In 1923 Louis de Broglie proposed an idea that was as revolutionary as it was simple:1 that one can “associate a periodical phenomenon with any isolated portion of matter or energy” and that a fixed observer will associate with that phenomenon a wave of wavelength λdB = h/mv that scales with Planck’s constant h, the object’s mass m, and its velocity v.

De Broglie’s waves came as a surprise because they represent states of matter that seem to defy classical logic. For instance, the textbook example of electron diffraction at a double slit can only be explained by assuming a delocalized quantum wave in free propagation—even though the electrons themselves are detected as localized clicks. Because every single particle can be described as a sum of two or more waves whose centers may be clearly separated, it is tempting to speak of the...

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