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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?

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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.

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  • $\begingroup$ Of course! I seem to have neglected that these are still probabilities of observing z+ and z-. Thank you! $\endgroup$ Commented Oct 10, 2020 at 21:47
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Outside the magnetic field, space is isotropic and angular momentum is conserved. Outside the field, there is no preferred axis. That would only appear at a measurement.

Classically: without a torque there is no precession.

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