= Solution
A <Stern-Gerlach experiment> sends a collimated beam through an inhomogeneous <magnetic field> and then measures its transverse position after a drift region. The <neutron> is electrically neutral but has a nonzero <magnetic moment> proportional to its <spin>, $\boldsymbol\mu=\gamma_n\mathbf S$. A predominantly $z$-directed field gives the interaction <Hamiltonian operator> and transverse <force>
$$
H_{\mathrm{int}}=-\boldsymbol\mu\cdot\mathbf B,\qquad
F_z=-\partial_zH_{\mathrm{int}}\simeq\mu_z\frac{\partial B_z}{\partial z}.
$$
Since a neutron has <spin one-half>, its spin component along the local field has only two <eigenvalues>, $S_z=\pm\hbar/2$. Hence there are two opposite force branches, corresponding to $\mu_z=\pm\gamma_n\hbar/2$, and a beam containing both spin components gives two displaced detector spots. This is <neutron Stern-Gerlach splitting>, not an electric-charge <Lorentz force>. A uniform <magnetic field> can cause <Larmor precession> but does not provide the required separating <magnetic field gradient>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2021/paper-1-neutron-stern-gerlach.png]
{title=An ideal neutron Stern-Gerlach arrangement separates two magnetic-moment projections}
{description=Original functional schematic: collimating slits, a region with a <magnetic field gradient>, and a position-sensitive neutron detector. The two curves represent the two allowed magnetic-moment branches, not a continuous distribution of classical spin orientations. Geometry and deflections are illustrative.}
The neutron's <gyromagnetic ratio> is negative, so its <magnetic moment> points opposite to its <spin>; this reverses which spin label goes to which spot without changing the two-way splitting. Equal populations occur for an initially unpolarized beam. A beam prepared entirely in one field-axis <spin eigenstate> gives only its corresponding branch, explaining the qualification “in general”.
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