Solution of the time dependent Schr\"odinger equation leading to Fowler-Nordheim field emission

We solve the time-dependent Schr"odinger equation describing the emission of electrons from a metal surface by an external electric field $E$, turned on at $t=0$. Starting with a wave function $\psi (x,0)$, representing a generalized eigenfunction when $E=0$, we find $\psi (x,t)$ and show that it approaches, as $t\to \mathrm{\infty }$, the Fowler-Nordheim tunneling wavefunction ${\psi }_E$. The deviation of $\psi$ from ${\psi }_E$ decays asymptotically as a power law $t^{-\frac{3}{2}}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.