  We present a method for calculating the time-dependent many-body wavefunction
that follows a local quench. We apply the method to the voltage-driven
nonequilibrium Kondo model to find the exact time-evolving wavefunction
following a quench where the dot is suddenly attached to the leads at $t=0$.
The method, which does not use Bethe ansatz, also works in other quantum
impurity models (we include results for the interacting resonant level and the
Anderson impurity model) and may be of wider applicability. In the particular
case of the Kondo model, 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 that satisfies the
Lippmann-Schwinger equation. We show that the electric current in the
time-evolving wavefunction is given by a series expression that can be expanded
either in weak coupling or in strong coupling, converging to all orders in the
steady-state limit in either case. The series agrees to leading order with
known results in the well-studied regime of weak antiferromagnetic coupling and
also reveals another universal regime of strong ferromagnetic coupling, with
Kondo temperature $T_K^{(F)} = D e^{-\frac{3\pi^2}{8} \rho |J|}$ ($J<0$,
$\rho|J|\to\infty$). In this regime, the differential conductance $dI/dV$
reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or
temperature.
