So I'm a little confused on the notation when working with wave functions constructed as a linear combination of an orthornormal basis set. Like on the form:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\Phi[/itex]=Ʃ_{n}c_{n}ψ_{n}

If I want to find the expectation value represented by the operator O for the state described by [itex]\Phi[/itex], I would calculate the inner product between [itex]\Phi[/itex] and O[itex]\Phi[/itex], like:

<[itex]\Phi[/itex]|O|[itex]\Phi[/itex]> = ∫dq [itex]\Phi[/itex]*(q)O[itex]\Phi[/itex](q) (assuming [itex]\Phi[/itex] is normalized so <[itex]\Phi[/itex]|[itex]\Phi[/itex]> = 1)

And now comes the question: When I insert the expanded wave function, why is 2 different indices used for the summations/basis functions:

<[itex]\Phi[/itex]|O|[itex]\Phi[/itex]> = ∫dq(Ʃ_{n}c_{n}*ψ_{n}*)O(Ʃ_{m}c_{m}ψ_{m})

This is how the derivations look like in most textbooks, and I dont understand the difference between n and m. I would think the indices should be the same, as it is the same wave function, [itex]\Phi[/itex].

Thanks in advance!

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# Expectation values for expanded wave functions

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