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 Quote by Born2bwire The Heisenberg uncertainty principle is a statistical relationship. It says nothing about the results of an individual measurement. Instead, it is talking about the statistical spread, the standard deviation, in the measured data that you would get should you do an ensemble of measurements. That is, if I have a 1000 jars with an identical experiment setup inside them. Let's say I have a photon emitter and a wall of tiny detectors on the other side. I measure the energy/momentum of the photon by the spectral emmission and plot it's endpoint by which detector gets set off. I make these measurements on these 1000 independent but identical jars. Then I look at the results. What I find is that the detector and spectral emmission will have some variance in the results. The product of the statistical deviations in these results must satisfy the Heisenberg uncertainty principle.
But the principle argues that these uncertainties are inherit in nature, not in the precision of the experiment. The way I understood it was that the principle would allow the uncertainty of either position or momentum to become arbitrarily small, but the effect would be that the other variable would become arbitrarily large. I guess my question really asks: How can the other variable become arbitarily large when it is limited by C?

Imagine that the momentum is measured so precisely that the principle implies an uncertainty in the position of 10 light years. But if the time it took to measure the momentum was half a year (very large jars), how can the position at the time of measurement possibly be uncertain to 10 light years? You already know that the position of the particle at the time of measurement must have been within 0.5 light years of your detector, otherwise you would have never been able to measure the momentum in the first place!

I think how I derive it from a thought experiment is irrelevant and will only take the discussion of track. The experiment is Heisenbergs microscope. Examples are all over the net.