Variational treatment of the linear potential

M. Bentaïba, L. Chétouani, Amel Mazouz

José Rachid Mohallem

Loïc Joubert-Doriol

In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on (i) the definition of a linearly independent working space and (ii) a variational construction of the propagator over finite time steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are carried out on a quartic double-well potential with Gaussian basis functions whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.

Giovanni P. Arrighini, Nicola L. Durante, Carla Guidotti

The construction of the exact quantum propagator of a particle subjected to a constant force offers a challenge to existing propagator methodology. Here, besides considering the traditional approach, which involves an expansion in the complete set of energy eigenstates, we illustrate (i) the merits of a rather more elegant procedure founded on the ‘‘disentanglement’’ of the time-evolution operator and (ii) the power of a general treatment based on a time-dependent variational approach.

Michele Cini, Luigi A. Radicati