In quantum mechanics, what information does a wave function contain, beyond a simple probability distribution of a particle’s location, and why does it require complex values?

Question

Layman’s explanations of wave functions often describe them as a complex-valued representation of the probability of a given particle existing in a given place at a given time. But the use of complex numbers suggests that it’s more than just a probability distribution. Does the wave function contain more information than a simple probability distribution, and if so, what is that and why does it require complex values?

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