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. The real-space pair-correlation function calculated in this way can, in principle, be used to construct improved exchange-correlation functionals to solve electronic structure problems. Furthermore, we find that the functional form of the probability of transitions between the confined ground state and the
nth excited state is invariant under scaling of the ratio ${Q}^{2}/{\ensuremath{\nu}}_{n}$, where
Q is the transferred momentum and ${\ensuremath{\nu}}_{n}$ is the corresponding excitation energy. Using the scaling variable ${Q}^{2}/{\ensuremath{\nu}}_{n}$ the maxima of the transition probabilities can also be expressed in terms of a
q-Gaussian.
Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.