What is the difference between a quantum wave function and a probability distribution of a particle’s location, and why does the wave function require imaginary numbers?

Question

Quantum wave functions are often described as something like a map of where a particle is more or less likely to be found when the wave function collapses. This seems a lot like a probability distribution. But it seems like the wave function is a more complex thing than a probability distribution – what’s the rest of the story?

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Answer 1

There are some quirks of quantum, that cannot be covered by a simple probability distribution.

For ex., it is possible in quantum to have two 100% probabilities, that sum up to 0%. Probability distribution cannot easily do this, but in wave function we can just use numbers `+1` and `-1`.

However, in quantum it is also possible to have *three* 100% probabilities, that sum up to 0%! Now, even `+1` and `-1` won’t save us: we need to go complex! In complex numbers we can find 3 such probabilities (we can also find 4, 5, 6, etc. – any number, actually).