我正在尝试搜索此 Latex 演示文稿的源代码：http://www.math.wisc.edu/~ynagel/tex/GPStalk.pdf你知道我怎样才能做出这样的演讲吗？
答案1
此次演讲显然是高级 LaTeX 短期课程，它是由电源点。以下是源代码：http://www.math.wisc.edu/~ynagel/tex/GPStalk.tex
\documentclass[
mode=present,
paper=screen,
orient=landscape,
display=slides,
style=default
]{powerdot}
\usepackage{epsfig}
\usepackage{amsbsy}
\usepackage{rotating}
\usepackage{graphicx}
\usepackage{natbib}
\usepackage{boxedminipage}
\usepackage{my_macros}
\usepackage{movie15}
\usepackage{rotate}
%\usepackage[monochrome]{color}
%\setlength{\unitlength}{1cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% title
\title{A Mixed Classical/Quantum Transport Model}
\author{Kyle Novak}
\date{28 Sept 2005}
%\institution{Department of Mathematics \\ University of Wisconsin--Madison}
%\email{[email protected]}
%\slideCaption{Kyle Novak}
%\Logo{\includegraphics[height=.9cm]{UW_crest_1color}}
\begin{document}
\maketitle
\section[slide=false]{Background}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}[toc=,bm=]{Overview}
\tableofcontents[content=sections]
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Motivation}
We want to study quantum scale phenomena using a largely classical scale model.\\[12pt]
\begin{minipage}[c]{1.25in}
\includegraphics[width=1in]{quantumdot.ps}
\end{minipage}
\begin{minipage}[c]{2.75in}
\begin{itemize}
\item{Nanotechnology}
\item{Electron transport in semiconductors}
\item{Tunneling diodes}
\item{Quantum dot structures}
\item{Quantum computing}
\end{itemize}
\end{minipage}
\\[.25in]
\begin{minipage}[c]{1.25in}
\includegraphics[width=1in]{coherence}
\end{minipage}
\begin{minipage}[c]{2.75in}
\begin{itemize}
\item{Multi-scale problems}
\item{High frequency limit, geometric optics, fluids}
\item{Numerical PDEs}
\end{itemize}
\end{minipage}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Schr\"odinger Equation}
\ceqnbox{pdlblue}{i\hbar\D{}{t}\Psi(x,t) = -\frac{\hbar^2}{2m}\Delta \Psi(x,t) + V(x) \Psi(x,t)}
Physical interpretation
\begin{itemize}
\item $|\Psi|^2$: Position probability density $\rho(x,t)$
\item $|\widehat{\Psi}|^2$: Momentum probability density
%\item Heisenberg relation $\sigma_x \sigma_p > \eps /2$
\item Wavepacket = particle
\end{itemize}
\pause
Wave packet solution
\[
\Psi(x,t;p_0) = \intI \phi(p-p_0)\psi_E(x,t)e^{iEt/\hbar} \,dp
\]
where $\psi_E$ solves %the Sturm-Liouville problem
$ -\frac{\hbar^2}{2m} \Delta \psi + V(x) \psi = E \psi$
with Hamiltonian $E = p^2/2m - V(x)$ along particle trajectory
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Gaussian Wave Packet}
For symmetry in position and momentum ($\sigma_x \sim \sigma_p = \sqrt{eps/2}$), we take
\[
\phi(p-p_0) = \exp(-(p-p_0)^2/2\eps)/(\pi\eps)^{1/4}
\]
Simplified behavior of wave packet
\begin{itemize}
\item Away from quantum region acts like a particle
\item Near a quantum region acts like a wave
%\begin{itemize}
% \item Resonance
% \item Partial transmission/reflection
% \item Interference
%\end{itemize}
\end{itemize}
\begin{center}
\includemovie[poster=Wavepacket.ps]{8.00cm}{3.0cm}{Wavepacket.mpg}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide\
\begin{slide}{Scale}
Consider characteristic length and time:
\begin{center}
$L \delta x$ and $L\delta t$ (where $\delta x = \lambda = \hbar/p$)
\end{center}
Rescale $x$ and $t$:
\[
i\eps\D{}{t}\Psi_t = (-\frac{\eps^2}{2m} \Delta + V(x)) \Psi
\WHERE \eps = \hbar /L
\]
What happens when $L\to\infty$ and $\eps \to 0$?
