The more general uncertainty principle, regarding Fourier transforms

The reason Trek has it is likely a common misunderstanding of QM: it's not only that we cannot record both position and momentum, the information does not exist in the first place . It's easier to see why if you think about Fourier transforms: ▶ this video TLDW: an infinitely long wave does not (cannot) have any definite location, but it does have a definite periodic wavelength; conversely, a single impulse noise (a shockwave from e.g. a bullet or an explosion) has a definite location (in the direction of motion and at any given point in time) but no meaningful wavelength. The more you constrain the possibility space of one, the looser the other becomes in a physical sense, not just the information you have about it .

The clearest, most intuitive explanation of the Fourier Uncertainty Principle that I have come across is by Grant Sanderson of 3Blue1Brown. ▶ this video

"The more general uncertainty principle, regarding Fourier transforms" by 3blue1brown ▶ this video From link : > Convolution is in fact multiplication in Fourier space (this is the convolution theorem [1]) which says that Fourier transforms convert convolutions to products. 1. link : >> In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. "QFT and iQFT; Inverted Quantum Fourier Transform" link

Yes, but that's because the momentum is the fourier transform of the position and vice versa So there's an uncertainty principle for signal processing, entirely because of the fourier transform and not because of quantum mechanics See this video ▶ this video

This may help answer some of your question, though I recognize it does not address it directly: ▶ this video (3b1b on the Heisenberg uncertainty principle and how it arises) Think of a free neutron as the wave envelope, rather than any of the bumps in the wave.

The uncertainty is not exclusive to QM and is a property of any ‘wavy’ system, it seems. It is inherent. Here is the example: ▶ this video — if you want to defeat QM, attack its wave-part. It would be nice to have much better (deterministic, measurable) physics, but some properties just emerge from a fundamental level and you can’t do much about it (monster group etc). Nature and maths don’t care about our confusion and convenience. >should have refuted it and looked for something better instead Afaik, the time when theories were driven by ideas is over. Today it’s petabytes of data that you can’t really argue with over a mailbox, only discuss. (I’m just a layman physicist with deep interest, but shallow understanding, so please don’t quote me on any of this)

In many cases, Wikipedia actually lists the analogy for classical probability theory. For several theorems, the proof for Hilbert spaces can be directly brought to the classical world. The main difference to keep in mind is that we have amplitudes, not probabilities, and they can be negative or complex. We recover typical probability theory by taking norms/squares of amplitudes. The uncertainty principle follows via doing signal analysis on Fourier-transformed wavefunctions. A fun professional treatment from Baez et al. is [0], and 3blue1brown has an excellent two-video visual explanation [1][2]. The uncertainty principle turns out to be a special case of the sampling theorem (yes, that one! [3]), which itself turns out to be a special case of a result in sheaf theory [4]. Measurements matter because "observables don't commute"; taking linear operators on the complex numbers or other Hilbert spaces can have lasting effects which can't easily be undone. Combine this with "conservation of probability", which is formally mostly abstract nonsense [5], and we get mostly to Aaronson's point of view. (I would go further using the Free Will Theorem. [6]) When Aaronson says that the no-cloning principle is provable using probability, but for complex numbers, he's referring to the standard proof [7]. There are two connections to draw to typical probability theory. The first, and bigger, connection is that random variables can't be cloned in typical probability theory either! The second

It is a neat trick actually. If you superimpose a few of these waves with frequencies that are fairly close, then you will see how they cancel out in the distance, while they reinforce each other in the center. This is called a "wave packet". 3blue1brown was somewhat useful, somewhat tangential video on the topic ▶ this video The more rigorous way to say "superimpose a few of these waves", is to say that there is some uncertainty in the frequency. This uncertainty in the frequency lets you have more certainty about the position (the wave packet is centered somewhere, instead of being completely distributed the way an infinite plane wave is)

The limitations of the Fourier transform are intimately tied to the uncertainty principle: ▶ this video Reality does appear to be limited by the limitations of the Fourier transform.

The 'measuring something disturbs it' idea described above is called the observer effect link The observer effect is a real thing, and even has interesting effects in quantum mechanics, e.g. the quantum zeno effect link Yet the observer effect is not the reason why we can't know an object's position and velocity at the same time. There are two ways we can see that this supposed explanation is a red herring: - We don't need to interact with (e.g. 'bounce a photon off') a quantum system in order to observe it ( link ). I particularly like the "quantum bomb detector" ( link ), which can tells us whether a photon detector connected to a bomb is working or not, without hitting it with a photon and hence triggering the bomb (50% of the time, at least). - The actual reason, the uncertainty principle ( link ), doesn't require any notion of observation at all, let alone interaction. It's a simple property of waves (the relationship between duration and frequency). 3blue1brown did a nice video on this ▶ this video

3Blue1Brown has an extremely good explanation[1] of the intrinsic uncertainty, and why it's separate from measurement uncertainty. (the previous episode[2] is a recommended prerequisite for background on how the Fourier Transform works) > emerges through a different, as yet unknown, mechanism. In 3Blue1Bron's explanation[1], he shows how the intrinsic uncertainty is an inherent trade-off of trying to measure both position and frequency. A short wave packet only a few wavelengths long correlates with a narrow (precise) range of positions, but also correlates well with a very wide range of frequencies due. A Heisenberg-like uncertainty exists any time you are working with weave packets with length near the wavelength. 3Blue1Brown gives a very good example using Doppler radar. [1] ▶ this video [2] link