When we observe a quantum object does it collapse into a point? Or does it collapse into a smaller wave of area that is out of our range of accuracy? My gut tells me the latter.
Answer
Indeed, the latter. What you measure is actually not a pure projection operator $|x\rangle \langle x|$ but something more smudged like $M_x = \int dy \, p_y |y\rangle \langle y|$. Further general measurements need not be matrices, they need to be linear operators, the most general physical form of which are completely positive maps.
This means there is some probability that your measurement assigned value $x$ to the position of an object which was actually at a different location. Operators of this form will not in general collapse the wavefunction into a point. But if you are not careful the information you did not collect will be erased by interaction with the environment, which can make your system nearly classical.
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