\begin{center}
\onslide*{2}{\includegraphics[height=1.25in]{ant_huge0}}
\onslide*{3}{\includegraphics[height=1.25in]{ant_big0}}
\onslide*{4}{\includegraphics[height=1.25in]{ant_small0}}
\end{center}
\end{slide}
\section[slide=false]{Semi-classical Limit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Wigner Equation}
Schr\"odinger equation
\[
\mathcal{S} \Psi(x,t) = \D{}{t} \Psi(x,t) + \frac{\eps}{2i} \Delta \Psi(x,t) - \frac{1}{\eps i} V(x) \Psi(x,t) = 0
\]
Wigner distribution function
\[
f(x,p,t) = \mathcal{W}[\Psi,\Psi] = \intI \conj \Psi(x-\tfrac12\eps y,t)\Psi(x+\tfrac12\eps y,t)e^{ipy}\,dy
\]
Wigner equation:
$\mathcal W[\mathcal S \Psi,\Psi] + \mathcal W[\Psi,\mathcal S \Psi] = 0$
\pause
\[
\D{}{t} f + p\grad f + \Theta = 0 \WHERE
\]
\[
\Theta = -\frac{1}{\eps i} \int \left[ V(x+\tfrac12\eps y) - V(x - \tfrac12 \eps y) \right] \hat f(x,y,t) e^{-ipy} \,dy
\]
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Semiclassical limit}
\[
\Theta = \grad_xV \cdot \grad_p f - \frac{1}{\eps i} \sum_{n=1}^\infty \frac{(-1)^n \(\frac{\eps}{2}\)^{2n}}{(2n+1)!} \grad_x^{2n+1} V \grad^{2n+1}_p f
\]
\pause
For $V(x)$ smooth, when $\eps \to 0$
\[
\Theta \to \grad_xV \cdot \grad_p f(x,p,t)
\]
Liouville equation:
\ceqnbox{pdlblue}{\dd{}{t}f = \D{}{t}f + \frac{p}{m} \grad_x f - \grad_xV \cdot \grad_p f = 0}
\pause
Characteristics $\equiv$ Hamiltonian system:
\begin{align*}
\dot{\mathbf{x}} &= \frac{\mathbf{p}}{m} \\
\dot{\mathbf{p}} &= -\grad_x V(x) = \mathbf{F}
\end{align*}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
%\begin{slide}{Schr\"odinger/Liouville}
%\begin{align*}
% \( \D{}{t} + \frac{\eps}{2mi}\Delta - \frac{1}{\eps i} V(x)\) \Psi(x,t) &= 0\\
% \(\D{}{t} + \frac{p}{m} \grad_x - \grad_x V \cdot \grad_p\) f(x,p,t) &= 0
%\end{align*}
% wave particle\\
% linear in $\Psi$ linear in $f$
% $p,x$ dependent independent\\
%\begin{align*}
% |\Psi(x,t)|^2 &= \intI f(x,p,t) \,dp\\
% |\hat \Psi(x,t)|^2 &= \intI f(x,p,t) \,dx
%\end{align*}
%\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Why not use the Schr\"odinger Equation?}
Liouville equation
\begin{itemize}
\item Arbitrary particle distribution, \emph{but}
\item No wave phenomena: tunneling, resonance, partial transmission/reflection, interference
\end{itemize}
\pause
Schr\"odinger equation
\begin{itemize}
\item Accurately models particle at any scale, \emph{but}
\item Single particle ($x$ and $p$ distribution are not independent)
\item Numerically, we must resolve the de~Broglie wavelength.
Typically, $\Delta x = O(\eps/p)$ or $\Delta x = o(\eps/p)$
\item Numerically, difficult to implement boundary conditions
\end{itemize}
\pause
\vspace{12pt}
\fcolorbox{pdlblue}{white}{
\begin{minipage}{.5in}
\emph{Idea!}
\end{minipage}
\begin{minipage}{2.5in}
Use Liouville equation globally. \\
Use Schr\"odinger equation locally.
