My question is clearly related to this one, but my interest is not specifically in Heisenberg's result. To quote from Wikipedia.

A nonzero function and its Fourier transform cannot both be sharply localized at the same time. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave.

Who first explicitly notes the fact stated in the first sentence (A nonzero function and its Fourier transform cannot both be sharply localized at the same time), and who was it that found that the Gaussian distribution makes the trade-off equal in the frequency and time domains?

  • $\begingroup$ I've linked back here in How close to reality is my handwaving argument about Fabry Perot interferometers? $\endgroup$
    – uhoh
    Nov 30, 2023 at 22:27
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    $\begingroup$ Wiener states in his autobiography that he made remarks to this effect in a 1925 Göttingen lecture attended by Born and Heisenberg, illustrating them with pitch/duration precision tradeoff in music, see Folland-Sitaram. The first proof of the inequality for variances is due to Kennard in 1927 (motivated by Heisenberg's heuristic application to QM, for Gaussian packets only). That the equality is attained for Gaussian densities was first proved by Hardy in 1933, see Fernández-Malinnikova. $\endgroup$
    – Conifold
    Dec 1, 2023 at 12:00

2 Answers 2


It seems that Heisenberg was the first to notice this phenomenon in 1927. He did not use the terminology of Fourier transform. Heisenberg's principle was rigorously stated and proved by Kennard in 1927 and Weyl in 1928 (see the references in Wikipedia).

The first formulation in pure mathematical literature is due to G. H. Hardy,

Hardy cites an unpublished remark by N. Wiener: "a pair of transforms and cannot both be very small", and does not mention Heisenberg. However, mathematical formulation of Hardy is different from Heisenberg's. Since then many results in the spirit of Wiener's principle were proven. See, for example the book by Khavin and Joricke, The Uncertainty Principle in Harmonic Analysis, Springer , 1994.

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    $\begingroup$ Interestingly, I discovered Dennis Gabor's 1946 paper on the Theory of Communication in which he develops the same result but is clearly unaware of Hardy. $\endgroup$ Dec 28, 2023 at 11:18
  • $\begingroup$ For a beautiful, new, and unifying view about many uncertainty principles, see the paper, The uncertainty principle: variations on a theme by Yuri Wigderson and Avi Wigderson, available at arxiv.org/abs/2006.11206 , $\endgroup$ Jan 9 at 9:58

Küpfmüller's uncertainty principle based on rectangular functions as bandlimiting frequency filters appeared already 1924, years before Heisenberg's.


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