In the present work we consider a time-dependent Schrödinger equation for systems invariant under the reparametrization of time. We develop the two-stage procedure of construction such systems from a given initial ones, which is not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schrödinger equation. T...

We have studied the charge capture and impact excitation processes in H+ on He+ collisions over a wide range of collision energies by solving the time-dependent Schrödinger equation with the classical trajectory approximation for the projectile. The time-dependent Schrödinger equation is solved by the split-operator method with a generalized pseudospectral (nonuniform grid) method in the energy...

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximatedwith the Chebyshev pseudo-spectral collocationmethod in space variable and the Crank–...

Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The CaldirolaKanai oscillator belongs to this class. For a general time-dependent h...