Sean Carroll

If a particle is in a superposition, do the different positions move through space at identical speeds?

And if not, could the different spatial positions of the particle experience different passages of time?

So this is actually an excellent question because there's sort of two ways of answering it.

One way says no, and one way says yes.

And there, you know, sort of depends on what you, how you interpret exactly what is going on in the question.

So the way that would say yes, different parts of the wave function experience different passages of time is, you know, wave functions spread out unless they're already infinitely spread.

To be a little bit precise about it, just to be, we'll be precisely correct and then we'll

be more hand-wavy about it, a quantum state of definite speed is a quantum state of definite momentum, right?

Those are the same thing.

And in quantum mechanics, just good old ordinary quantum mechanics of a single particle, the only kinds of states that you can get, the only kind of wave function you can get with a definite velocity or momentum

is spread out all over space.

It is a plane wave that is not localized to any position whatsoever.

This is a reflection of the uncertainty principle.

If you know the momentum exactly, you know the position not at all, right?

So any localized quantum wave function necessarily involves parts of it moving at different velocities.

And there is a rigorous version of that statement, which is to say I can take the quantum wave function as a function of position

I can transform it into a quantum wave function as a function of momentum, that is to say as a superposition of different states with definite momentum.

And then what I would find is that it's not perfectly localized in momentum.

There's different momenta and they move different speeds and that's the answer to the question.

So in a very real sense, wave functions spread out because different parts of the wave function are moving at different velocities.