The generalized uncertainty principle has been described as a general consequence of incorporating a minimal length from a theory of quantum gravity. We consider a simple quantum mechanical model where the operator corresponding to position has discrete eigenvalues and show how the generalized uncertainty principle results for minimum uncertainty wave packets.

The properties of quantum mechanics with a discrete phase space are studied. The minimum uncertainty states are found, and these states become the Gaussian wave packets in the continuum limit. With a suitably chosen Hamiltonian that gives free particle motion in the continuum limit, it is found that full or approximate periodic time evolution can result. This represents an example of revivals of wave packets that in the continuum limit is the familiar free particle motion on a line.Finally we examine the uncertainty principle for discrete phase space and obtain the correction terms to the continuum case.