Physics Asked on July 5, 2021
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 $Bhat{z}$ 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 $langle S_{x} rangle propto cosomega t$ and $langle S_{y} rangle propto sinomega t$
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: $$ |psi(t)rangle=c_{uparrow}e^{-iomega t}|uparrowrangle_z+c_{downarrow}e^{iomega t}|downarrowrangle_z$$ 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.
Answered by RMPsp on July 5, 2021
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.
Answered by user137289 on July 5, 2021
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