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Two methods for studying quantum discrete dynamical systems (i.e. with a finite or countably infinite number of degrees of freedom) are described and compared for: (i) The discrete self-trapping (DST) equation, (ii) The Ablowitz-Ladik (AL) equation, and (iii) A fermionic polaron (FP) model. The first method, called the quantum inverse scattering method (QISM), emphasizes the relationship between the quantum theory and classical integrability. The second method, which we call the number state method (NSM), can be extended to systems that are not classically integrable. Thus the two methods are not equivalent.
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