Ian Jauslin
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authorIan Jauslin <jauslin@ias.edu>2019-05-14 11:28:19 -0400
committerIan Jauslin <jauslin@ias.edu>2019-05-14 11:28:19 -0400
commitf34f94a8fd01975cf2e17a8a8a7e04665aed1cca (patch)
tree102901914bcfd8f117afa2d9127c429d7ad64573 /Jauslin_SMM121_2019.tex
As presented at tthe 121st Statistical Mechanics Meeting on 2019-05-14HEADv1.0master
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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}