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With many sources on the internet it has sort of become diluted as to how and why the Heisenberg uncertainty principle still makes sense. One claims that it is due to quantum superposition and that a particle can have multiple positions and velocities at the same time while other sources claim that this happens due to the fact that we cannot pinpointed the exact position of the particle as we would need to increase the frequency of the electromagnetic wave in order to see it, but that this will cause it to alter it and some others claim that the principle has nothing to do with measurement. So are any of the right and how?

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One claims that it is due to quantum superposition and that a particle can have multiple positions and velocities at the same time.

This is not the HUP. This just tells you why there is quantum uncertainty at all. i.e. because of quantum superposition, in general if you were to measure some observable across many similarly prepared states, you would get a spread in the measurements. You would not get the same measurement each time. What the HUP (or any uncertainty relation) does is tells you how the product of these spreads must be limited for certain pairs of observables. You do not need to invoke pairs of observables or the HUP to discuss superposition, and superposition is needed for the HUP just because it tells you why there are any spreads in the first place for the HUP (or any uncertainty principle) to be about in the first place.

As an opinion of mine with this point, I really do not like explaining superposition as "the particle has multiple positions and velocities at the same time." Having something that tells you about the potential outcomes of a measurement does not mean the particle simultaneously has all of those outcomes before the measurement. It is a decent explanation for the very shallow descriptions needed for the lay person, but as soon as you start going any deeper into QM this explanation seems nonsensical (at least to me).

while other sources claim that this happens due to the fact that we cannot pinpointed the exact position of the particle as we would need to increase the frequency of the electromagnetic wave in order to see it, but that this will cause it to alter it

While this was an (the?) original argument / motivation for the HUP, you can show that the HUP arises directly from the axioms of QM. In other words, you don't need to bring in "disturbing the system" to explain the HUP.

and some others claim that the principle has nothing to do with measurement.

Go with this one. The HUP is not a statement of "the particle has a definite position and a definite momentum, we just don't know what it is." The HUP just sets a limit on the product of the uncertainty (spread) of measurements of position and momentum for similarly prepared systems. In other words, it is something tells us what to expect of measurements of our system, not something that tells us what the measurements do to the system.

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Sub-atomic particles are (or are associated with) wave packets of finite size. Fourier analysis tells us that uncertainty is a property of wave packets. If the packet is long, you can get a reasonably accurate measurement of wavelength, frequency, (and energy). If it is short (or confined) you cannot.

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  • $\begingroup$ Yeah but the HUP is not really related to Fourier analysis: Fourier-type relations are classical, not quantum in nature. $\endgroup$ – ZeroTheHero Apr 21 '20 at 17:15
  • $\begingroup$ Gee, I have a book on quantum mechanics, and its all wave equations. $\endgroup$ – R.W. Bird Apr 21 '20 at 18:27
  • $\begingroup$ @ZeroTheHero How do 'classical' Fourier-type relations differ from the quantum $\hat{x}-\hat{p}$ relation? $\endgroup$ – Nihar Karve Oct 30 '20 at 14:11
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    $\begingroup$ @NiharKarve The Fourier relations do not depend on a commutator. Moreover whereas the HUP is a strict lower bound $\Delta A\Delta B \ge \frac{1}{2}\vert \langle [A,B]\rangle\vert$ the Fourier ones are approximates $\Delta A\Delta B\sim $(something). $\endgroup$ – ZeroTheHero Oct 30 '20 at 17:31
  • $\begingroup$ @ZeroTheHero but defining $\Delta f$ as $\int x^2 |f(x)|^2 \ dx$, isn't $\Delta f \Delta\hat{f} \ge \frac{1}{16\pi^2}$ (along with some mild assumptions) similar? $\endgroup$ – Nihar Karve Oct 30 '20 at 18:09
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Heisenberg himself believed for a long time that it was purely a measurement problem, that in measuring a particle's properties we inevitably disturbed them.

Others argued that it was a fundamental quantum phenomenon. The principle of complementarity was discovered, in which properties come in complementary pairs and the more precisely we confine one the less precise the other becomes.

Ultimately, the quantum weirdness folk won the day in the laboratory and Heisenberg came round. When you constrain a quantum "particle" in one way it really does squish out sideways in wildly fuzzy quantum-statistical fashion. But you will still find quantum realists and hidden-variable theorists around and of course the Internet loves them.

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