"Understanding Commuting Observables Proof

[dingo_d]

Homework Statement

In the proof that two observables $$\hat{O}$$ and $$\hat{O}'$$ commute iff they admit a common basis of eigenvectors, I'm not understanding one part.

Homework Equations

If $${|a_k\rangle}$$ is basis in Hilbert space we have:

$$(OO')_{jk}=\langle a_j|\hat{O}\hat{O}'|a_k\rangle=\sum_n\langle a_j|\hat{O}|a_n\rangle\langle a_n|\hat{O}'|a_k\rangle=\sum_n\hat{O}_{kn}\hat{O}'_{nj}$$


Now I'm confused a bit. And maybe I'm not understanding the matter that well (I'm only starting to study QM), but why is the last part

$$\sum_n\hat{O}_{kn}\hat{O}'_{nj}$$ and not $$\sum_n\hat{O}_{jn}\hat{O}'_{nk}$$?

I mean it 'feels' to me that the latter should be true, because it is in the brackets (I mean the indices)...

 

[vela]

You're right. The indices are messed up in the last step.

 

[dingo_d]

Oh, so it's a typing error... Well that's yay! for me (this time my intuition wasn't wrong ^^)