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Lower bounds on the product of the variances of a function and of its Fourier transform are pretty standard in physics texts. See e.g. the discussion leading t

Lower bounds on the product of the variances of a function and of its Fourier transform are pretty standard in physics texts. See e.g. the discussion leading to Eq. 16.34 in Feynman's lectures of physics https://www.feynmanlectures.caltech.edu/III_16.html https://www.feynmanlectures.caltech.edu/III_16.html The weird thing is not that there is a relationship between a function and its Fourier transform. That's pretty "elementary" math, as you observe. The weird thing is that physically meaningful quantities such as position and momentum should be related via Fourier transform in the first place. No amount of math can prove this fact---you need experiments. By the way, in Heisenberg-type lower bounds, you allow both the function and its Fourier transform to be nonzero everywhere, yielding the result that the Gaussian distribution minimizes the product of the variances. But you can also ask a different question: assuming that I want the function to be band-limited, i.e., I want the Fourier transform to be strictly 0 outside an interval [-B, B], which function minimizes the dispersion in the time domain? This question yields the beautiful theory of the "prolate spheroidal functions", which are far from elementary. This kind of question is useful e.g. in signal processing: if you are allowed to look at N samples of an audio signal, what is the best low-pass filter that you can design?

graycat 5y ago
Thanks. And thanks for the URL to Feynman's lectures -- I lost my paper copy in a move. Yup, I understand that early on in quantum mechanics, we will get to both energy and momentum of these particles that have wave functions that evolve as in Schrõdinger's equation. Then I was wondering what math was going to take a wave function and pop out energy and momentum. I began to suspect something that smelled like it popped out of someone's back side. Good to hear that the theory was guessing and experiment confirmed, likely not the first time. You are farther into digital filtering than I ever got: For a while I was doing such things for the US Navy with data they collected in sea trials. At the time the fast Fourier transform (FFT) was a hot topic. Actually it was later that I studied Fourier theory with some care. I'm going after quantum mechanics just out of curiosity and with the basic assumption that it's not quite right and I want to check carefully. Maybe it's not really the final theory. Occasionally I get torqued: E.g., okay, sure, there is a Hilbert space (as I recall, Rudin shows that, really, there is only one) and each wave function is a point in that space, but no way will I easily accept that all the wave function FORM a Hilbert space: That is, via Rudin and more, a Hilbert space is a complete inner product space where here complete means that every Cauchy convergent sequence converges. Then I recall the common, old examples that nice, smooth, likely even infinitely differentiable, functions can converge to a square wave with its jump discontinuities. But the physics people assure me that each wave function is differentiable and also continuous. And that's a point of small irritation -- of course they are continuous; it's an elementary exercise to show that every differentiable function is continuous. And I got torqued at Feynman where in his Lectures he has that a particle of unknown position has position probability density uniform everywhere -- no it doesn't; it can't; there can be no such density since its integral would not be 1, actually either 0 or infinity. I don't even like the common integration from minus infinity to infinity: Rudin develops the Riemann integral very carefully but only on closed intervals of finite length, that is, on compact sets. Sure the integral from minus infinity to infinity can be defined as a limit, an improper integral, but then we have a problem: Start with some standard Rudin material that there can be an infinite series that does converge but is conditionally convergent and then with rearrangements can have the series converge to anything might want. Well, the same could hold for integrating from minus infinity to infinity -- the result get can depend on just how the limiting operation is done. E.g., integrate on Monday, Tuesday, and Wednesday, then on Saturday, Friday, Thursday, then Sunday, and continue this pattern for each week. So, without more assumptions, that improper integral is not so good. So, sure, measure theory and the Lebesgue integral clean up this mess, have some assumptions and derivations that do permit integrating from minus infinity to infinity. Right, I can be picky. Uh, who's to say that God is not?

graycat 5y ago

Thanks. And thanks for the URL to Feynman's lectures -- I lost my paper copy in a move. Yup, I understand that early on in quantum mechanics, we will get to both energy and momentum of these particles that have wave functions that evolve as in Schrõdinger's equation. Then I was wondering what math was going to take a wave function and pop out energy and momentum. I began to suspect something that smelled like it popped out of someone's back side. Good to hear that the theory was guessing and experiment confirmed, likely not the first time. You are farther into digital filtering than I ever got: For a while I was doing such things for the US Navy with data they collected in sea trials. At the time the fast Fourier transform (FFT) was a hot topic. Actually it was later that I studied Fourier theory with some care. I'm going after quantum mechanics just out of curiosity and with the basic assumption that it's not quite right and I want to check carefully. Maybe it's not really the final theory. Occasionally I get torqued: E.g., okay, sure, there is a Hilbert space (as I recall, Rudin shows that, really, there is only one) and each wave function is a point in that space, but no way will I easily accept that all the wave function FORM a Hilbert space: That is, via Rudin and more, a Hilbert space is a complete inner product space where here complete means that every Cauchy convergent sequence converges. Then I recall the common, old examples that nice, smooth, likely even infinitely differentiable, functions can converge to a square wave with its jump discontinuities. But the physics people assure me that each wave function is differentiable and also continuous. And that's a point of small irritation -- of course they are continuous; it's an elementary exercise to show that every differentiable function is continuous. And I got torqued at Feynman where in his Lectures he has that a particle of unknown position has position probability density uniform everywhere -- no it doesn't; it can't; there can be no such density since its integral would not be 1, actually either 0 or infinity. I don't even like the common integration from minus infinity to infinity: Rudin develops the Riemann integral very carefully but only on closed intervals of finite length, that is, on compact sets. Sure the integral from minus infinity to infinity can be defined as a limit, an improper integral, but then we have a problem: Start with some standard Rudin material that there can be an infinite series that does converge but is conditionally convergent and then with rearrangements can have the series converge to anything might want. Well, the same could hold for integrating from minus infinity to infinity -- the result get can depend on just how the limiting operation is done. E.g., integrate on Monday, Tuesday, and Wednesday, then on Saturday, Friday, Thursday, then Sunday, and continue this pattern for each week. So, without more assumptions, that improper integral is not so good. So, sure, measure theory and the Lebesgue integral clean up this mess, have some assumptions and derivations that do permit integrating from minus infinity to infinity. Right, I can be picky. Uh, who's to say that God is not?