DescriptionWe present a new method for calculating the time-dependent many-body wavefunction that follows a local quench. We use the method to find the exact wavefunction of several quantum impurity models, focusing mainly on the nonequilibrium Kondo model driven by a bias voltage. The Bethe ansatz is not used, and integrability does not appear to play any role. We show that the long time limit (with the system size taken to infinity first) of the time-evolving wavefunction is a current-carrying nonequilibrium steady state, which we can also find directly with our method. We evaluate the steady state current in the regimes of strong and weak coupling, finding a new universal regime of emph{strong ferromagnetic} coupling with Kondo temperature $T_K = D e^{-frac{3pi^2}{8}
ho |J|} (J<0, ho|J|oinfty). In this regime, the differential conductance dI/dV reaches the unitarity limit 2e^2/h asymptotically at emph{large} voltage or temperature.