  Strong quantum zero-point motion (ZPM) of light nuclei and other particles is
a crucial aspect of many state-of-the-art quantum materials. However, it has
only recently begun to be explored from an $\textit{ab initio}$ perspective,
through several competing approximations. Here we develop a unified description
of muon and light nucleus ZPM and establish the regimes of anharmonicity and
positional quantum entanglement where different approximation schemes apply.
Via density functional theory and path-integral molecular dynamics simulations
we demonstrate that in solid nitrogen, $\alpha\unicode{x2013}$N$_2$, muon ZPM
is both strongly anharmonic and many-body in character, with the muon forming
an extended electric-dipole polaron around a central, quantum-entangled
[N$_2\unicode{x2013}\mu\unicode{x2013}$N$_2$]$^+$ complex. By combining this
quantitative description of quantum muon ZPM with precision muon quadrupolar
level-crossing resonance experiments, we independently determine the static
$^{14}$N nuclear quadrupolar coupling constant of pristine
$\alpha\unicode{x2013}$N$_2$ to be $-5.36(2)$ MHz, a significant improvement in
accuracy over the previously-accepted value of $-5.39(5)$ MHz, and a validation
of our unified description of light-particle ZPM.
