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Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the Hilbert state) matters? >What are some actual end-of-calculation quantum mechanical results, which are supposed to mirror some experimental values, where the constant $c$ in the explicit representation of the distribution $c\ \delta(x-y)$, which representas a sharp localization used in calculations, matters? If I write $c$ and I don't set it to 1, how does it enter the result in the end?

If you want to say the norm emerges as a limit of normed states, then justify the norm for the calculation to the point of extracting the specific physical values the theory is supposed to compute.

Also, I guess the "if you would position-localize" statement is a little handwaving, or does this have any practical parallel. (Given that it's not really possible, is it? Especially once relativity comes into play.)

I ask because on the one hand the momentum and position eigenstates/operator are the one you learn first, but in comparison to spins, there bring some mathematical difficulties. And in a quantum mechanics course, most example calculations will be about bounded problems. I guess there might be non-relativistic scattering processes where a delta is actually used to compute some scattering angle. But nonrelativistic scattering is eighter taught in mathematical physic courses, which focus on the structure of the theory (and I don't know practical physical results coming from there), or they are just the results obtained as classical limits of QFT calculations.

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Most expectations $\langle A\rangle:=\psi^* A\psi/\psi^*\psi$ are undefined in unnormalizable states. Thus the basis for doing statistics is gone.

In particular, in an (unnormalizable) position eigenstate, you cannot speak sensibly about any observable except functions of position and operators commuting with the position operator (such as spin). But once this is enough, one would describe the system in a position-independent way, and the psoition observable is completely eliminated.

An example is the position of the center of mass, which is eliminated by restricting to the rest frame, before bound state calculations can be done.

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