Numerical Studies On Wave-particle Duality And Anderson Localization Phenomenon

For a quantum electron moving in a lattice system, intuitively, it exhibits fully particlelike behavior if it completely localizes on a site, while fully wavelike behavior if it uniformly distributes on all sites. However, it is generally in intermediate states and it has both particlelike and wavelike properties, i.e., the wave-particle duality. On the other hand, Anderson localization is one of the most important and interesting phenomena in solid-state physics, which predicts electron localization in disordered systems. For electron systems, Anderson transitions, i.e., metal-insulator transitions or delocalization-localization transitions have received most attentions. We discuss how to measure particlelike, wavelike and localization properties of one-dimensional single electron lattice systems.Firstly, we introduce a distinguishability D and a visibility V to measure a single electron’s particlelike and wavelike behaviors, respectively. They satisfy the relation D2+V2=1. For one-dimensional nonuniform lattice systems, numerical results show that D2 is less than, equal to, and greater than V2 for delocalized, critical, and localized states, respectively. In this sense, the Anderson transition can be viewed as a transition from relatively more particlelike behavior to relatively more wavelike ones.Secondly, we propose a novel quantity to measure the degree of localization properties of various types of one-dimension single electron states. The quantity includes information about the spatial variation of probability density of quantum states. Numerical results show that it can distinguish localized states from delocalized ones, so it can be used as a fruitful index to monitor the localization-delocalization transition. Comparing with existing measures, such as geometric average density of states, inverse participation ratio, and quantum information entropies, our proposed quantity has some advantages over them.