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To add: we can also apply Fourier transforms to probability measures, giving "characteristic functions", not only to time-series. It takes more than that to get to QM and ℏ, but it's a helpful first step.
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Maybe it is worth pointing out that there are three distinctly different uncertainty principles: (i) Bounds to second moments in measurements of pairs of observables performed on iid copies of a state.
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Many quantum properties can be brought back to just a few, very precise statements, such as the Uncertainty Principle being a simply consequence from the wave-particle duality. More educators should stress this, levying QM from its "weirdness".
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