We solve the time-dependent Schrodinger 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 $ttoinfty$, the Fowler-Nordheim tunneling wavefunction $psi_E$. The deviation of $psi$ from $psi_E$ decays asymptotically as a power law $t^{-frac32}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.
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