The simplest illustration of it is the famous double-slit experiment. This involves shining a laser at an object with two tiny slits in it, very close together. The light waves passing through each slit interfere with each other and form a characteristic pattern, consisting of a series of dots that receive a lot of light, with the gaps between the dots getting very little light. The weird part is that this pattern still shows up if you just fire one individual photon at the slits at a time – it’s more likely to hit the places where those dots were, and less likely to hit the gaps between the dots. Effectively the photon behaves as if it passes through both the slits and interferes with itself.
All kinds of variations of this experiment have been carried out over the decades, including ones in which the laser is replaced with a stream of electrons, atoms, or even fairly large molecules, and they all show the same behaviour.
The only coherent way to understand experiments like this is to model fundamental particles as if, instead of being in a single state, they are in a “superposition” of different states simultaneously. For example, at a given time, an electron does not have a single location – it is effectively smeared out across a region of space. This all sounds vague, but it has a concrete mathematical description and has been tested with all kinds of experiments. The only big remaining mystery is the so-called “measurement problem”, which is essentially the question of how exactly our large-scale world, in which things seem to have defininte locations, is built up out of quantum systems in which everything is in a superposition of different states. There are various proposed answers, but it has turned out to be very difficult to do experiments to probe this question.
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