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

Spin diffusion in the low-dimensional molecular quantum Heisenberg antiferromagnet Cu(pyz)(NO$_{3}$)$_{2}$ detected with implanted muons