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\documentclass{ian-presentation}
\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{array}
\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Exact solution of the time dependent Schrödinger\par
\hfil equation for photoemission from a metal surface
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Ovidiu Costin}, {\bf Rodica Costin}, and {\bf Joel L. Lebowitz}\par
\vfil
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject
\setcounter{page}1
\pagestyle{plain}
\title{Electron emission}
$$
V(x)=U\Theta(x)
,\quad
E_{\mathrm F}=k_{\mathrm F}^2<U
$$
\hfil\includegraphics[height=5cm]{potential_square.pdf}
\vfill
\eject
\title{Thermal emission}
$$
V(x)=U\Theta(x)
,\quad
k^2>U
$$
\hfil\includegraphics[height=5cm]{potential_square_thermal.pdf}
\vfill
\eject
\title{Field emission}
$$
V(x)=\Theta(x)(U-Ex)
$$
\hfil\includegraphics[height=5cm]{potential.pdf}
\vfill
\eject
\title{Photoemission}
$$
V_t(x)=\Theta(x)(U-E_tx)
,\quad
E_t=2\epsilon\omega\cos(\omega t)
$$
\hfil\includegraphics[height=5cm]{potential_square_photonic.pdf}
\vfill
\eject
\title{Photoemission}
\begin{itemize}
\item Photoelectric effect: early observations: \href{https://doi.org/10.1002/andp.18872670827}{[Hertz, 1887]}, \href{https://doi.org/10.1002/andp.18882690206}{[Hallwachs, 1888]}, \href{https://doi.org/10.1002/andp.19003070611}{[Lenard, 1900]}.
\item When a metal is irradiated with ultra-violet light, electrons are ionized, with kinetic energies at integer multiples of $\hbar\omega$.
\item \href{https://doi.org/10.1002/andp.19053220607}{[Einstein, 1905]}: interpretation: quanta of light ({\it photons}) of energy $\hbar\omega$ are absorbed by the electrons, whose kinetic energy is raised by $n\hbar\omega$, and can escape the metal.
\end{itemize}
\vfill
\eject
\title{Photoemission}
\vskip-10pt
\begin{itemize}
\item Time dependent potential:
$$
V_t(x)=\Theta(x)(U-2\epsilon\omega\cos(\omega t)x)
$$
\vskip-10pt
\item Schr\"odinger equation
$$
i\partial_t\psi(x,t)=-\Delta\psi(x,t)+V_t(x)\psi(x,t)
$$
\vskip-10pt
\item Magnetic gauge:
$$
\Psi(x,t)
:=\psi(x,t)e^{-ix\Theta(x)A(t)}
,\quad
A(t):=\int_0^t ds\ 2\epsilon\omega\cos(\omega s)
=
2\epsilon\sin(\omega t)
$$
satisfies
$$
i\partial_t\Psi(x,t)=\left((-i\nabla+\Theta(x)A(t))^2+\Theta(x)U\right)\Psi(x,t)
$$
\end{itemize}
\vfill
\eject
\title{Periodic solution}
\vskip-10pt
\begin{itemize}
\item \href{https://doi.org/10.1103/PhysRevA.72.023412}{[Faisal, Kami\'nski, Saczuk, 2005]}
$$
\Psi_{\mathrm{FKS}}(x,t)=\left\{\begin{array}{ll}
e^{ikx}\exp\left(-ik^2t\right)+\Psi_R(x,t)&\mathrm{\ if\ }x<0\\
\Psi_T(x,t)&\mathrm{\ if\ }x>0
\end{array}\right.
$$
$$
\Psi_R(x,t)=\sum_{M\in\mathbb Z}R_Me^{-iq_Mx}\exp\left(-iq_M^2t\right)
,\quad
q_M=\sqrt{k^2+M\omega}
$$
$$
\Psi_T(x,t)=\sum_{M\in\mathbb Z}T_Me^{ip_Mx}\exp\left(-iUt-i\int_0^td\tau\ (p_M-A(\tau))^2\right)
$$
$$
p_M=\sqrt{k^2-U+M\omega-2\epsilon^2}
$$
\item $\Psi(x,t)$, $(-i\nabla+\Theta(x)A(t))\Psi(x,t)$ are continuous.
\end{itemize}
\vfill
\eject
\title{Initial value problem}
\begin{itemize}
\item Initial condition:
$$
\Psi(x,0)=
\left\{ \begin{array}{l@{\ }l}
e^{ikx}+R_0e^{-ikx} & x<0\\
T_0 e^{-\sqrt{U-k^2}x} & x>0
\end{array}\right.
$$
$R_0$ and $T_0$ ensure that $\Psi$ and $\partial\Psi$ are continuous.
\item In progress: $\Psi(x,t)$ behaves asymptotically like $\Psi_{\mathrm{FKS}}$:
$$
\psi(x,t)
=\psi_{\mathrm{FKS}}(x,t)+\left(\frac{t}{\tau_{\mathrm{FKS}}(x)}\right)^{-\frac32}+O(t^{-\frac52})
.
$$
\end{itemize}
\end{document}
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