My lecturer has written [itex]A | \alpha_n> = a_n |\alpha_n> => A = \sum_n a_n | \alpha_n>< \alpha_n |[/itex](adsbygoogle = window.adsbygoogle || []).push({});

and

[itex]B | \alpha_k> = b_k |\alpha_k> => B = \sum_k b_k | \alpha_k>< \alpha_k |[/itex]

Where A is a hermitian operator. I understand he's used the properties of the unitary projector operator here, but is ok to include the subscript n three times in a sum? It goes on to say:

[itex] AB = \sum_{n,k} a_n b_k |\alpha_n>< \alpha_n | \alpha_k >< \alpha_k| = \sum_n a_n b_n | \alpha_n>< \alpha_n| [/itex]

Now to me I can't see how the indices are correct there, since the middle [itex]< \alpha_n | \alpha_k > = \delta_{nk}[/itex] can only get rid of the b_k index and not the one in the vector as well.

Can anyone shed any light please?

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# Quantum Operators Indices Confusion

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