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We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.

We obtain a partial differential equation for a pulse travelling inside a Bose-Einstein condensate under conditions of electromagnetically induced transparency. The equation is valid for a weak probe pulse. We solve the equation for the case of a three-level BEC in Λ configuration with one of its ground state spatial profiles initially constant. The solution characterizes, in detail, the effect that the evolution of the condensate wave function has on pulse propagation, including the process of stopping and releasing it.

The Lagrangian formulation has been an extensive tool for the analysis of physical systems. In particular, we have applied the Lagrangian procedure to deduce the dynamics and stability for an electric pendulum system. We have considered two cases, a repulsive and attractive electric interactions as perturbations to the classical simple pendulum model. We study both cases, the repulsive and attractive electric interactions that can be considered as perturbations to the classical simple pendulum model. We have contrast both situations studying their restrictions, phase trajectories and stability points for this purpose.