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I remember seeing multiple sources (don't ask me to recall them) remarking on their surprise that Fourier didn't arrive at some version of an uncertainty principle, given how naturally it arises from the transforms.
There is some indirect evidence* that Sommerfeld was onto this relationship as early as 1916, which he arrived to by applying Fourier-type reasoning to Bohr's model.
*B. Bayman** lectures, Grad Classical Physics course, U of MN, '98/99.
(**BB was Wigner's postdoc?)
@phalpern?
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