Comments by "Lepi Doptera" (@lepidoptera9337) on "The more general uncertainty principle, regarding Fourier transforms" video.

Light is a quantum mechanical phenomenon, so you are doing QM here whether you like it or not. It just happens that the uncertainty principle has nothing to do with quantum mechanics. It's a general property of functions and linear operators, no matter what they are describing. Something like a Gaussian does, of course, have a unique parametrization and we can define "the center" in a unique way. That is simply not the case for arbitrary functions. Where is the center of a plane wave, for instance? The entire notion of it having a beginning, a center and an end is useless. What's the physical meaning of the center of a hollow sphere? There is no physical interaction at "the center", it's empty. All the physical interaction would be with the shell, even in classical physics. One simply has to learn to live with the fact that "center of mass" is a very, very limited concept for a very limited class of problems, classical or not.
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Relativity. E^2=(pc)^2 + (m0c^2)^2 or, to write this the way physicists who use c=1 would write it: E^2=p^2+m0^2. In quantum mechanics we are concerned about quanta of energy and that's the general relationship between rest mass, momentum and energy.
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The uncertainty principle is a fairly trivial mathematical lemma about certain types of linear operators. There is nothing really difficult or exciting about it. The mathematical proof takes a quarter page or something, at least for L2 functions. One can prove it with some more effort for more general cases. That's a common property of functional analysis lemmata. Many of them are quite trivial (at least for a mathematician) for well behaved functions and have technically difficult to prove analogs for less well behaved functions. Physicists are almost always working in L2 for these kinds of problems, so the visualization for what is happening is lengthier than the actual proof that it can't be any other way. Mathematicians, on the other hand, are almost universally interested in "strange functions" for which these relationships are much, much harder to establish. The "glorification" of Heisenberg's uncertainty relation in particular is therefor more of an educational side effect that stems from a lack of modern calculus teaching in high school. We could include the more trivial aspects of functional analysis (like this result) at the high school level, if we wanted to. Unfortunately our high school curriculum is stuck in the late 19th century and does not want to come into the light of the 21st.
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@bakersmileyface I didn't say that I can do it without looking it up or without thinking about it for a really long time. I am not a mathematician and I haven't done this stuff in forty years. What I am saying is that I know that the average high schooler can be taught how this stuff works with the proper teaching method. Our schools are, unfortunately, not well prepared to teach students modern material. The way we teach calculus is roughly from around the 1860s (epsilontics, Weierstrass) or earlier (inifnitesimals, 1670, Mercator, Leibniz) and it is very pedestrian. These things here are from the 1920s. We are another 100 years later, so why are we still teaching like it's the baroque period? Shall we also hand out wigs and powder to the students? :-)
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Yes, it would be if you are approaching it from the mathematical side, which tells you very little about physics. I would suggest you try to understand the physics, first, before you learn the math. That makes it a lot clearer what is going on. Quantum mechanics is about quanta of energy that get exchanged between systems. It is NOT about objects. It is NOT about particles and it is NOT about waves. When systems exchange quanta of energy, they also exchange momentum, angular momentum and charges like electric charge or leptonic charge. Energy and momentum are not quantized, but angular momentum and charges are. They only come in integer multiples of a minimal quantity (usually expressed in Planck units and electron charge, but these are arbitrary numbers due to the choice of our units - in rational units they should be set equal to 1). In an experiment the detection of one quantum is basically one "click" of a detector. In the ideal scenario such a detector can tell us where and when a quantum was detected and how much energy, momentum, angular momentum and charge it had. A single detection carries basically no relevant physical information about the systems that emitted these quanta, so we have to bunch an infinite number of them into a quantum mechanical ensemble and then we can form frequentist counts aka histograms aka probability distributions with them. Because we are now working in an ensemble theory, we can assume that each member of the ensemble is isolated and that means that sub-ensembles are statistically independent of each other. This leads to the usual Kolmogorov axioms for probability theory. It turns out that these axioms for statistically independent ensembles can also be satisfied with complex and quaternion based functions (and probably product algebras built from these basic elements, but that's less often used). The resulting functions are normalized elements in Hilbert spaces (normalized because the number of members in an ensemble does not change during the evolution of the ensemble), which basically leads to unitary dynamics, i.e. rotations in finite and infinite dimensional linear spaces. In the trivial finite dimensional case these rotations can be described by Heisenberg's unitary matrices, in the infinite dimensional case we need linear operators and partial differential equations. And that is basically what quantum mechanics is: the theory of unitary rotations, except that there is not much physics in that because that part is basically a simplifying assumption (that all the copies of the system we start out with are still there when we detect the results). In experiments that's far from true. We actually lose most quanta of energy in most of our experiments. The experimentalists are simply fudging that loss with an arbitrary (experimentally measured) number that we call "quantum efficiency" and the theorists never get to see that fudge, so many of them think that "unitarity" is some god-like property of quantum physics. It's nothing like that. It's just what's left over after the experimentalists have cleaned the data. If you keep that in mind you will start to understand why the mathematical structure of quantum mechanics is so meaningless. It's an artificial construct, not a property of observational reality. It has that in common with Hamiltonian mechanics, but that's another rant. ;-)
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