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It is well known that, due to the uncertainty principle, the Planck constantsets a resolution boundary in phase space and the resulting trade-off inresolution between incompatible measurements has been thoroughly investigated.It is also known that, in the classical regime, sufficiently coarsemeasurements of position and momentum can simultaneously be determined.However, the picture of how the uncertainty principle gradually disappears aswe transition from the quantum to the classical regime is not so vivid. In thepresent work we clarify this picture by studying the associated probabilitiesthat quantify the effects of the uncertainty principle in the framework offinite-dimensional quantum mechanics on a lattice. We also study how theseprobabilities are perturbed by the granularity of the lattice and show thatthey can signal the discontinuity of the underlying space.

Transition to the classical regime in quantum mechanics on a lattice and implications of discontinuous space