wave-particle duality explained
A single electron, or a single grain of light, behaves like a wave spread wide when nobody looks, and becomes a particle that lands on one point the moment it is detected. The experiment that shows both faces on one screen is the double slit. Particles arrive one by one and pile up as dots, and when those dots gather, bright and dark stripes appear. The more the dots gather, the more the wave pattern shows.
Each dot is the mark of a particle arriving at one place, but the pattern of the whole pile is the interference of a wave. Narrow the slit spacing d and the stripe spacing widens, and that relation is Δy = λL/d. λ is the de Broglie wavelength λ = h/p, the distance from one crest to the next, and the wave number is k = 2π/λ. Send the particles one at a time, however sparsely, and the same stripes still pile up, so each particle interferes with itself rather than with another. Each press of fire drops a cluster of dots along the interference strength. The place where the dots gather is what builds the stripes, and the stripes stay even when the arrivals are sparse.
Even with a single slit the wave nature stays. Light or an electron passing a narrow gap spreads into one wide central band, and that shape is a sinc² envelope. Narrow the gap and the spread grows wider still. Narrowing the position blurs the momentum, which is one face of the uncertainty principle, and it is also the diffraction limit that caps the resolution of every lens and microscope. Move the slit width and a narrower gap spreads even wider. The trade, that a narrower gap spreads wider, is the heart of this picture, and it sits under every instrument that tries to resolve a small feature.
What happens if a path detector is switched on to learn which slit was used? With the detector off the stripes stay alive and the thing looks like a wave, and with it on the stripes vanish and only flat bands remain. The detector does not do this by shoving the particle roughly. Learning which path was taken is itself what erases the interference, and wiping that path information later brings the stripes back, which experiments have confirmed. Flip the toggle ON/OFF and the act of observing itself erases the interference. Knowing the path is what wipes the pattern, and forgetting that path is what brings the pattern back.
Before detection the particle is not a point. It travels as a wave packet, a lump of wave gathered tight. As time passes and the distance grows, the lump spreads sideways and its height drops. It spreads because the wavelength pieces inside it travel at slightly different speeds, so the position of a free particle grows less certain as time goes on. Raise the travel distance and the packet shows how it widens and blurs. The farther it travels, the more the lump blurs, and that blur is the position growing less sure.
The two faces are complementary pictures of the same quantum. Put the toggle toward the wave and a smooth wave appears. Put it toward the particle and it becomes one point carrying a momentum arrow. Both pictures are needed, yet they never appear together at one moment, and that is complementarity. Which face shows up depends not on the object itself but on which experiment we choose, and Niels Bohr set that thought down as the complementarity principle. Walk the toggle and the two sides of the same object take turns. One face at a time.