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arXiv · 0705.1388

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

Naomichi Hatano·Keita Sasada·Hiroaki Nakamura·Tomio Petrosky
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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.

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Naomichi Hatano, Keita Sasada, Hiroaki Nakamura, Tomio Petrosky. 2007-05-10. Some properties of the resonant state in quantum mechanics and its computation. https://doi.org/10.1143/ptp.119.187

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