= Solution
A <Stern-Gerlach experiment> uses an oven and slits to produce a collimated <atomic beam>, conventionally of neutral <silver> atoms, and sends it through an inhomogeneous <magnetic field> onto a position-sensitive detector. The unpaired outer electron supplies the effective <spin one-half> <magnetic dipole moment>; neutral atoms avoid the ordinary Lorentz deflection of a charged beam. For an analysing direction $z$, the spatial gradient produces a force approximately $F_z=\mu_z\partial_zB_z$. The resulting separated beam positions measure the allowed projections of <angular momentum>. A <spin one-half> beam gives two branches, rather than the continuous distribution expected from classical dipoles with arbitrary orientations. A uniform <magnetic field> can rotate a <spin> but does not itself provide this spatial separation.
To measure the rotation, first use a $z$-directed <Stern-Gerlach experiment> and an aperture to transmit only the $+z$ branch. Pass this prepared <spin eigenstate> through a region of approximately uniform field in the $x$ direction, then use a second $z$ analyser to separate and count the two populations. For a field region of length $\ell$ and beam speed $v$, the precession angle is $\theta=\lambda B\ell/v$, and the predicted fractions are
$$
\boxed{P_+=\cos^2(\theta/2),\qquad P_-=\sin^2(\theta/2).}
$$
Varying $B$ or the transit time gives complementary oscillations, demonstrating coherent <Larmor precession>. Velocity selection and a sufficiently uniform rotation field keep different particles from accumulating different phases and washing out the contrast. Detection should compare the branch fractions after allowing for losses and analyser efficiency.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-1-spin-rotation.png]
{title=Preparing a spin, rotating it in a transverse field and analysing its final projection}
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