Tunneling of interacting particles through the potential barriers: computer simulation by quantum molecular dynamics method

Yu. E. Lozovik ; A. V. Filinov ; A. S. Arkhipov

Matematičeskoe modelirovanie, Tome 15 (2003) no. 7, pp. 18-36

Résumé

The tunneling of two interacting particles represented by wave packets through the potential barrier is considered. To solve this problem we use the method of computer simulation of nonstationary quantum processes (quantum molecular dynamics) based on the Wigner formulation of quantum mechanics. The main idea of the method is to use the ensemble of classical trajectories to solve quantum Wigner-Liouville equation. Those trajectories are determined by the equations analogous to classical equations of motion, with the only distinction that the force has extra “quantum” term. That term depends on the Wigner function and its derivatives. In the context of the method Wigner function is approximated in the small region near every point of the phase space by local many-dimensional Gauss distribution. Local moments of the ensemble of trajectories (covariance coordinate-coordinate, momentum-momentum, coordinatemomentum, average coordinates and momenta) define the parameters of that Gauss distribution and through that the “quantum” term of the force. In the framework of the method the quantum effects arise due to two sources. First, trajectories are not independent, the force for each of them is determined by the distribution of ensemble of the trajectories in phase space. Second, initial distribution has specific form because it is built with the help of initial Wigner function and latter must satisfy some rules, for example the principle of uncertainty must held. The influence of the interaction between the particles on the tunneling is investigated. For the configuration when both particles initially are situated on the same side of the barrier and move to it we have shown that the stronger the reflection between the particles the greater is the probability to find a particle after the barrier. The dependence of the tunneling time of wave packet on the strength of interaction is discussed.