Stern-Gerlach Experiment (spatial consistency of the magnetic moment)

[nmbr28albert]

In the Stern-Gerlach experiment, we have an inhomogeneous magnetic field aligned with the axis labeled z. The two distinct deflections are caused by the force from two distinct magnetic moments $\mu = \gamma S$. I understand that the spin larmor precesses around the field and that the expectation value for $S_z$ remains constant, but when we collapse the spin by placing a screen to detect to detect the particles, how are we certain that after collapsing the spin state (I think the spin is collapsed before going through the field?) the spin vector cannot change spatially, therefore changing the z component of spin. I suspect my misunderstanding is due to thinking about the "directional" nature of spin the wrong way.

 

[WannabeNewton]

The spin is only measured at the detector, it is not measured in the magnetic field, and in fact we do not measure the spin directly even at the detector. What we measure is the position of the particle on the screen and from this we infer whether or not the particle is spin up or spin down due to the double-splitting of the incident beam along the axis of the magnetic field. So even if there is spin precession it will not affect the purpose of the experiment. That being said, the heuristic treatment of the Stern-Gerlach experiment usually given in QM books when motivating spin leaves a lot to be desired. It is mostly based on classical arguments and as such basically uses nothing from the QM formalism.

For starters, the classical argument uses the idea that the particle experiences a force $\vec{F} = -\vec{\nabla}(-\vec{\mu}\cdot\vec{B})$ due to the inhomogeneous magnetic field coupling to its magnetic moment. This is fine if we are using Newton's 2nd law but when talking about the double-splitting of the incident beam due to the inhomogeneous magnetic field, one must use the Schrodinger equation (or rather an augmentation therefore, see below). Then it is not even clear how one would interpret the double-splitting due to the time evolution of the state $|\psi \rangle$ of each particle under $i\hbar \partial_t |\psi \rangle = H |\psi \rangle$ in terms of a classical force $F = \mu_z \partial_z B_z$.

A proper treatment of the Stern-Gerlach experiment in fact requires the Pauli equation. So the $|\psi \rangle$ above is in fact a 2-component spinor and $H$ is the Hamiltonian on the space of such spinors. Seeing the Stern-Gerlach effect calculated using the Pauli equation will make it clear to you what role, or rather lack thereof, the spin precession of $|\psi \rangle$ under the spinor time evolution $i\hbar \partial_t |\psi \rangle = - (\mu \cdot B) |\psi \rangle$ plays in the effect.

For this I suggest reading chapter 3 of https://www.physics.byu.edu/research/theory/Docs/JaredsThesis05.pdf.