= Solution
\b[Operators, quantization and rotations.] <Angular momentum> is the generator of spatial rotations. In <quantum mechanics>, its components are <Hermitian operators> on their appropriate domains and obey
$$
[\widehat J_i,\widehat J_j]=i\hbar\epsilon_{ijk}\widehat J_k,
\qquad [\widehat J^2,\widehat J_i]=0.
$$
These <angular momentum commutation relations> prevent all three components from having simultaneous sharp values. For example, the <Robertson uncertainty principle> gives $\Delta J_x\Delta J_y\ge\hbar|\langle J_z\rangle|/2$. One conventionally diagonalizes $\widehat J^2$ and a selected component $\widehat J_z$:
$$
\boxed{\widehat J^2|j,m\rangle=\hbar^2j(j+1)|j,m\rangle,
\qquad\widehat J_z|j,m\rangle=\hbar m|j,m\rangle.}
$$
The allowed labels are $j=0,\tfrac12,1,\tfrac32,\ldots$ and $m=-j,-j+1,\ldots,j$. An angular-momentum multiplet has $2j+1$ states. The magnitude $\hbar\sqrt{j(j+1)}$ differs from the largest possible component $\hbar j$; the remaining components fluctuate even in a state with maximal $m$.
The <angular momentum lowering operator> and its raising counterpart, $\widehat J_\pm=\widehat J_x\pm i\widehat J_y$, satisfy
$$
\widehat J_\pm|j,m\rangle
=\hbar\sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle.
$$
The ladder terminates at $m=\pm j$. A rotation through angle $\theta$ about unit vector $\widehat{\mathbf n}$ acts through the <unitary operator> $\exp[-i\theta\widehat{\mathbf n}\cdot\widehat{\mathbf J}/\hbar]$. Thus quantized components and the continuous rotation of a state are compatible.
\b[<Orbital angular momentum>.] The orbital contribution is associated with motion in space:
$$
\widehat{\mathbf L}=\widehat{\mathbf r}\times\widehat{\mathbf p}
=-i\hbar\,\mathbf r\times\boldsymbol\nabla.
$$
For a spinless spatial <wavefunction>, the simultaneous angular eigenfunctions are the <spherical harmonics> $Y_{\ell m}(\theta,\phi)$, with $\ell=0,1,2,\ldots$ and $m=-\ell,\ldots,\ell$. Single-valued ordinary scalar spatial <wavefunctions> give integer orbital labels. In a central <potential energy>, $[\widehat H,\widehat L^2]=[\widehat H,\widehat L_z]=0$, so stationary states may be labelled by these angular <quantum numbers>. The angular kinetic contribution appears in the radial equation as $\ell(\ell+1)\hbar^2/(2mr^2)$, the <centrifugal potential>. Noncentral interactions need not conserve these orbital labels.
\b[Intrinsic <spin> and addition.] <Spin angular momentum> is an intrinsic internal degree of freedom with the same commutation algebra. It is not the <angular momentum> of a small classical charged sphere literally rotating about its centre. For an <Electron>, $s=1/2$ and $m_s=\pm1/2$, so $\widehat S_i=(\hbar/2)\sigma_i$ in terms of the <Pauli matrices>. Its squared <spin> is $3\hbar^2/4$. Other particles can have integer or half-integer <spin>, including <spin> zero.
For one particle with orbital and <spin> degrees of freedom, $\widehat{\mathbf J}=\widehat{\mathbf L}+\widehat{\mathbf S}$; the two component sets commute with each other. <Addition of angular momentum> permits $j=|\ell-s|,|\ell-s|+1,\ldots,\ell+s$. The coupled states $|j,m_j\rangle$ are linear combinations of the uncoupled $|\ell,m_\ell\rangle|s,m_s\rangle$ with $m_j=m_\ell+m_s$, determined by <Clebsch-Gordan coefficients>. A rotationally invariant <spin-orbit interaction> can conserve $J^2,J_z$ while mixing the separate component labels.
<Angular momentum> also carries a <magnetic moment>. For an electron's orbital motion and intrinsic <spin>,
$$
\boldsymbol\mu_L=-\frac e{2m_e}\mathbf L,\qquad
\boldsymbol\mu_S=-g_s\frac e{2m_e}\mathbf S,\qquad g_s\simeq2.
$$
The minus signs reflect negative <electric charge>. With the <Bohr magneton> $\mu_B=e\hbar/(2m_e)$, a <spin> projection $m_s$ has $\mu_z=-g_s\mu_Bm_s$. A <magnetic field> can therefore distinguish angular-momentum states through their interaction <energy> $-\boldsymbol\mu\cdot\mathbf B$.
\b[Stern–Gerlach measurement.] In the <Stern-Gerlach experiment>, atoms emerge from a heated source, pass through collimating apertures, traverse a strongly inhomogeneous <magnetic field>, and strike a position-sensitive screen. The beam is neutral, so its separation is interpreted through the atomic <magnetic moment> rather than the ordinary charged-particle <Lorentz force>. Near a chosen quantization direction, the interaction gives
$$
U=-\mu_zB_z,\qquad F_z\simeq\mu_z\frac{\partial B_z}{\partial z}.
$$
A uniform field can exert a <torque> but would not separate the beam by this translational force; the gradient is essential. Different allowed magnetic-moment components become correlated with different trajectories. A classical ensemble with continuously varying orientations would yield a continuous spread, whereas discrete quantum components yield separated channels.
In the modern electronic description of ground-state <silver>, filled shells cancel their electronic angular momenta and the unpaired $5s$ electron has $L=0$, $S=1/2$, $J=1/2$. The effective electronic measurement therefore has two components, approximately $\mu_z=\pm\mu_B$, rather than an orbital $\ell=1/2$, which is not allowed. For positive field gradient the electronic $m_J=+1/2$ component has negative $\mu_z$ and deflects downward in this coordinate convention. Nuclear <spin> and <hyperfine structure> are neglected in this simplified two-channel account. Historically the original splitting preceded the electron-spin interpretation, so the modern explanation should not be confused with the original investigators' interpretation.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-1-c10-spin-analyser.png]
{title=A neutral atomic beam separates into two electronic magnetic-moment channels in a field gradient}
More generally a prepared <spin> state $a|+z\rangle+b|-z\rangle$ gives the two outcomes with probabilities $|a|^2,|b|^2$. Selecting one channel prepares a state for further measurements. An ideal second analyser along the same axis then gives a definite result, whereas one along a perpendicular axis gives two equally likely outcomes for a spin-one-half <eigenstate>. This connects the apparatus directly to incompatible <spin> components and the <Born rule>. \b[<Angular momentum> is quantized through its operator algebra; the Stern–Gerlach gradient turns that discrete internal information into spatially distinguishable outcomes.]
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