Im not sure what your point is (maybe I am just very bone-headed). I know how to expand the product of 2 sums. Thats not the problem. The problem is inserting 2 different expressions for the same wave function. Like putting it your way:
\Phi = \psi1 + \psi2 + \psi3...\psin
When calculating...
But why use 2 different sums for the same wave function? \Phi is equal to Ʃncnψn, not Ʃmcmψm. Or cleary it is, but then n = m?? Then why not just only use n in the first place?
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:
\Phi=Ʃn cnψn
If I want to find the expectation value represented by the operator O for the state described by \Phi, I would...