# Does spin precession stop immediately after leaving Stern-Gerlach apparatus and split into the orthogonal direction of precession?

A small concept I'm a tiny bit confused about. Sakurai in the introduction to quantum mech introduces the Stern Gerlach experiment. According to his discussion, an $$S_{x}+$$ particle that goes into a SG apparatus with $$B\hat{z}$$ will split 50-50 into $$z+$$ and $$z-$$ directions.

However, if we consider time dynamics of the state, Sakurai says a $$S_{x}+$$ state will start to precession and the expectation value of $$\langle S_{x} \rangle \propto \cos\omega t$$ and $$\langle S_{y} \rangle \propto \sin\omega t$$

My question is this: Is the spin precession only valid when the spin is under the action of the magnetic field? Once the spin exits the magnetic field, shouldn't it split into $$z+$$ and $$z-$$ with basically no knowledge of whether or not it was precessing?

Yes, the spin precession happens because of the external magenetic field. In the $$z$$ basis, it adds a phase to the states like: $$|\psi(t)\rangle=c_{\uparrow}e^{-i\omega t}|\uparrow\rangle_z+c_{\downarrow}e^{i\omega t}|\downarrow\rangle_z$$ but it does not change the probabilities of outcome $$c_{\uparrow}$$ and $$c_{\downarrow}$$, that are given by the initial condition. So in this experiment, once the particles exit the magnetic field, they split into $$+z$$ and $$-z$$ without any track of the precession because it didn't affected to the outcome probabilities.