I've been given an assignment where I have to prove that the angular momentum operators $L_j = \varepsilon_{jkl}q_{k}p_{l}$ commute with the Hamiltonian, given as $H = \frac{p^2}{2m} + V(r)$.
Now, I can prove that $[L_j, \frac{p^2}{2m}] = 0$, and we've been given that $V(r) = \sum_{i\in Z}C_i r^n$, which basically means that I just have to show $[L_j, r] = 0 = [L_j, r^{-1}]$.
Now, using the fact that I can get the following from $[r^2, p_j]$, and then subsequently show that $[L_j, r] = 0$; $$[r^2, p_j] = [q_{i}q_{i} , p_j] = 2i q_j$$ $$=> [r^2, p_j] = 2r[r, p_j] = 2i q_j$$ $$=> [r, p_j] = \frac{i q_j}{r}$$ $$=> [p_j, r] = \frac{-i q_j}{r}$$
Then, $$[L_j, r] = [\varepsilon_{jkl} q_k q_l, r] = \varepsilon{jkl}(q_k[p_l, r] + [q_k, r]p_l) = \varepsilon_{jkl}(q_k \frac{-i q_j}{r}) = \frac{-i \varepsilon_{jkl} q_j q_k}{r} = \frac{-i \varepsilon{kjl} q_k q_j}{r} = \frac{i \varepsilon_{jkl} q_k q_j}{r}$$ $$=> [L_j, r] = 0$$
However, I am completely lost/struggling to prove $[L_j, r^{-1}] = 0$. Any help would be fantastic!!
$$ \left[L_{j},r\right] = \sum_{k\ell j}\epsilon_{k\ell j}\left[x_{k}p_{\ell},r\right] = \sum_{k\ell j}\epsilon_{k\ell j}\left\{% x_{k}\left[p_{\ell},r\right] + \overbrace{\left[x_{k},r\right]}^{=\ 0}\,\,\, p_{\ell} \right\} $$
$$ \left[p_{\ell},r\right] = -{\rm i}\hbar\,{\partial \over \partial x_{\ell}}\ r + r\,\,{\rm i}\hbar\,{\partial \over \partial x_{\ell}} = -{\rm i}\hbar\,{\partial r \over \partial x_{\ell}} = -{\rm i}\hbar\,{x_{\ell} \over r} $$
$$ \left[L_{j},r\right] = -\,{{\rm i}\hbar \over r}\sum_{k\ell j} \epsilon_{k\ell j}x_{k}x_{\ell} = -\,{{\rm i}\hbar \over r}\sum_{k\ell j} \epsilon_{\ell kj}x_{\ell}x_{k} = {{\rm i}\hbar \over r}\sum_{\ell kj} \epsilon_{k\ell j}x_{k}x_{\ell} = -\left[L_{j},r\right] $$
$$ \color{#ff0000}{\large% \left[L_{j},r\right] = -\left[L_{j},r\right] \qquad\Longrightarrow\qquad \left[L_{j},r\right] = 0} $$