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Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories for the particles that travel through it depending on their spin.

My Question: At what point is the spin determined when it is in superposition? When the particle starts to feel the magnetic field? Or only when the trajectory is measured in the detector?

This is a similar question, however it does not answer my question.

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3 Answers 3

up vote 2 down vote accepted

The spin wavefunction unitarily evolves into either an up state or down state by decoherence with the environment, a.k.a. measurement.

Edit

When the particle enters a magnetic field, the wavefunction evolves (unitarily) according to $$i\hbar \partial_t |\psi\rangle = \frac{e}{m} \mathbf{B} \cdot \mathbf{S} |\psi\rangle$$ so the up and down amplitudes just evolve in different ways. In the case of the Stern-Gerlach apparatus, $\mathbf{B}$ is non-uniform, so the electron's wavefunction also evolves in space. You can write the general spin-position wavefunction as

$$|\psi\rangle = \int\!dx\, \left( \psi_\uparrow(x)\, |x\rangle |\uparrow\rangle + \psi_\downarrow(x)\, |x\rangle |\downarrow \rangle \right)$$ so the interaction with the magnetic field basically changes the coefficients $\psi_\uparrow(x)$ and $\psi_\downarrow(x)$.

Now, in principle, wavefunctions only ever evolve unitarily ("smoothly"), because bad things happen when they don't. So even when the electron hits a detector, the system remains in some sort of superposition. The problem is that now we aren't only considering the degrees of freedom of the electron, but that of the detector as well (and the experimenter, and her environment, etc.) So the wavefunction I wrote above becomes much more complicated:

$$|\mathrm{System}\rangle = \mathrm{stuff} \otimes |\uparrow\rangle + \mathrm{more\,stuff}\otimes |\downarrow \rangle $$

After the measurement, in principle, (because of linearity of quantum mechanics) the "$\mathrm{stuff}$" part evolves completely independently of the "$\mathrm{more\,stuff} $" part, and the experimenter can't tell that she herself is in a superposition of two outcomes (Schrodinger's cat). In practice, however, once you have many many degrees of freedom, states like these tend to be very unstable and quickly decay into a state where the superposition is lost. This is called decoherence.

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But what is a measurement? Can the entering in a magnetic field be considered a measurement? after all its acting on the particle. –  stebu92 Jan 4 at 21:25
    
Please see my edit. –  lionelbrits Jan 5 at 0:16
    
I hoped there was a simpler answer. I've only had an introductory course to quantum mechanics. But it appears the question has more implications than I expected. Thanks for your extensive answer! –  stebu92 Jan 5 at 13:24

Your question is an interesting part of the open problem in the foundations of quantum mechanics, the Measurement Problem. It can't really be addressed experimentally, and the different interpretations of QM say diffefent things. (In Many Worlds, for example, when done properly, it's the experimenter that goes into an entangled superposition with the particle, and the spin is never determined.)

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The spins are determined when they hit something (i.e. "decoher" by interacting with a macroscopic entity.). You can infer a trajectory for them that extends back to the beginnings of the inhomogeneous field.

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