what are wave functions
That's quite a difficult question to answer. However I guess the best way to describe it is to say that the wave function is a way to describe the state of a system in quantum mechanics.
To elaborate a little more:
You see when quantum theory was first being developed physicists realized that classical kinematics did not make sense. The idea of a well defined position, among other things, didn't really make any sense.
One formulation of a way to fix this problem was purposed by a physicist by the name of Erwin Schrodinger who purposed the idea of a differential equation, the solution to this differential equation was the wave function.
A wave function itself is actually something that is rather abstract in the sense that a wave function isn't anything physical.
Actually most of the time a wave function is even complex valued! Because the wave function can be complex valued you can image why this is a problem: an imaginary position doesn't really make sense.
To fix this another physicist came up with a statistical interpretation of the wave function and essentially came up with the idea that the wave function squared is something that can be physical, and also formulated the idea that the wave function squared is a probability of that physical quantity.
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