Kronecker-Delta / Levi-Civita tensor relation

Question

I have a quite simple computation question:

Given is the following relation:

$$ \epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} $$

And the computation steps:

$$i\hbar\left(\epsilon_{njk}\epsilon_{nmi} \ r_kp_m - \epsilon_{klj}\epsilon_{kin} \ r_np_l \right) \ \ (1)$$ $$i\hbar\left[ \left(r_ip_j - r_np_n\delta_{ij} \right) - \left(r_jp_i - r_np_n\delta_{ij} \right) \right] \ \ (2)$$

Where $r_{index}$ and $p_{index}$ are just vector components.

How do I get from $(1)$ to $(2)$ ?

This is from a Quantum Mechanics Problem but my Problem is rather mathematical.

Answer 1

We have \begin{align*} \epsilon_{njk}\epsilon_{nmi} r_kp_m &= (\delta_{jm}\delta_{ki}-\delta_{ji}\delta_{km})r_kp_m\\ &=r_ip_j-\delta_{ij}r_kp_k \end{align*} and similarly $$ \epsilon_{klj}\epsilon_{kin} \ r_np_l=\delta_{ij}r_np_n-r_jp_i. $$

Answer 2

You start with the Cauchy-Binet identity $$ \langle a\times b,c\times d\rangle=⟨a,c⟩⟨b,d⟩-⟨a,d⟩⟨b,c⟩ $$ and then insert the set of given vectors $r,p$ and two placeholders $u,v$ in two ways $$ ⟨u×r,p×v⟩=⟨u,p⟩⟨r,v⟩−⟨u,v⟩⟨r,p⟩,\\ ⟨p×u,v×r⟩=⟨p,v⟩⟨u,r⟩−⟨p,r⟩⟨u,v⟩. $$ Then in the difference one gets $$ ⟨u×r,p×v⟩-⟨p×u,v×r⟩=⟨u,p⟩⟨r,v⟩-⟨p,v⟩⟨u,r⟩=⟨p×r,u×v⟩. $$