  We solve rigorously the time dependent Schr\"odinger equation describing
electron emission from a metal surface by a laser field perpendicular to the
surface. We consider the system to be one-dimensional, with the half-line $x<0$
corresponding to the bulk of the metal and $x>0$ to the vacuum. The laser field
is modeled as a classical electric field oscillating with frequency $\omega$,
acting only at $x>0$. We consider an initial condition which is a stationary
state of the system without a field, and, at time $t=0$, the field is switched
on. We prove the existence of a solution $\psi(x,t)$ of the Schr\"odinger
equation for $t>0$, and compute the surface current. The current exhibits a
complex oscillatory behavior, which is not captured by the "simple" three step
scenario. As $t\to\infty$, $\psi(x,t)$ converges with a rate $t^{-\frac32}$ to
a time periodic function with period $\frac{2\pi}{\omega}$ which coincides with
that found by Faisal, Kami\'nski and Saczuk (Phys Rev A 72, 023412, 2015).
However, for realistic values of the parameters, we have found that it can take
quite a long time (over 50 laser periods) for the system to converge to its
asymptote. Of particular physical importance is the current averaged over a
laser period $\frac{2\pi}\omega$, which exhibits a dramatic increase when
$\hbar\omega$ becomes larger than the work function of the metal, which is
consistent with the original photoelectric effect.
