  We present the results of muon-spin relaxation measurements of spin
excitations in the one-dimensional quantum Heisenberg antiferromagnet
Cu(pyz)(NO$_{3}$)$_{2}$. Using density-functional theory we propose muon sites
and assess the degree of perturbation the muon probe causes on the system. We
identify a site involving the muon forming a hydroxyl-type bond with an oxygen
on the nitrate group that is sensitive to the characteristic spin dynamics of
the system. Our measurements of the spin dynamics show that in the temperature
range $T_{\mathrm{N}}<T<J$ (between the ordering temperature $T_{\mathrm{N}}$
and the exchange energy scale $J$) the field-dependent muon spin relaxation is
characteristic of diffusive transport of spin excitations over a wide range of
applied fields. We also identify a possible crossover at higher applied fields
in the muon probe's response to the fluctuation spectrum, to a regime where the
muon detects early-time transport with a ballistic character. This behavior is
contrasted with that found for $T>J$ and that in the related two-dimensional
system Cu(pyz)$_2$(ClO$_4$)$_{2}$.
