Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field

Abstract

We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space $x⩽0$, the Schrödinger equation of the system is $i{\partial }_t\psi =-\frac{1}{2}{\partial }_x^2\psi +\Theta (x)(U-Ex\cos \omega t)\psi$, $t>0$, $x\in \mathbb{R}$, where $\Theta (x)$ is the Heaviside function and $U>0$ is the effective confining potential (we choose units so that $m=e=\hslash =1$). The amplitude $E$ of the external electric field and the frequency $\omega$ are arbitrary. We prove existence and uniqueness of classical solutions of this equation for general initial conditions $\psi (x,0)=f(x)$, $x\in \mathbb{R}$. When the initial condition is in $L^2$ the evolution is unitary and the wave function goes to zero at any fixed $x$ as $t\to \infty$. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-$L^2$ initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all $t>0$. For these we show that the solution approaches in the large $t$ limit a periodic state that satisfies an infinite set of equations formally derived by Faisal, et. al. Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior. It shows a steep increase in the current as the frequency passes a threshold value $\omega ={\omega }_c$, with ${\omega }_c$ depending on the strength of the electric field. For small $E$, ${\omega }_c$ represents the threshold in the classical photoelectric effect.

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