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1. $$\int_a^b {| \psi(x) |^2 dx}$$

The above integral is said to give the probability of finding a particle between x=a and x=b

2. $$\int_{-\infty}^{+\infty} {x | \psi(x) |^2 dx}$$

The above integral is said to yield the expectation value for finding the particle at point x. The problem is: My quantum physics book and other sources say that this expectation value and expectation values in general are what tie the calculations to actual physical observables. That is a problem because, if the first integral that I listed gives the probability of finding the particle in a certain region, then wouldn't that integral be the one that connects the calculations to the physical observable of position (and not the expectation value integral)? In other words, what exactly does this expectation value tell you? My hypothesis about the difference between the two integrals is that the first one gives you the probability of finding the particle in set region while the expectation value tells you something about a specific point. If my hypothesis is correct, what exactly does it tell you about said point? In a thread I made once before, someone stated that it was not the probability of finding the particle at said point.

3. | \psi(x) |^2

My chemistry book says that the square of the magnitude of the wave function evaluated at x gives you the probability of finding the particle at point x. I, having already known about the first integral that I mentioned, felt that simply squaring the wave function is way too simple. If you can get precision to a single point by just squaring the wave function, then integrating over a region seems almost useless. Does the square of the magnitude of the wave function alone really give you the probability of finding the probability at a point, or do you think that my AP chemistry book was simply simplifying it for students who have not taken calculus? Also, if I am right in saying that this is an over simplification, then what exactly does plugging a point directly into a wave function tell you anyway?