Why do we need the position operator?

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Main Question or Discussion Point

As I understand it, |Ψ|2 gives us the probability density of the wavefunction, Ψ. And when we integrate it, we get the probability of finding the particle at whichever location we desire, as set by the limits of the integration. But when we use the position operator, we have integrand Ψ*xΨ dx, which gives us the probability of finding the particle at a given place.

It seems to me as though, both of the above operations are doing the exact same thing. Then why do we have a position operator?

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which gives us the probability of finding the particle at a given place
No. It gives an expectation value.

One is a probability density, the other is an expectation value. Different things ! Just like a Gauss distribution and the corresponding average are different things.

DarMM