Saturday, 18 November 2017

classical mechanics - Shouldn't the Uncertainty Principle be intuitively obvious, at least when talking about the position and momentum of an object?



Please forgive me if I'm wrong, as I have no formal physics training (apart from some in high school and personal reading), but there's something about Heisenberg's Uncertainty Principle that strikes me as quite obvious, and I find it strange that nobody thought about it before quantum mechanics development began, and still most people and texts explain it in quantum mechanics terms (such as citing wave/particle dualism, or the observer effect)... while actually it should appear blatantly obvious in classical mechanics too, at least regarding the position and momentum variables, due to the very definition of speed.


As everyone knows, the speed of an object is the variation of its position over an interval of time; in order to measure an object's speed, you need at least two measurements of its position at different times, and as much as you can minimize this time interval, this would always create an uncertainty on the object's position; even if the object was exactly in the same place at both times, and even if the time was a single nanosecond, this still wouldn't guarantee its speed is exactly zero, as it could have moved in the meantime.


If you, on the contrary, reduce the time interval to exactly zero and only measure the object's position at a specific time, you will know very precisely where the object is, but you will never be able to know where it came from and where it's going to, thus you will have no information at all about its speed.


So, shouldn't the inability to exactly measure the position and speed (and thus the momentum) of an object derive directly from the very definition of speed?




This line of reasoning could also be generalized to any couple of variables of which one is defined as a variation of the other over time; thus, the general principle should be:



You can't misure with complete accuracy both $x$ and $\frac{\Delta x}{\Delta t}$



For any possible two points in time, there will always be a (however small) time interval between them, and during that interval the value of $x$ could have changed in any way that the two consecutive measurements couldn't possibly show. Thus, there will always be a (however low) uncertainty for every physical quantity if you try to misure both its value and its variation over time. This is what should have been obvious from the beginning even in classical mechanics, yet nobody seem to have tought about it until the same conclusion was reached in quantum mechanics, for completely different reasons...





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