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The uncertainty principle states that knowing the position of a particle with great accuracy would result in a very low certainty in the momentum of the particle.
Suppose I have an experiment where I keep measuring the position of the particle every nanosecond. Couldn't I deduce from all of these datapoints both the position and the velocity and therefore, the momentum?
Is this a contradiction with the uncertainty principle?
Every time you measure a particle, you change the wavefunction of the particle. Different measurement results will leave the particle in different states. The outcome of a measurement is random. This means that you have no guarantee of any repeated measurement being even remotely related to the original wavefunction, since measurement itself changes the wavefunction in a random way, and the more precise a measurement is, the more it can disturb the wavefunction.
There is some data that can be extracted from these kinds of measurements (for example, the expectation value of an observable), but you won't be able to break the Heisenberg Uncertainty Principle.
The uncertainty of the location would grow with the number of sampling points, given constant time distance. In this case, many data points could yield the momentum with greater (statistical) accuracy, but the time resp. location will lose its accuracy.
F.e., with 1000 samples the momentum could be calculated with a certain better accuracy. But the location resp. the time, when this momentum is valid, is smeared over the path of this particle within 1 microsecond. The changes in the momentum for a few nanoseconds are smoothed out to an average value valid for the total interval of 1 microsecond.
Heisenberg did not only derive the momentum-location uncertainty, but also the uncertainty for time-energy resp. time-frequency, since E=hf.
One and only 1 sample point in the time domain is equivalent to a Dirac impulse functional. It has the maximal accuracy in time, since it defines one single time point.
But in the frequency domain, this sample results in a constant, i.e. all frequencies are equally contained in this sample yielding maximal uncertainty.
In mathematical words: The Fourier or Laplace transform of a Dirac functional yields a constant - and vice versa, i.e a constant in the time domain yields a Dirac functional in the frequency domain.
If the number of sampling points in the time domain is increased, the uncertainty in the frequency domain would be decreased. Obviously the uncertainty in the time domain would be increased, the frequency/energy information would be no more able to be mapped to a single time point.
Caveat: The time point of a sample is - in theory, not in reality - smaller then the Planck time.
