= Solution
<angular momentum> generates rotations. Its components obey $[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$, and $J^2$ commutes with each component. A compatible basis has $J^2=\hbar^2j(j+1)$ and $J_z=m\hbar$, where $m=-j,-j+1,\ldots,j$. <ladder operators> change $m$ by one, terminate at $\pm j$, and give $2j+1$ magnetic substates. Different Cartesian components generally cannot all be sharp simultaneously.
<orbital angular momentum> is $\mathbf L=\mathbf r\times\mathbf p=-i\hbar\mathbf r\times\nabla$. Its angular eigenfunctions are <spherical harmonics>, with integer $\ell$ arising from their spatial rotation properties and single-valuedness. <spin> is intrinsic <angular momentum>, not literal rotation of a finite charged sphere. It may be half-integer; <spin> $1/2$ is represented by $\mathbf S=\hbar\boldsymbol\sigma/2$. For several contributions, $\mathbf J=\mathbf L+\mathbf S$ obeys the same algebra. Coupling angular momenta $j_1,j_2$ permits $J=|j_1-j_2|,\ldots,j_1+j_2$, with <Clebsch-Gordan coefficients> relating product and coupled bases. <spin-orbit coupling> can distinguish these total-angular-momentum sectors.
The Stern-Gerlach apparatus sends a collimated neutral atomic beam through an inhomogeneous <magnetic field> and records its deflection. A <magnetic moment> experiences $F_z\simeq\mu_z\,\partial B_z/\partial z$; field uniformity would produce precession but no analogous net separating force. A classically continuous distribution of moment orientations would broaden the beam continuously. Gerlach and Stern instead observed separated silver-beam components. Reversing or removing the field and comparing the undeflected profile controls the magnetic origin; collimation, velocity spread and finite resolution affect the spots. The original experiment demonstrated space quantization; its modern silver interpretation uses the unpaired <electron> <spin> with nearly zero <orbital angular momentum>, rather than a direct measurement of a free <electron>'s trajectory.<Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld by Walther Gerlach and Otto Stern>[Experimental setup, recorded silver deposits and field-on/off comparison]{ref}
For two spin-$1/2$ particles, the <triplet state> sector has $S=1$ and symmetric <spin> states
$$
|1,1\rangle=|\uparrow\uparrow\rangle,\quad
|1,0\rangle=\frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt2},\quad
|1,-1\rangle=|\downarrow\downarrow\rangle.
$$
The <spin singlet state> has $S=0$ and antisymmetric <spin> state
$$
\boxed{|0,0\rangle=\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}}.
$$
Their total $S^2$ values are $2\hbar^2$ and zero, while $\mathbf S_1\cdot\mathbf S_2$ has values $\hbar^2/4$ and $-3\hbar^2/4$. The singlet is rotationally invariant and gives anticorrelated <spin> measurements along a common axis. For identical <fermions> the complete state is antisymmetric: a symmetric spatial state pairs with a singlet <spin> state, whereas an antisymmetric spatial state pairs with triplet <spin>. \b[Orbital, <spin> and coupled angular momenta share an algebra, but their physical origins and allowed representations differ.]
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