so true. I have a similar beef about the popular explanation for uncertainty principle: "well you see the light hits the particle very hard so we know where it was but we don't know where its gone now". urgh.
As someone who doesn't know any better than the explanation you have a beef with, I would love if you could explain in layman's terms why it's wrong and what a more accurate understanding is. I always thought that analogy was exactly how it worked, but it seems I have been misled unawares.
It presupposes that position and momentum have definite states that are just uncertain to us, while in most interpretations of quantum mechanics (e.g. Copenhagen and many worlds) the particle exists as a wavefunction, lacking a specific position or momentum, but instead existing as a probability density function in this space.
The uncertainty principle here then relates to how much this probability density function 'peaks' in position space or momentum space. A higher peak in one space results in a wider spread in the other. This is because position and momentum are Fourier transforms of each other.
So is it true that there are multiple properties of a particle---such as location, position, and maybe its spin---that are all described as wave functions, and therefore they can all be entangled? Can anything that is described by a wave function be entangled?
Since position and momentum (assuming that's what you meant, since you said location and position which are synonyms) have this sort of dual relationship, I don't think it makes sense to talk about entanglement with respect to them - they intrinsically have to be related to each other, and the position state (i.e., function) a particle is in fully determines its momentum state.
But it is possible to imagine usually unrelated properties of a particle being entangled, e.g. a two-peaked position function, spin up if it's over here and spin down if it's over there. So that's possible. Usually when discussing entanglement, though, we're talking about 'distinct'* particles. Electron A's spin entangled with electron B's spin. Not that it has to be spin, of course. But that's a common case because of how naturally this sort of entanglement occurs, for example, in atoms where electrons have to form spin pairs.
* This is complicated by QFT where particles are not exactly distinct, but exist as excitations in a particle field. E.g. there aren't two electrons but the electron field is excited by two quanta. At least, that's my understanding; I never went to grad school for physics, so I'm limited to undergraduate knowledge and some extracurricular reading.
> Since position and momentum (assuming that's what you meant, since you said location and position which are synonyms)
Yep, sorry, artifact of the editing process.
> they intrinsically have to be related to each other, and the position state (i.e., function) a particle is in fully determines its momentum state.
Sure, but the momentum doesn't determine the position (due to the constant of integration) so you can have two particles with the same momentum functions and different locations, and that leads to my next question...
> Usually when discussing entanglement, though, we're talking about 'distinct'* particles.
That's what I actually meant to ask but didn't phrase clearly: since position and momentum are described by wave functions, can you entangle the positions of two particles? or entangle their momentum?
> Sure, but the momentum doesn't determine the position (due to the constant of integration) so you can have two particles with the same momentum functions and different locations, and that leads to my next question...
There's no constant of integration since the integral will be over all of space (or momentum space).
> That's what I actually meant to ask but didn't phrase clearly: since position and momentum are described by wave functions, can you entangle the positions of two particles? or entangle their momentum?
