The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position S x and momentum S p Shannon information entropies for the low-lying states = n 0, 1 are calculated. Some interesting features of the information entropy densities ρ x ( )and ρ p ( ) s as well as the probability densities ρ x ( ) and ρ p ( ) are demonstrated. We find that the choices of the values for those parameters have to satisfy the condition on n max . We also notice that the ρ p ( ) and ρ p ( ) s are symmetric to the momentum p and the ρ x ( ) or ρ p ( ) is equal or greater than 1 at some positions r or momentum p. In addition, the Bialynicki-Birula-Mycielski inequality is tested from different cases and found to hold for these cases.

We consider position-dependent effective mass particles in double heterostructures, subject to the action of several potentials. A comparative study of the energy spectra of these particles is carried out, emphasizing the effect of different boundary conditions associated with the kinetic energy operator. Energy spectra that cannot be expressed analytically, have been estimated by means of the corresponding reflection coefficient poles. The heterostructure model adopted assumes a middle region, where the potential and mass are some finite distributions of the position, but outside of it its behavior is constant. Finally, we discuss the gain of a double parabolic quantum well in a laser.

We study the ($1+1$)-dimensional Dirac equation for charge carriers in some heterostructures. Both, the mass profile and the modulated Fermi velocity of the quasi-particle, are considered position dependent. We have used mass and Fermi velocity that admit only approximate analytical solutions. However, we also calculate numerically the \textit{exact} energy spectra of each heterostructure through the corresponding reflection coefficient poles.

This work presents a novel methodology to analytically solve the stationary Schrödinger equation in presence of a couple of two-dimensional semi-infinite rectangular potential barriers, when the incident wave is a finite-width monoenergetic wave packet. Such methodology does not depend at all on the incident wavefront of the packet and is based on the transfer-matrix method, but unlike the latter, our transfer matrix is built partly in real space and partly in Fourier space. A spectrum of angular plane waves is used to represent the incident, reflected and transmitted beams. As a particular case, we study the transmission of Hermite-Gaussian wave packets through the barrier system. A detailed analysis of the transmission coefficient is carried out as a function of both the parameters of the incident beam (which in turn are directly related to the shape of the incident packet) and the parameters of the barriers. We also briefly discuss the behavior of the probability density of three transmitted beams.