  The investigation of the dynamics of quantum many-body systems is a concerted
effort involving computational studies of mathematical models and experimental
studies of material samples. Some commonalities of the two tracks of
investigation are discussed in the context of the quantum spin dynamics of
low-dimensional magnetic systems, in particular spin chains. The study of
quantum fluctuations in such systems at equilibrium amounts to exploring the
spectrum of collective excitations and the rate at which they are excited from
the ground state by dynamical variables of interest. The exact results obtained
via Bethe ansatz or algebraic analysis (quantum groups) for a select class of
completely integrable models can be used as benchmarks for numerical studies of
nonintegrable models, for which computational access to the spectrum of
collective excitations is limited.
