Ian Jauslin

## Exact solution of the 1D time-dependent Schrödinger equation for the emission of quasi-free electrons from a flat metal surface by a laser

2019

### Abstract

We solve exactly the one-dimensional Schrödinger equation for $\psi(x,t)$ describing the emission of electrons from a flat metal surface, located at $x=0$, by a periodic electric field $E\cos(\omega t)$ at $x>0$, turned on at $t=0$. We prove that for all physical initial conditions $\psi(x,0)$, the solution $\psi(x,t)$ exists, and converges for long times, at a rate $t^{-\frac32}$, to a periodic solution considered by Faisal et al. Using the exact solution, we compute $\psi(x,t)$, for $t>0$, via an exponentially convergent algorithm, taking as an initial condition a generalized eigenfunction representing a stationary state for $E=0$. We find, among other things, that: (i) the time it takes the current to reach its asymptotic state may be large compared to the period of the laser; (ii) the current averaged over a period increases dramatically as $\hbar\omega$ becomes larger than the work function of the metal plus the ponderomotive energy in the field. For weak fields, the latter is negligible, and the transition is at the same frequency as in the Einstein photoelectric effect; (iii) the current at the interface exhibits a complex oscillatory behavior, with the number of oscillations in one period increasing with the laser intensity and period. These oscillations get damped strongly as $x$ increases.

PDF:

LaTeX source:
• tarball: 19ccjl-1.0.tar.gz
• git repository: 19ccjl-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).

### Data

The data for the figures in this paper was obtained using the program 'photocomp', which I wrote, but is not released yet. Stay tuned!

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