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

摘要

We solve exactly the one-dimensional Schrodinger equation for psi(x,t) describing the emission of electrons from a flat metal surface, located atx= 0, by a periodic electric field E cos(omega t) at x> 0, turned on att= 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(-3/2), to a periodic solution considered by Faisalet al(2005Phys. Rev.A72023412). 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 h 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.