![]() ![]() Sun,“Uncertainty principles for Wigner-Ville distribution associated with the linear canonical transforms,” Abstract and Applied Analysis, vol. Rassias,“On the Heisenberg-Weyl inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol. Wang,“On signal moments and uncertainty relations associated with linear canonical transform,” Signal Processing, vol. Wolf, Integral transforms in science and engineering. Sheridan,“Fast numerical algorithm for the linear canonical transform,” Journal of the Optical Society of America A, vol. Ho,“New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms,” Signal Processing, vol. Wang,“Frames in linear canonical transform domain,” Acta Electronica Sinica, vol. (Using the DTFT with periodic data)It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence. Wang,“Multi-channel filter banks associated with linear canonical transform,” Signal Processing, vol. It completely describes the discrete-time Fourier transform (DTFT) of an -periodic sequence, which comprises only discrete frequency components. Collins,“Lens-system diffraction integral written in terms of matrix optics,” The Journal of the Optical Society of America, vol. ![]() Xu,“Three uncertainty relations for real signals associated with linear canonical transform,” IET Signal Processing, vol. Grafakos, Classical and modern Fourier analysis. Beckner,“Pitt’s inequality and the uncertainty principle,” Proceedings of the American Mathematical Society, vol. Beijing: Tsinghua University Press, 2009. Wang, Fractional Fourier transform and its applications. Smith,“Extensions of the Heisenberg-Weyl inequality,” International Journal of Mathematics and Mathematical Sciences, vol. Sitaram,“The uncertainty principle: A mathematical survey,” The Journal of Fourier Analysis and Applications, vol. Li, Linear canonical transform and its applications. Now, here's the fun part: you can show that the spacial distribution of protons in the nucleus is the Fourier transform of the momentum distribution.Īnd bingo, a measurement of the size of the nucleus.ĭo it with a polarized target and you can get info on the shape as well.T. ![]() (This is what nuclear physicists call "quasi-elastic scattering".) Now, if (1) we are shooting a beam of electrons at a stationary target, (2) we have a precision measurement of the momenta of the incident and scattered electrons and the ejected proton, (3) we are willing to neglect excitation energy of the remnant nucleus, and (4) we assume that $p$ mostly did not interact with the remnant after being scattered, we know the momentum of the proton inside the nucleus at the time it was struck.Ĭollect enough statistics on this and we have sampled the proton momentum distribution of the nucleus. Where $A$ represents that target nucleus and $B$ the remnant after we bounce a proton out. What we do is scatter things off of the component parts of the nucleus. Now, electron microscopy can just about provide vague picture of a medium or large atom as a out-of-focus ball, but there is no hope of employing that technique to something orders of magnitude smaller. What is the shape and size of a atomic nucleus?įrom Rutherford we learned that the nucleus is rather a lot smaller than the atom as a whole. Given that leftaroundabout and vonjd have addressed the fundamental place of the Fourier transform in the formalism, let me talk a little about an experimental application. ![]()
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