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I'm reading Hawking's A Briefer History of Time and it's explained that Heisenberg demonstrated his uncertainty principle based on fundamental limitations of positional meaurememt by basically bouncing photons off something and measuring how long they take to bounce back. (Maybe I didn't explain that quite right) This uncertainty is so great that to measure position down to the atomic scale would throw off our ability to measure velocity by kilometers. Or something like that.

Hearing this, my immediate thought was: Well wouldn't he have to prove that every other possible method of measuring position suffers the same limitation?

For example, the gravitational pull exerted by an atom is extremely small, and we don't fully understand gravity (haven't proven a theory of quantum gravity). To my knowledge, we haven't invented an instrument to measure the gravitational pull exerted by one particular atom upon another with great precision.

But in the future if we did invent such an instrument, we could, in the vacuum of space, measure the gravitational pull attributed to the atom who's position we desire to measure from three or more points in space, triangulating the measured pulls, and arrive at a precise position for said atom.

It seems to me that to prove the H.U.P. one would also need to prove the imprecision of measuring any other interactive forces between two particles, not just measurement of light. Since we don't have perfect theories of how all the forces work on quantum scales, such proofs, at least for now, seem impossible.

I assume I just misunderstand the problem.

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Any system that obeys the Schrodinger equation will be subject to a HUP. In particular, any two measurable variables (operators in QM) which are conjugate to one another will have their own UP. (e.g. position and momentum)

HUP can be proven mathematically within this framework, so the extent to which it applies to the universe is the extent to which the standard framework of QM applies. Of course there are shortcomings to the current model, but as a very well verified theory, it seems likely that there are HUPs that occur in nature. Of course, you point to gravity as a potential complication, and QM doesn't include gravity in a fully satisfactory way right now. But, in the example you give, the effect of gravity of one atom on another is so small compared to the other forces that it's fair to argue that 'physics is understood' in this case and that QM (or QFT) is the right approach.

I'm not sure if there's more to say than this. The idea of proving in an empirical way in which you seem to be demanding is unreasonable in my mind. In a strictly technical sense, it can only be proven mathematically within the context of the theory. QM is the best theory we have for certain circumstances, and in these circumstances, the HUP will apply.

If it's helpful to get some more intuition, the HUP relates to a more general mathematical phenomena that occurs between any two functions for which one is the Fourier transform of the other (known as Parseval's theorem). HUP can be seen as relating to the wave-like nature of the theory of QM.

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  • $\begingroup$ What I'm curious about is whether there exists a similar explanation for why it's true trying to measure position and velocity by measuring gravitational pull. Or other force interactions with the particle. And if that explanation exists, I'm curious to see it cited so I can understand. Based on your answer, to understand I would have to go learn quantum mechanics math. But Hawking was able to explain the limitation as it relates to light in just a few paragraphs. I'm looking for that kind of explanation, but say, for gravity. Or strong nuclear force. Etc. We could use any force to measure. $\endgroup$
    – john doe
    Jul 14, 2021 at 20:31
  • $\begingroup$ Ok, gotcha. Yeah I'm not aware of a source that does that. $\endgroup$
    – Ben Lansdell
    Jul 14, 2021 at 21:29
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    $\begingroup$ @johndoe To be clear, Hawking wasn't able to explain it in just a few paragraphs. Hawking told a story in just a few paragraphs. The story illustrates the fact that the HUP is not inconsistent with experience, even though it might seem inconsistent with naive extrapolations of everyday experience. But it's not an explanation. The only real explanation available today is the derivation from the principles of quantum theory, which is necessarily mathematical. $\endgroup$
    – Chiral Anomaly
    Jul 15, 2021 at 0:26
  • $\begingroup$ @ChiralAnomaly I research quantum mechanics every single week. I'm mot good at math, I've never had a college education and when I look at a quantum mechanics equation I see alien text. But I watch every PBS Spacetime video, every Anton Petrov video, and many more, read books by Penrose and Hawking all without ever understanding a single formula. One can understand complex quantum physics concepts without knowing any math at all. And plenty of people explain it in a visual way rather than as a formula. $\endgroup$
    – john doe
    Jul 15, 2021 at 1:41
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    $\begingroup$ @johndoe I applaud your curiosity! My earlier comment was meant to validate what you wrote in the question: "... wouldn't he have to prove that every other possible method of measuring position suffers the same limitation?" You're right. The number of methods of measurement is endless, so clearly we can't prove it by exhaustion. We can explore other methods of measurement to understand how the HUP works out in those specific cases, but that's still just (maybe compelling) evidence, not proof. $\endgroup$
    – Chiral Anomaly
    Jul 15, 2021 at 3:04

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