\end{minipage}
}
\end{slide}
\section[slide=false]{Mixed Model}{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{How do we do it?}
Coupling a quantum barrier with a Liouville
\begin{itemize}
\item Solve the time-independent Schr\"odinger equation for the a local barrier/well
\item Use the solution to determine scattering information
\item Solve the Liouville equation everywhere else
\item Use scattering information to connect across the barrier
\end{itemize}
Previous research
\begin{itemize}
\item N. Ben Abdallah, P. Degond and I.M. Gamba (2002)
\item S. Jin and X. Wen (2005)
\end{itemize}
Simplifying assumptions
\begin{itemize}
\item We work in 1-d
\item Particle moves instantaneously across the barrier
\item Barrier is sufficiently local
\item Particle has no phase information (no long range interaction)
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
%\begin{slide}{Outline}
%\begin{itemize}
% \item[] Solve the Schr\"odinger equation\\[12pt]
% \item[] Determine scattering information\\[12pt]
% \item[] Solve the Liouville equation\\[12pt]
% \item[] Connect across the barrier\\[12pt]
%\end{itemize}
%\end{slide}
\section[slide=false]{Schr\"odinger Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Transfer Matrix}
\onslide*{1}{
Solve $\frac{\eps^2}{2m} \Delta \Psi - V(x) \Psi = E \Psi$ where
\[
V(x) = \begin{cases}
V_1, & x\in\mathcal{C}_1\\
V_\mathcal{Q}(x), & x\in\mathcal{Q}\\
V_2, & x\in\mathcal{C}_2
\end{cases}
\]
\begin{center}
\psfrag{A}{}
\psfrag{B}{}
\psfrag{V1}{$\mathcal{C}_1$}
\psfrag{V2}{$\mathcal{C}_2$}
\psfrag{VQ}{$\mathcal{Q}$}
\includegraphics[width=3in]{single_barrier}
\end{center}
}
\onslide*{2-3}{
\begin{flushleft}
$
\text{In } \mathcal{C}: \Psi(x) = \begin{cases}
a_{1} e^{i\kappa_{1} x} + b_{1} e^{-i\kappa_{1} x}, & x\in\mathcal{C}_1\\
b_{2} e^{i\kappa_{2} x} + b_{2} e^{-i\kappa_{2} x}, & x\in\mathcal{C}_2
\end{cases}
$
\begin{flushright}
$\WHERE \kappa_j = \sqrt{p^2 - 2mV_j}/\eps$
\end{flushright}
\end{flushleft}
\begin{center}
\psfrag{A}[c]{$\omat{a_1 \rightarrow\\ b_1\leftarrow}$}
\psfrag{B}[c]{$\omat{ \rightarrow a_2 \\\leftarrow b_2}$}
\psfrag{V1}{$\mathcal{C}_1$}
\psfrag{V2}{$\mathcal{C}_2$}
\psfrag{VQ}{$\mathcal{Q}$}
\includegraphics[width=3in]{single_barrier}
\end{center}
}
\onslide*{3}{
In $\mathcal{Q}$: linear, 2nd-order BVP, so $\pmat{a_2\\b_2} = \mathsf{M} \pmat{a_1 \\ b_1}$
}
\onslide*{4-5}{
Multiple barriers\\[12pt]
\begin{center}
\psfrag{A}[c]{\small$\omat{a_1 \rightarrow\\ b_1\leftarrow}$}
\psfrag{B}[c]{\small$\omat{ \rightarrow a_n \\\leftarrow b_n}$}
\psfrag{M1}{$\mathsf{M}_1$}
\psfrag{M2}{$\mathsf{M}_2$}
\psfrag{...}{$\dots$}
\psfrag{MN}{$\mathsf{M}_n$}
