4 ms·

I think that's not quite it. The relevant theorem is that variance(position-space function f) * variance(Fourier transform of f) >= some constant. A wavefunct

I think that's not quite it. The relevant theorem is that variance(position-space function f) * variance(Fourier transform of f) >= some constant. A wavefunction is a wavefunction, it doesn't have more frequency components at the classical scale. It's just that what looks like a large position spread at the quantum scale is pretty tiny compared to classical length scales, and what looks like a large momentum spread (variance of the Fourier transform) is pretty tiny compared to classical-scale momenta. You shouldn't think of a wavefunction as being perfectly localized, with infinitely spread out frequency components, because that would mean the particle's momentum is infinitely uncertain. Instead, think of a Gaussian function, whose Fourier transform is a Gaussian. The widths of those two Gaussians are inversely proportional to each other. Also, you should think about continuous Fourier transforms, not discrete Fourier series. Periodic wavefunctions are only the norm in situations like crystals where the environment itself is periodic.

xeonmc 4y ago
Isn't continuous Fourier Transform simply the limit of Fourier series as the fundamental frequency approaches zero though? That's what I meant when I said "all of free space", it's the same idea with the emergence of "continuous" band structures in bulk materials -- although their sizes aren't strictly infinite, for all practical purposes it's long enough that the energy levels are sufficiently close-together to be considered a continuum. And yes, I stand corrected. My analogy with overtones bandwidth vs spatial localization instead pertains to the position-momentum tradeoff, it is indeed incorrect to overgeneralize it to what the classical limit means. The classical limit, as you correctly pointed out, is more about the tradeoff between the two being practically negligible in the length/impulse scale being dealt with in the classical regime -- effectively a "rounding error", so to speak.

xeonmc 4y ago

Isn't continuous Fourier Transform simply the limit of Fourier series as the fundamental frequency approaches zero though? That's what I meant when I said "all of free space", it's the same idea with the emergence of "continuous" band structures in bulk materials -- although their sizes aren't strictly infinite, for all practical purposes it's long enough that the energy levels are sufficiently close-together to be considered a continuum. And yes, I stand corrected. My analogy with overtones bandwidth vs spatial localization instead pertains to the position-momentum tradeoff, it is indeed incorrect to overgeneralize it to what the classical limit means. The classical limit, as you correctly pointed out, is more about the tradeoff between the two being practically negligible in the length/impulse scale being dealt with in the classical regime -- effectively a "rounding error", so to speak.