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So far, I have seen two meanings of the Heisenberg Uncertainty principle through two derivations. One interpretation is that the observation of a particle requires the interaction of an electron which transfers momentum to the particle via the Compton effect. Using results from classical mechanics and optics, the uncertainty in the observed position and momentum resulting from an interaction with a single photon is:

$$\Delta p \Delta x\ \geq2h$$

As I understand it, this interpretation is that precise simultaneous observation of the momentum and position of a particle is impossible due to practical limitations.

Another interpretation takes the particle to be a wave packet and derives a relation of the max to half amplitude range in x and the range in reciprocal wavelengths of the constituent waves. This relation is:

$$\Delta x\Delta k\geq\frac{1}{4\pi}$$

Using results from Quantum Mechanics, it can be shown that:

$$\Delta p\Delta x\geq\frac{h}{4\pi}$$

As I understand it, this interpretation is that precise simultaneous observation of the momentum and position of a particle is meaningless as there is a range in the position of the particle within the packet and a range in the reciprocal wavelengths and therefore momenta which could result in this position.

But why must these results agree? For example, could we not live in a world where precise simultaneous measurement of the position and momentum is possible with the particle simply collapsing to a possible position in the packet (i.e., there really is a range of position in the usual sense of the word) so the measurement is useless as per interpretation 2? Or, could we not live in a world where the opposite was true, precise measurement was impossible but it was meaningful to define precise momentum and position?

I accept Physics simply “is the way it is” as far as scientists (and not philosophers) are concerned but this seems like too big of a coincidence to simply be a coincidence and I feel I’m lacking the intuition to understand what the uncertainty principle really means. Perhaps these interpretations are manifestations of a different and more fundamental interpretation (with a more fundamental derivation)?

physBa
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    The Uncertainty Principle isn't just about a limitation in our ability to measure what is going on in a particle. It means that the particle doesn't have a precise position or momentum. More on this here - Does the collapse of the wave function happen immediately everywhere? – mmesser314 Jul 04 '22 at 17:56
  • @mmesser314 what does it mean for a particle to not have a precise position or momentum? Does the wavefunction not collapse to a single point? Or "would" it collapse to a single point if not for the fact any interaction creates ambiguity in where this point is? – physBa Jul 04 '22 at 18:50
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    "Does the wavefunction not collapse to a single point?" Imagine one million completely identical copies of an experiment, where all particles are in exactly the same state (i.e. has exactly the same wave function). When measured, each individual particle will collapse to a single point, but the one million identical particles will not all collapse to the same point. This is surprising since they were in exactly the same state before the position was measured. – Marius Ladegård Meyer Jul 04 '22 at 19:06
  • @MariusLadegårdMeyer so the unmeasured particle has no exact position but at measurement it collapses to a point and then it does have an exact position? – physBa Jul 04 '22 at 19:08
  • Yes! But after collapse it is now in a different state. – Marius Ladegård Meyer Jul 04 '22 at 19:10
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    And the information in the wave function is precisely the probabilities for the different possible points it could collapse to. – Marius Ladegård Meyer Jul 04 '22 at 19:11

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