From f34f94a8fd01975cf2e17a8a8a7e04665aed1cca Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Tue, 14 May 2019 11:28:19 -0400
Subject: As presented at tthe 121st Statistical Mechanics Meeting on
 2019-05-14

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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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