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

@article{Costin2018SolutionOT,
  title={Solution of the time dependent Schr{\"o}dinger equation leading to Fowler-Nordheim field emission},
  author={Ovidiu Costin and Rodica D. Costin and Ian Jauslin and Joel L. Lebowitz},
  journal={Journal of Applied Physics},
  year={2018}
}
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 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… 
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