On the Mapping of Time-Dependent Densities onto Potentials in Quantum Mechanics

摘要

The mapping of time-dependent densities on potentials in quantum mechanics iscritically examined. The issue is of significance ever since Runge and Gross(Phys. Rev. Lett. 52, 997 (1984)) established the uniqueness of the mapping,forming a theoretical basis for time-dependent density functional theory. Weargue that besides existence (so called v-representability) and uniquenessthere is an important question of stability and chaos. Studying a 2-levelsystem we find innocent, almost constant densities that cannot be constructedfrom any potential (non-existence). We further show via a Lyapunov analysisthat the mapping of densities on potentials has chaotic regions in this case.In real space the situation is more subtle. V-representability is formallyassured but the mapping is often chaotic making the actual construction of thepotential almost impossible. The chaotic nature of the mapping, studied for thefirst time here, has serious consequences regarding the possibility of usingTDDFT in real-time settings.