Solution of the time dependent Schrödinger equation leading to Fowler-Nordheim field emission

We solve the time-dependent Schrödinger 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 ψ(x,0), representing a generalized eigenfunction when E=0, we find ψ(x,t) and show that it approaches, as t→∞, the Fowler-Nordheim tunneling wavefunction ψE. The deviation of ψ from ψE decays asymptotically as a power law t-32. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds. We plot the short-time evolution of the current and density.