Schroedinger Equation

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

Theoretical and numerical investigations are carried out for the amplitude modulation of dust-ion acoustic waves (DIAW) propagating in an unmagnetized weakly coupled collisionless fully ionized plasma consisting of isothermal electrons, warm ions and charged dust grains. Modulation oblique (by an angle $\theta$ ) to the carrier wave propagation direction is considered. The stability analysis, based on a nonlinear Schrödinger-type equation (NLSE), exhibits a sensitivity of the instability region to the modulation angle $\theta$ , the dust concentration and the ion temperature. It is found that the ion temperature may strongly modify the wave’s stability profile, in qualitative agreement with previous results, obtained for an electron-ion plasma. The effect of the ion temperature on the formation of DIAW envelope excitations (envelope solitons) is also discussed.

We design non-singular cloaks enabling objects to scatter waves like objects with smaller size and very different shapes. We consider the Schrodinger equation which is valid e.g. in the contexts of geometrical and quantum optics. More precisely, we introduce a generalized non-singular transformation for star domains, and numerically demonstrate that an object of nearly any given shape surrounded by a given cloak scatters waves in exactly the same way as a smaller object of another shape. When a source is located inside the cloak, it scatters waves as if it would be located some distance away from a small object. Moreover, the invisibility region actually hosts almost-trapped eigenstates. Mimetism is numerically shown to break down for the quantified energies associated with confined modes. If we further allow for non-isomorphic transformations, our approach leads to the design of quantum super-scatterers: a small size object surrounded by a quantum cloak described by a negative anisotropic heterogeneous effective mass and a negative spatially varying potential scatters matter waves like a larger nano-object of different shape. Potential applications might be for instance in quantum dots probing.

The purpose of this paper is to provide an analytic treatment of the hydrogen atom in its ground energy state by considering the hydrogen atom as a quantum system rather than a classical system, through a theoretical investigation. This analysis involves the method and the solution to the Schrödinger wave equation for the said hydrogen atom. Furthermore, a graphical analysis is conducted of the plots of the electron density function and the radial distribution function of the theoretical hydrogen atom in question. The research question that will be investigated is, `What conclusions can be made regarding the information obtained by applying the Schrödinger equation on the hydrogen atom and investigating its relevant wavefunctions?'

By examining the intrinsic properties of the hydrogen atom and the spherical coordinate system, the necessary information can be obtained to solve the Schrödinger wave equation for the hydrogen atom. The solutions to this equation are two pieces of important information that could not be found by simply analyzing the properties of the hydrogen atom; the energy of the single electron and the most probable radius for an electron to be located at surrounding the atom or the Bohr radius. Two other important functions, the electron density function and the radial distribution function, regarding the quantum nature of the hydrogen atom are then derived, plotted and analyzed.

The solution to the Schrödinger equation for the hydrogen atom was that the energy of the electron was 13.6eV and that the Bohr radius was 5.29 x 10^-11m. There were also many significant consequences that were then reached from analyzing the plots of the two aforementioned functions such as the fact that there is a probability of the electron in the hydrogen atom existing anywhere in the universe for when the radius is greater than zero.

We derive an exact analytical solution to the time-dependent Schr\"odinger equation for transmission of a Gaussian wave packet through an arbitrary potential of finite range. We consider the situation where the initial Gaussian wave packet is sufficiently broad in momentum space to guarantee that the resonance structure of the system is included in the dynamical description. We demonstrate that the transmitted wave packet exhibits a transient behavior which at very large distances and long times may be written as the free evolving Gaussian wave packet solution times the transmission amplitude of the system and hence it reproduces the resonance spectra of the system. This is a novel result that predicts the ultimate fate of the transmitted Gaussian wave packet. We also prove that at a fixed distance and very long times the solution goes as $t^{-3/2}$ which extends to arbitrary finite range potentials previous analysis on this issue. Our results are exemplified for single and multibarrier systems.

We are concerned with the following equation −ε 2 ∆u + V (x)u = f (u), u(x) > 0 in R 2. By a variational approach, we construct a solution u ε which concentrates, as ε → 0, around arbitrarily given isolated local minima of the confining potential V : here the nonlinearity f has a quite general Moser's critical growth, as in particular we do not require the monotonicity of f (s)/s nor the Ambrosetti-Rabinowitz condition.

We consider a diffusion process on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schroedinger equations on a quantum graph are discussed.

For a Dirac operator in \( {\Bbb R}^3 \) , with an electric potential behaving at infinity like a power of |x|, we prove the existence of resonances and we study, when \( c \rightarrow + \infty \) , the asymptotic expansion of their real part,and an estimation of their imaginary part, generalizing an old result of Titchmarsh.

It is shown that a generalized nonlinear Schrödinger equation proposed by Malomed and Stenflo admits, for a specific range of parameters, resonant soliton interaction. The equation is transformed to the'resonant'nonlinear Schrödinger equation, as originally introduced to ...