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Friedrich Hund organized complicated atomic spectral lines through relationships between the angular momenta and energy levels of many-electron configurations. In modern notation, a term symbol ${}^{2S+1}{L}_J$ records total electron spin $S$, total orbital angular momentum $L$ and coupled total angular momentum $J$. For example, $S=1$ permits three spin projections, whereas $S=0$ permits one. The classification reduces a large pattern of lines to relationships between fewer quantum states. This modern spin notation should be distinguished from the historical language of his original spectral analysis.^[ref]

The magnetic spectra of a scalar field and the Dirac equation do not fix every higher-spin spectrum. In SI units, a charged mode with a specified gyromagnetic ratio corresponding to $g=2$ has the model branchesHere $n=0,1,\dots$ is the Landau level index and $m_s$ is the dimensionless spin projection measured along $q\mathbf{B}$. The power $c^2$ in the magnetic term follows because $|q|\hslash B$ has dimensions of squared momentum in SI units; the Gaussian-unit field convention uses a different power of $c$.

For spin one, the aligned $n=0,m_s=1,p_z=0$ branch becomes $E^2=m^2{c}^4-|q|\hslash |B|c^2$. A negative value signals an instability of the assumed background in this model. This extrapolation requires the stated magnetic coupling: spin alone does not determine $g$ for an arbitrary higher-spin field.

A neutrino is a spin-one-half fermion. Its spin projection along its momentum defines neutrino helicity.

The uncertainty of a spin projection is measured by its variance. In a maximally polarized spin coherent state, the longitudinal variance is zero and each transverse variance is $S/2$ in units $\hslash =1$. The transverse fluctuations do not imply finite magnon occupation in an isotropic ferromagnet's polarized vacuum.

The spin projection is the component of a particle's spin angular momentum along a specified spatial direction. For the $z$ direction, an eigenstate of the angular momentum operator $S_z$ has measured value $m_s\hslash$, where $m_s=-s,-s+1,\dots ,s$ and $\hslash$ is the reduced Planck constant. For spin one-half, the two possible results are $+\hslash /2$ and $-\hslash /2$.

The direction belongs to the measurement arrangement: a state with definite $S_z$ generally has uncertain $S_x$. A Stern-Gerlach experiment separates the corresponding components through a spatially varying magnetic field. This is a component measurement, rather than a claim that the full spin angular momentum points along one of two classical directions.