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Presumably, to know momentum accurately, one needs to know velocity accurately. So, we need to measure distance and time accurately. But, with instruments of GIVEN precision, we can measure velocity with ARBITRARILY high accuracy simply by measuring over greater distances (and times), because we would have greater resolution, as it were. The trade-off is that our highly accurate measurement for velocity (thus momentum) would be an average over a great distance, so the position is relatively low resolution. I'm wondering if this "classical" explanation is just an analogy, or if there is a underlying principle which is at work at the quantum level as well. The principle might be that if property A includes in its definition a change in property B, then they both cannot be known to arbitrarily high accuracy.

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Uncertainty Principle says it's impossible to DETERMINE with certainty both position and momentum, whether through direct or indirect measurements, or through theoretical calculation. One dimensional harmonic oscillator, for example, in classical mechanics if the position is determined, you could calculate the momentum without performing any measurement (assuming the initial conditions are known). But this is not the true in QM.

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    $\begingroup$ maybe one should add "due to the probabilistic nature of the solutions of quantum mechanical equations". $\endgroup$ – anna v Feb 9 '18 at 5:25
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    $\begingroup$ It's a yes-or-no question. $\endgroup$ – Tom B. Feb 9 '18 at 7:27

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