon_{ikl}x_k[p_l,x_k]=-i\hbar\epsilon_{ikl}x_k\delta_{lk}=-i\hbar\epsilon_{ikl}x_l$

Any insights?

Edit: resolved, see comments below.

  • 3
    $\begingroup$ Your mistake is repeating indices too many times. Einstein summation only works when there are exactly two copies of the index, so when a term with three of the same index pops up, as you did here, a mistake has been made. $\endgroup$ – Sean E. Lake Feb 12 '17 at 1:15
  • $\begingroup$ @SeanLake so would I need to write $[L_i,x_k]=[\epsilon_{ijl}x_jp_l,x_k]$? That doesn't seem to work either. $\endgroup$ – Spuds Feb 12 '17 at 1:24
  • $\begingroup$ @SeanLake nevermind, I think I got it by using $[L_i,x_k]=[\epsilon_{ilm}x_lp_m,x_k]$. $\endgroup$ – Spuds Feb 12 '17 at 1:34

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