Comment by graycat | Hacker News Reader

Comment by graycat | Hacker News Reader

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THANKS!!! I keep trying to get a good understanding of at least the beginnings, i.e., foundations, of quantum mechanics and of the Heiserberg uncertainty principle, but I get frustrated at everything I can find on the Internet including the lectures of courses at MIT. So, for me, the physics is an obscure disaster.

But Fourier theory!!!! That's FINE!!! I went carefully through the very careful treatment of Fourier series in the third edition of the W. Rudin Principles of Mathematical Analysis and also his very careful treatment of the Fourier transform in his Real and Complex Analysis, also distributions in his Functional Analysis. Also read and used the Blackman and Tukey The Measurement of Power Spectra, wrote some corresponding software that pleased some people in the US Navy, etc. Then I took a course at the level of the second Rudin book, and in the class when the prof got to the Fourier transform one of the students, with a physics background blurted out "That's the Heisenberg uncertainty principle!"

So, now as I look at the MIT lectures, etc., I expect to see a good, solid, clear connection with the Fourier transform -- but so far I've found NOTHING.

Yup, so, thanks for confirming that, yup, the Heisenberg uncertainty principle is not some weird and obscure feature of nature but, really, just an elementary result of any elementary but precise math derivation of the Fourier transform!!!!

But, looks like I will have to make the connections with the basics of quantum mechanics myself.

Yup, multiply in the time domain is the same as doing a convolution in the frequency domain -- that is, if take the Fourier transform of a function zero outside of a finite interval, then in effect have multiplied the function by a box, 1 in the interval and 0 otherwise, and then for the transform are doing a convolution, that is, a weighted sum, of the transform of the box which is just a version of sin(x)/x, that is, with a peak at the origin and falling off rapidly away from the origin. That is, the weighted sum of the convolution is a "smearing out". So, on to the connection with the physics ..., hopefully.

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

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?

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?

If you think of the Fourier transform as letting you express a function as a sum of sine waves, then in quantum mechanics, you could also choose to express a wavefunction as a sum of energy levels—or as a sum of momentums, or a sum of positions.

https://en.wikipedia.org/wiki/Conjugate_variables

Thanks!

Yes, it has begun to dawn on me that since

d/dt e^(iwt) = iwe^(iwt)

(frequency w, time t) that

e^(iwt)

is both an eigenvector of simple versions of Schrödinger's equation and also, essentially via Fourier theory, one axis of an orthogonal basis of a linear algebra (or Hilbert space if prefer that since it permits infinitely many orthogonal basis vectors) representation of quite general wave function solutions to the equation. And just by including more basis vectors, can get whatever level of accuracy we might want.

That is, the terns in the Fourier transform act like the coefficients in the linear combination of basis vectors, the linear combination that approximates the wave function solution of Schrödinger's equation.

So, essentially we are into harmonic motion, e.g., swinging pendulums, whether we wanted to be or not.

Then we are ready to guess that for treatment of the electron in a hydrogen atom we will be into something like Sturm-Liouville theory of two point boundary value problems and standing waves, vibrating violin strings, etc. That is, an electron in what it does with a hydrogen nucleus will be like a vibrating violin string.

Thanks.

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