I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$
$$[L_i,r_j] ~=~ i\hbar\; \epsilon_{ijk}r_k,$$
$$[L_i,L_j] ~=~ i\hbar\; \epsilon_{ijk}L_k.$$
Now I am trying to calculate
$$[L_i,(p\times L)_j].$$
I do not know how to reduce it. When I try I am left with too many dummy indices that I don't know what to do. For example, using
$$[AB,C] = A[B,C]+[A,C]B,$$
and expanding the cross product in terms of Levi-Civita symbols
$$[L_i,\epsilon_{jmn}p_m L_n],$$
but I don't know how to proceed correctly from here. For example, I tried,
$$ =~ \epsilon_{jmn}(\;p_m[L_i,L_n]+[L_i,p_m]L_n ).$$
Is this correct? If so, in the next step I used the known commutation relations
$$ =~i\hbar\; \epsilon_{jmn}(\; \epsilon_{ink}p_mL_k+\epsilon_{imk}p_kL_n).$$
Once again, I am stuck and do not know how to evaluate this further. Could someone tell me what I am doing wrong, or if not, how to proceed?
Best Answer
You should use the Levi-Civita reduction formula
$$\epsilon_{ijk}\epsilon_{ilm}=\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}$$
and using the fact that $(a\times b)_i=\epsilon_{ijk}a_jb_k$ you should be done.