What is the relationship between spin and angular momentum in quantum mechanics?

[Ontophobe]

The "color force" doesn't actually change the color of quarks because they don't actually have any color. We use the word "color" non-literally here. Now, I'm told that quarks don't "actually" "spin" either, so again, I infer that it's a term used out of convenience, but I'm also told that they do "actually" have angular momentum. How am I to divorce the concept of angular momentum from spin? In what sense do spinless entities have angular momentum? What do we even mean by "actually" in this particular context?

 

[Khashishi]

Angular momentum is a conserved quantity, so when we talk about angular momentum of particles and of macroscopic objects, we are talking about the same conserved quantity. To flip the spin of a large number of particles, you have to apply a torque somehow, so if you are in a closed system, the opposite reaction torque must go somewhere, perhaps causing the macroscopic structure to start rotating.

How do we tell if something is spinning? For a macroscopic object, we look at the object at successive points in time, and we see that it has rotated. But this technique doesn't work for a point particle like a quark or electron, since rotating a point does nothing. We need another definition of spin for quantum objects

 

[Nugatory]

Now, I'm told that quarks don't "actually" "spin" either

What you have been told (correctly) is that they are not little tiny balls rotating about their axis the way the Earth rotates about its axis. However, they do have an internal angular momentum that contributes to the total angular momentum of the system of which they are part.

How am I to divorce the concept of angular momentum from spin?

You don't. Instead you have to divest yourself of the idea that only little balls rotating around an axis can have internal angular momentum.

 

[mikeyork]

The difference between intrinsic spin and angular momentum is that spin is an SU(2) property that exists independently of any space-time frame. Angular momentum, on the other hand, is a space-time concept. Spin can be interpreted as angular momentum only in a space-time frame. Angular momentum in a CM frame can be interpreted as spin in an SU(2) context.

 

[Ontophobe]

What is a CM frame and and SU(2) context?

 

[mikeyork]

CM = center of momentum. (Net momentum is 0.)
SU(2) is a symmetry group that specifies the behavior of certain quantum numbers and how they combine (interpreted as angular momentum in a space-time context). It is mathematically related to the 3d spatial rotation group.
You should be able to find a more detailed explanation in any standard QM textbook.