Undergrad Understanding Spin & Angular Momentum in Quantum Mechanics

[Silviu]

Hello! I got a bit confused about the fact that the whole the description of spin (and angular momentum) is done in the z direction. So, if we are told that a system of 2 particles is in a singlet state i.e. $$\frac{\uparrow \downarrow -\downarrow \uparrow }{2}$$ does this mean that measuring the spin of the first one on $\textbf{any}$ axis (not only z) will ensure that measuring the spin of the second one will give the opposite result? Or is this form true only for the z axis, and if we want to check on other axis, we need to project the up and down of the z axis onto the orthonormal basis of this other system of axis and work from there? Thank you!

 

[vanhees71]

The correctly normalized singlet state is
$$|S=0,M=0 \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle-|1/2,+1/2 \rangle).$$
It's true that if measuring the spin component in an arbitrary direction of particle 1 then the spin component of particle 2 in the same direction is opposite. It doesn't matter which basis you use to describe the spin-singlet state. The reason is that for total spin $S=0$ the state doesn't change under rotations, i.e., you cannot distinguish any spin direction from any other.