>>11779063The electron is not a tiny dot. An "electron cloud" is not a cloud of many tiny dots. Your premise of what a "quantum particle" is is fundamentally flawed. I don't mean that as an insult; it's not your fault that your high school chemistry teachers lied to you. In fact, they probably didn't really know the correct way either.
>>11779107>>11779217I often read explanations like these, and while there is a hint of truth in them, they are fundamentally founded on misconceptions themselves. Yes, measurement disturbs the state. But even if there were no projective measurement dynamics in quantum mechanics, there would still always be uncertainties.
The position and momentum observables do not commute. Full stop, there's your answer. The operator variance is 0 only when is an eigenstate of . If I have 2 noncommuting observables, , then they have no common eigenbasis, hence you will always have cases in which necessarily implies that .
Now, operationally, when I do an experiment, yes, I project the state into the eigenbasis of whichever observable corresponds to my "measurement method" (formally, a POVM). But the existence of nonzero operator variances does not at all depend on the fact that this is how quantum measurements work. While I wouldn't call it coincidence (there are indeed some deep connections), it's incredibly convenient that the Born rule allows us to use the statistics of our experimental data to exactly replicate the operator variance (in principle, of course).
