Uncertainty Principle versus spin alignment

[hvo]

TL;DR

How can spin vectors be perfectly parallel/antiparallel if uncertainty principle says neither its direction and magnitude can be determined simultaneously?

Hello,

So I know that the magnetic moments of atoms are dependent on the spin and orbital angular momenta of its electrons. Both of these quantities are limited by the uncertainty principle so that neither of their direction and magnitude can be known simultaneously with arbitrary precision. The only quantities that aren't restricted by the uncertainty principle are the magnitude and the vector's projection along a z axis.

So how can magnetic moments align parallel to one another? I've tried to read lectures and books on this but none mentioned the uncertainty principle and they all have pictures of perfectly aligned spins. I've considered that these vectors I see in textbooks are actually projections along a common axis, but then why is spin wave considered a separate phenomena?

 

[Nugatory]

Both of these quantities are limited by the uncertainty principle so that neither of their direction and magnitude can be known simultaneously with arbitrary precision.

The direction of the angular momentum vector is is not known exactly, only the z-component of the direction is known. That's allowed by the uncertainty principle.

(And why is the z-component of the angular momentum the one that we know? That's just by convention: we we pick the one direction in which we will choose to know the exact value of the component in that direction and call that direction the z axis).

 

[Haorong Wu]

Since $\left [ {\bf S} ^2, S_z \right ]=0$, then the magnitude of the spin angular momentum and its z-axis component can be simultaneously determined. As @ Nugatory mentioned, the direction of z-axis is chosen by convention. In fact, $\left [ {\bf S }^2, S_{\hat n} \right ]=0$ where $S_{\hat n}$ is along an arbitrary direction, so the magnitude of the spin angular momentum and its component along arbitrary direction can be simultaneously determined.