Quantum entanglement, quantum measurement, spin and position

Question

By uncertainty principle, we know that determining particle's position at some location is limited. So we cannot determine the position of a particle at some exact point location as this would make the standard deviation of momentum reach infinite.

But as far as I know, spin is exactly determined when a particle gets measured - although before measurement it can exist in superposition of different states.

So, why are these two quantities(?) different? (Also, as far as I know, spin is part of wavefunction that includes position, right?)

Answer 1

There is no limit to measuring the position of a particle to arbitrary accuracy, or measuring the momentum to arbitrary accuracy. It's just that we can't measure them both to arbitrary accuracy at the same time. This is because the operators for position and momentum do not commute.

The uncertainty principle only applies to pairs of observables whose operators don't commute. For example the operators for spin and position commute, so we can measure the spin of a particle and it's position to arbitrary accuracy at the same time.

The uncertainty principle does apply to spin. For example we can measure the spin in the $z$ direction and the $x$ or $y$ directions, but we can't measure both $S_z$ and $S_x$ or $S_y$ at the same time because the operators don't commute. However we can measure total spin and $S_z$ (or $S_x$ or $S_y$) at the same time because the operators for total spin and spin in a particular direction do commute.