"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 ^^)