there are a few answered questions regarding the commutator of any two 3D angular momentum operator components $[L_i, L_j]$ , however, I am trying to go through fully using index notation so that I can arrive at the generic expression $$L_k=i \hbar \epsilon_{kij}x_ip_j$$ where $x$ and $p$ are the position and momentum operators in 3D.
Using $$[AB,C] = A[B,C] + [A,C]B$$ and cancelling out self-commutators, as well as using the canonical commutation relations I come to this line (skipped some steps):
$$[L_i, L_j] = \epsilon_{iab}\epsilon_{jcd}[x_ap_b,x_cp_d] = … = \epsilon_{iab}\epsilon_{jcd}(x_ap_d\delta _{bc} – x_bp_c\delta _{ad}) $$
Now if I try to contract the levi-civitas for each of the two terms in the brackets, I just get more kronecker deltas, with which I don't know what to do. If someone is feeling really generous today, could you go through the final steps to the solution so that I can understand what is happening (all the answers I have found just skip this assuming the reader will understand). Thank you!
Best Answer
You have mixed indices in the end of your line. Correctly: $$-\epsilon_{iab}\epsilon_{jcd}(x_ap_d\delta _{bc} - x_cp_b\delta _{ad}).$$ So further, $-\epsilon_{iab}\epsilon_{jcd}(x_ap_d\delta _{bc} - x_cp_b\delta _{ad})=\epsilon_{iab}\epsilon_{jcd}x_cp_b\delta _{ad}-\epsilon_{iab}\epsilon_{jcd}x_ap_d\delta _{bc}=\\=\epsilon_{idb}\epsilon_{jcd}x_cp_b - \epsilon_{iac}\epsilon_{jcd}x_ap_d=\epsilon_{iac}\epsilon_{jdc}x_ap_d-\epsilon_{ibd}\epsilon_{jcd}x_cp_b=\\=(\delta _{ij}\delta _{ad}-\delta _{id}\delta _{aj})x_ap_d-(\delta _{ij}\delta _{bc}-\delta _{ic}\delta _{bj})x_cp_b=\\=\delta _{ij}x_ap_a-x_jp_i-\delta _{ij}x_bp_b+x_ip_j=x_ip_j-x_jp_i=\epsilon_{ijk}L_k.$