You flip a coin and it comes up heads, you’re not surprised. You flip a coin and it comes up tails, you’re not surprised. You flip a coin and it comes up both heads and tails simultaneously and you suddenly realize that the coin isn’t what you thought it was. In this analogy heads is “photons are waves” and tails is “photons are particles.”
The actual experiment involves taking two narrow slits near together and firing photons through them. If they make two distinct patches on your measuring screen, then they’re particles; if they make an interference pattern then they’re waves. They make an interference pattern. Neat, photons are waves! Easy enough. Then you find a way to fire photons one at a time through the slit so they can’t possibly interfere with each other… and somehow they still make an interference pattern. This makes no sense if photons are waves *or* particles, but it also doesn’t make any sense (according to your current model) for them to be *both*. So the only thing you can conclude is that they’re something else entirely, something that isn’t a particle or a wave but can act like both of those things. And you have no idea what that might be.
We’ve been studying them for a long time after that so now we have *some* idea but it’s still a little bit shaky. There have also been a number of follow-up experiments that revealed further brain-breaking weirdness.
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