Application study of virtual time evolution-split operator method to harmonic oscillator

Abstract:

We present virtual time evolution-split operator method for solving the two-dimensional time- dependent Schrodinger equation. In this method, the Hamiltonian is accessed by employing the two representations of the wave function. One is a coordinate representations, in which the coordinate dependence of the wave function is discretized using a discrete variable constructed from the Coulomb wave function. Another is the momentum representation, the time function is expanded in the corresponding. As an example, the present method is applied to the harmonic oscillator. It is found that there exists coulomb interaction when two electrons are stored in the harmonic oscillator potential well. We explore the qualitative change of two electrons ground state wave function by altering the strength of the harmonic oscillator potential. When harmonic oscillator potential is weak, the wave functions are not overlapped, which can be used an evidence for presentence of Wingner lattice. With the increase of harmonic oscillator potential, the wave functions begin to overlap, which is similar to the mokecules formation.