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Suppose a beam of particles is prepared with spin in the $+z$ direction (e.g. by a Stern-Gerlach apparatus). Suppose this beam then enters a Stern-Gerlach apparatus with orientation $\theta$ relative to $+z$. Quantum Mechanics predicts that with probability $\cos^2(\theta/2)$ a particle will be deflected in the $+\theta$ direction, and with probability $\sin^2(\theta/2)$ a particle will be deflected in the $-\theta$ direction.

I have not been able to find any published experimental tests of this prediction for arbitrary values of $\theta$. If you know of any, please provide the appropriate citation.

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Not surprisingly, such questions date back to the original Stern-Gerlach experiments. An early experiment, perhaps not exactly what you seek, is reported by TE Phipps and O Stern (yes, the same Stern) in Zeitschrift fur Physik 73(3-4) 185-191 (1932). Another, more recent article, is K. Brodsky et al., Europhysics Letters 44(2) 137-143 (1998) on 'Single and double interaction zone with comoving fields in Stern-Gerlach atom interferometry' - this is probably way more sophisticated that you seek, but shows that double Stern-Gerlach experiments are not uncommon, but perhaps not called that.

A nice article that covers most of the related theory in an accessible way is AR Mackintosh, 'The Stern-Gerlach experiment, electron spin and intermediate quantum mechanics', Eur. J. Phys 4 97-106 (1983). This is theory only.

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    $\begingroup$ Thank you for your reply. However, none of these cites provide empirical verification of the predictions. Since I am essentially asking for verification of Malus' law for polarizers but applied to spin particles in a Stern-Gerlach apparatus, and since many college labs have equipment for students to experience the beam splitting phenomena, surely someone has done the experiment for arbitrary $\theta$. If not, then they should because it would be a valuable publication. $\endgroup$
    – Dale Stahl
    May 29, 2015 at 16:54
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I don't know if precisely this experiment has been done, but experiments close to it have been done in various interference experiments with atoms and neutrons. However, I would like to suggest that the question comes from a misunderstanding of how physics makes progress. It is neither necessary nor useful to demonstrate directly every application of some concept in physics. Rather, we use the fact that the subject is a logical construct that hangs together, and we build up experimental support for the whole structure. In the case of quantum theory and spin angular momentum, there is a huge wealth of experimental support for the theory, whether or not anyone ever does precisely an experiment with a sequence of two Stern-Gerlach-style apparatuses using magnetic field gradients to redirect particles according to their spin state.

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    $\begingroup$ The sequential SG is of great pedagogical interest. Many QM courses begin by describing a seq. SG and deducing basic principles of QM from it. This approach is very simple, yet very effective and imo even very aesthetic – you can deduce (almost) the entirety of quantum logic from just a single experiment. However, if the experiment hasn't been performed yet, the foundations of QM seem much less convincing to the students, hence my interest in the experiment. I personally do not doubt the foundations of QM, but a successful realization of the seq. SG would make it easier to convince others :) $\endgroup$
    – csha
    May 23, 2021 at 0:19
  • $\begingroup$ Could you provide some more information about the experiments you mention, please? I've found papers about some more advanced SG experiments on neutron beams, but they were very technical and I couldn't make out what was it that was actually tested. $\endgroup$
    – csha
    May 23, 2021 at 0:21

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