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?


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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You get the probability distribution for a given state by taking the squared magnitude of the wavefunction, projected onto the measurement basis. (By projected onto the measurement basis, for an example of quantum spin, if you’re measuring spin up/spin down, and your particle is pointing right, that state “spin right” is 1/sqrt(2) (|up> + |down>), meaning that when you project it to the measurement basis, you have 1/sqrt(2) in the up direction, and 1/sqrt(2) in the down direction, rather than 1 in the “right” direction.)

So it’s similar in concept to the square root of a probability distribution.

However, we still haven’t addressed complex numbers.

The complex numbers are essentially there to encode extra information. For example, in the double slit experiment, which it sounds like you might know about, the probability of finding light or electrons in a certain place is 0 because the wavefunction cancels through the two slits.

Probability density functions can’t cancel out because they’re strictly positive, but the wavefunction can because it’s made of complex numbers.

**Edit:** Also, the wavefunction has information about much more than just position, so it’s not really just a probability distribution of position.

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).