  We calculate the single-particle momentum distribution of a quantum
many-particle system in the presence of the Coulomb interaction and a confining
potential. The region of intermediate momenta, where the confining potential
dominates, marks a crossover from a Gaussian distribution valid at low momenta
to a power-law behavior valid at high momenta. We show that for all momenta the
momentum distribution can be parametrized by a $q$-Gaussian distribution whose
parameters are specified by the confining potential. Furthermore, we find that
the functional form of the probability of transitions between the confined
ground state and the $n^{th}$ excited state is invariant under scaling of the
ratio $Q^2/\nu_n$, where $Q$ is the transferred momentum and $\nu_n$ is the
corresponding excitation energy. Using the scaling variable $Q^2/\nu_n$ the
maxima of the transition probabilities can also be expressed in terms of a
$q$-Gaussian.
