Mapping from current densities to vector potentials in time-dependent current density functional theory

Abstract

Under reasonable assumptions the time-dependent particle density $n(\stackrel{P\vec}{r},t)$ and the current density $\stackrel{P\vec}{j}(\stackrel{P\vec}{r},t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\stackrel{P\vec}{r},t)$ and $\stackrel{P\vec}{A}(\stackrel{P\vec}{r},t)$ and is initially in the quantum state $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and starting from an initial state $\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{'}}(0)⟩$, which yields the same density and current as $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$. Here we show that given the initial state of this other many-particle system, the potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.