  Weyl's formulation of quantum mechanics opened the possibility of studying
the dynamics of quantum systems both in infinite-dimensional and
finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger,
a self-consistent theoretical framework describing physical systems
characterised by a finite-dimensional space of states has been created. The
used mathematical formalism is further developed by adding finite-dimensional
versions of some notions and results from the continuous case. Discrete
versions of the continuous Gaussian functions have been defined by using the
Jacobi theta functions. We continue the investigation of the properties of
these finite Gaussians by following the analogy with the continuous case. We
study the uncertainty relation of finite Gaussian states, the form of the
associated Wigner quasi-distribution and the evolution under free-particle and
quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis
is put on the recovery of the known continuous-limit results when the dimension
$d$ of the system increases.
