Sean Carroll

That's accurate in quantum mechanics as we know it, but it's not very helpful if you're not already up on what all those words mean.

The other way of explaining what the quantum state is is to say it's a superposition of possible measurement outcomes.

So if you have the down-to-earth idea that there is a wave function that depends on x, the position of a particle, then that wave function squared gives you the probability of seeing the particle at that point if you were to measure it, okay?

So the quantum state, either way that you want to talk about it, that second way is a little bit more physically transparent, but also a little bit tricking you into thinking that there's some implicit notion of space in the quantum state that may or may not be there, depending on what kind of model you're actually looking at.

So the classical limit is a matter of emergence.

So emergence happens when you have one theory, the lower level or the microscopic theory, in this case quantum mechanics.

And there is a map to another theory that in general is many to one.

So many different theories.

states in the microscopic theory get mapped under the emergence map to the same state in the macroscopic theory in this case the macroscopic theory is just classical mechanics either newtonian or relativistic or whatever you want to have so what we're saying is for a wave function there can be wave functions that are more or less localized around some location in space

And when that happens, you can map them to an idealized point particle at that location in space.

And then there's a theorem due to Paul Ehrenfest that says under the right conditions, and we're not gonna go into what the right conditions are, but they are there.

Under the right conditions, the average value of the wave function, the expectation value, roughly speaking, where it is peaked, right, around, if it's localized in space, then you put a dot at sort of the middle of the wave function.

And Ehrenfest's theorem says that dot will obey the classical equations of motion.

So you can derive Newton's laws of motion from quantum mechanical laws underlying it.

So there's actually two and that's the classical limit.

So secretly there were two big steps there and they're both important and sort of we tend to ignore one and pay attention to the other one.

The first one is you have to start with a wave function that is more or less localized around some classical point.

Like when I say I have a wave function and it's really concentrated near some point, some wave functions are like that, some are not, right?

Some could be spread out or some could be like, oh, there's a bunch of it here, but there's also a bunch of it over there very, very far away.

Those are not classical-looking wave functions.