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**Determining the commutation relation of operators -- Einstein summation notation**

## Homework Statement

Determine the commutator [itex] [L_i, C_j] [/itex].

## Homework Equations

[itex]L_i = \epsilon_{ijk}r_j p_k[/itex]

[itex]C_i = \epsilon_{ijk}A_j B_k[/itex]

[itex] [L_i, A_j] = i \hbar \epsilon_{ijk} A_k[/itex]

[itex] [L_i, B_j] = i \hbar \epsilon_{ijk} B_k[/itex]

## The Attempt at a Solution

To be clear, the goal of this procedure is to become familiar with Einstein summation notation. That said, I've broken open [itex]C_j = \epsilon_{jmn}A_m B_n[/itex] and expanded the commutator accordingly. My problem is opening up [itex]L_i[/itex] in a meaningful way that gives me a nice identity with four deltas. Will I have to expand two epsilons with 6 different indices into deltas, or is there any way to get the epsilons to share an index that will give me the result (which I know to be [itex] [L_i, C_j] = i \hbar \epsilon_{ijk}C_k[/itex])?