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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 investiga- tion 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 exci- tations 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.

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