New computational methodologies in time dependent quantum

Many physical phenomena are often described by partial differential equations (PDEs) which are usually not analytically solvable. The behavior and general properties of physical systems described by the PDEs can be ussually guessed from a mathematical reasoning. We are interested, however, in visualization and quantification of these effects - thus in their numerical simulations. When realistic systems are considered, equations in quantum physics are diffcult objects to deal with. Mainly, because of high oscillations, high dimentionality, nonlinearities, and singularities.

During the lifetime of the project, we will be interested in PDEs describing the evolution of the hydrogen atom subjected to a time dependent electric field and in Dirac equations.

A hydrogen atom immersed in a time dependent electric field brings numerical difficulties thanks to a singularity at origin of coordinates and posible effects of resonances. On the other hand, Dirac equations are chalenging (numerically) because the Pauli matrices give a richer structure to the PDEs involved when magnetic and electric potential are considered.

The main goal of this project is to introduce new numerical methodologies specified for the equations of interest.