I think the double slit experiment is a really good one. It is simple enough to set up, but it really gets to the heart of what quantum mechanics is about. The point is that to get the interference pattern on the screen, the light has to go through both slits at once. The experiment can also be done with single electrons, but that is harder.
There is a lot of depth to understand in this experiment.
another good example of quantum behaviour is the photoelectric effect. It involves shining a light source (normally a UV light) onto some material and detecting the electrons that are emitted. I think this might be on the GCSE or A-level syllabus so your science teacher might have a demonstration in the lab.
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