Nonperturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field
Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz
2022
Abstract
We analyze nonperturbatively the onedimensional Schrodinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the halfspace $x\geqslant 0$, the Schrodinger equation of the system is $i\partial_t\psi=\frac12\partial_x^2\psi+\Theta(x) (UE x \cos(\omega t))\psi$, $t>0$, $x\in\mathbb R$, where $\Theta(x)$ is the Heaviside function and $U>0$ is the effective confining potential (we choose units so that $m=e=\hbar=1$). The amplitude $E$ of the external electric field and the frequency $\omega$ are arbitrary. We prove existence and uniqueness of classical solutions of the Schrodinger 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 [Phys. Rev. A 72, 023412 (2005)] under the assumption that the solution is periodic. Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution is very complicated. 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.
This paper presents a mathematical proof of the results anticipated in [CCJL19].
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 This article was peer reviewd and published in Communications in Mathematical Physics, volume 402, pages 20312078, 2023
 arXiv preprint: arXiv:2209.07570.
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[CCJL18]: Solution of the time dependent Schrodinger equation leading to FowlerNordheim field emission
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[CCJL19]: Exact solution of the 1D timedependent Schrodinger equation for the emission of quasifree electrons from a flat metal surface by a laser
Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz, 2019
(published in Journal of Physics A: Mathematical and Theoretical, volume 53, issue 36, number 365201, 2020)
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