The "central mystery" of quantum mechanics is the wave-particle duality, as exemplified by the famous double-slit experiment.
In this idealized Demonstration, a special laser (not yet on the market!) can be tuned to any wavelength of visible light and triggered to emit pulses of up to 3500 photons. Small numbers of photons (or individual photons), after passing through the slits, produce scintillations at apparently random points on the screen. But as the number of photons per pulse is increased, a pattern of light and dark bands gradually emerges, with spacings determined by the wavelength and slit separation, which you can control with sliders.
The intensities, in fact, trace out a classical diffraction pattern, first observed in Young's double-slit experiment. The trajectories of individual photons cannot be predicted. But each photon somehow "knows" about the entire diffraction pattern, which is exhibited only in the statistical behavior of a large number of photons. Remarkably, electrons and other particles behave analogously, in accord with their de Broglie wavelengths. This constitutes the fundamental basis for the quantum theory of matter.
