I believe the point of your problem is to prove that d^2/dx^2 is Hermitian. You only need to substitute and show that the difference <f|Lg> - <Lf|g> is zero using integrals. All functions in quantum physics should be square integrable because if they weren't, then the inner product, and hence...
The statement you can make about a wave function before measurement is that it is a superposition or sum of basis wave functions, such that when a measurement is made, that wave function collapses into one of the basis.
Another words it is a wave packet made of many waves each with its own...