Some properties of the resonant state in quantum mechanics and its computation

TL;DR

This paper investigates the properties of resonant states in quantum mechanics, demonstrating particle conservation with an expanding volume, and introduces new numerical methods for resonance computation and time evolution analysis.

Contribution

It presents a novel numerical approach for finding resonance poles and analyzing resonant states without relying on complex scaling techniques.

Findings

Particle number conservation with expanding volume

Super-convergent iterative method for resonance poles

Numerical trick for time evolution in finite regions

Abstract

The resonant state of the open quantum system is studied from the viewpoint of the outgoing momentum flux. We show that the number of particles is conserved for a resonant state, if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles would decay exponentially in a fixed volume of integration. Moreover, we introduce new numerical methods of treating the resonant state with the use of the effective potential. We first give a numerical method of finding a resonance pole in the complex energy plane. The method seeks an energy eigenvalue iteratively. We found that our method leads to a super-convergence, the convergence exponential with respect to the iteration step. The present method is completely independent of commonly used complex scaling. We also give a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Since the wave function of the resonant state is diverging away from the scattering potential, it has been previously difficult to follow its time evolution numerically in a finite area.