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 probes 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$_4$)$_{2}$.
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