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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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Related: and links therein. – Qmechanic Mar 10 '14 at 14:02
@Qmechanic, thanks but I'm not talking about limits in our ability to measure things here. I just think the very definition of speed implies that you can't exactly measure it and position at the same time. – Massimo Mar 10 '14 at 14:06
@Massimo: so it doesn't seem normal to you that one can measure both the value of a curve $y = f(x)$ AND its tangent at its tangent? Is it because the problem is always presented as if we were doing a single measurement? What if one imagined simply sampling positions at regular time intervals such that Shannon's sampling theorem applies and THEN measure position and velocity at any point of the reconstructed trajectory with arbitrary accuracy? Just in case it is nor clear: even that strategy doesn't lead to statistically null variances in position and momentum. – gatsu Mar 10 '14 at 15:41

2 Answers 2

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

No, not necessarily.

in order to measure an object's speed, you need at least two measurements of its position at different times,

A police radar gun can be used to measure the speed of a object at a single point in time.

It can also be used to measure its position in space at the same time.

Using Heisenberg's Uncertainty Principle:

$$ \sigma_x \sigma_\rho \geq \frac{\hbar}{2} $$

Which leaves a minimum accuracy of speed of a one tonne car measured to $1 nm$ accuracy at $5 \times 10^{-29}ms^{-1}$. So classically for all intents and purposes one can measure a car to an arbitrary accuracy of both position and momentum at any point in time without invoking HUP.

This is made even easier when you assume that the measurement of position does not affect its position or momentum, which classically is true for the car, so you can measure them separately in any order.

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Well, of course you can measure both quantities at the same time, but only up to a certain degree of precision, however high. Never exactly. – Massimo Mar 10 '14 at 14:24
Of course, this becomes practically irrelevant for large (i.e. bigger than atoms) objects, just like most quantum physics. But it's the principle that counts. – Massimo Mar 10 '14 at 14:32
True, (your first comment) but that is the measurement error and it will exist for all measurements, the limit is not given classically except by the precision of the instruments. The HUP tells us that in the microcosm no matter how precisely we measure there exist pairs of variables, called conjugate, which are connected through the HUP indeterminancy, not measurement error. – anna v Mar 10 '14 at 14:33
I'm not talking about measuremt errors here. I'm talking about how speed is defined, as the variation of position over time; which implies that either you precisely know the position and thus can't know how it varies, or you precisely know its variation and thus you can't know its exact value. – Massimo Mar 10 '14 at 14:37
Of course what's intuitively obvious depends on your intuition, but it should at least be obvious that a delta-function looks nothing like a sine wave, and nearly obvious that it looks nothing like the sum of any small number of sine waves. So in that sense, yes, the qualitative outline of the uncertainty principle was certainly clear all along. It takes a little more work to quantify it. – WillO Mar 10 '14 at 14:39

"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?"

Yes, if you are talking about instantaneous speed. There is no speed at a point because the definition requires two points.

No, if you are talking about average speed, which unsurprisingly is what everyone means when they are talking about speed in the macroscopic regime.

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And thus, the very fact that speed needs to be measured as an average between two points explicitly forbids measuring it to an arbitrary level of precision; and this limit becomes increasingly relevant the smaller the objects and quantities involved become. – Massimo Mar 10 '14 at 14:45
Really, this limitation should have immediately become self-evident as soon as $\Delta t$ was used as a denominator... – Massimo Mar 10 '14 at 14:47

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