\includegraphics[width=3in]{multiple_barriers}
$\pmat{a_n\\b_n} = \underbrace{\mathsf{M_n} \mathsf{M}_{n-1} \cdots \mathsf{M_1}}_{\displaystyle \mathsf{M}}\pmat{a_1 \\ b_1}$
\end{center}
}
\onslide*{5}{
\vspace{12pt}
Two simple barriers
\begin{itemize}
\item Step ($\mathsf{D}$)
\item Translation ($\mathsf{P}$)
\end{itemize}
}
\onslide*{6}{
Arbitrary barrier
\begin{center}
\includegraphics[width=3in]{steps}
\end{center}
\begin{align*}
\mathsf{M}_j &= \mathsf{P}^{1/2}_{j+1}\mathsf{D}_j\mathsf{P}^{1/2}_j\\
\mathsf{D}_j &= \tfrac12\pmat{1+\kappa_{j-1}/\kappa_j & 1-\kappa_{j-1}/\kappa_j \\ 1-\kappa_{j-1}/\kappa_j & 1+\kappa_{j-1}/\kappa_j }\\
\mathsf{P}_j &= \pmat{\exp(i\Delta x \kappa_j) & 0 \\ 0 & \exp(-i\Delta x \kappa_j)}
\end{align*}
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Scattering Matrix}
\begin{center}
\psfrag{A}[c]{$\omat{a_1 \rightarrow\\ b_1\leftarrow}$}
\psfrag{B}[c]{$\omat{ \rightarrow a_2 \\\leftarrow b_2}$}
\psfrag{V1}{}
\psfrag{V2}{}
\psfrag{VQ}{}
\includegraphics[width=3in]{single_barrier}
\end{center}
\[
\begin{array}{cc}
\text{Transfer matrix}&\qquad\text{Scattering matrix}\\
\pmat{a_2\\b_2} = \mathsf{M} \pmat{a_1 \\ b_1}&\qquad \pmat{b_1\\a_2} = \mathsf{S} \pmat{a_1 \\ b_2}\\
\end{array}
\]
\begin{align*}
\mathsf{M} &= \pmat{m_{11} & m_{12} \\ m_{21} & m_{22}}\\
\mathsf{S} &= \pmat{r_1 & t_2 \\ t_1 & r_2} = \pmat{-m_{21}/m_{22} & 1/m_{22} \\ \det\mathsf{M}/m_{22} & m_{12}/m_{22}}
\end{align*}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Current Density}
Schr\"odinger equation
\[
\mathcal{S} \Psi(x,t) = \D{}{t} \Psi(x,t) + \frac{\eps}{2i} \eps \Delta \Psi(x,t) -\frac{1}{\eps i} V(x) \Psi(x,t) = 0
\]
Consider:
\[
2 \Re[\conj \Psi \mathcal{S} \Psi ] = \conj \Psi \mathcal{S} \Psi + \Psi \conj{\mathcal{S} \Psi} =0
\]
\pause
Continuity equation
\[
\D{}{t}\rho(x,t) + \nabla \cdot J = 0
\]
where probability current density $J = \eps \Im[\conj \Psi \grad \Psi]$
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Scattering Coefficients}
\begin{minipage}{2.5in}
$\Psi = \begin{cases}
a_{1} e^{i\kappa_{1} x} + b_{1} e^{-i\kappa_{1} x}, & x\in\mathcal{C}_1\\
a_{2} e^{i\kappa_{2} x} + b_{2} e^{-i\kappa_{2} x}, & x\in\mathcal{C}_1
\end{cases}$
\end{minipage}
\begin{minipage}{1.4in}
\begin{center}
\psfrag{A}[c]{$\begin{smallmatrix} a_1 \rightarrow \\ b_1 \leftarrow \end{smallmatrix}$}
\psfrag{B}[c]{$\begin{smallmatrix} \rightarrow a_2 \\ \leftarrow b_2 \end{smallmatrix}$}
\psfrag{V1}{}
\psfrag{V2}{}
\psfrag{VQ}{}
\includegraphics[width=1.4in]{single_barrier}
\end{center}
\end{minipage}
So,
\[
J(x) = \eps \Im[\conj \Psi \grad \Psi] =
\begin{cases}
\kappa_1 (|a_1|^2 - |b_1|^2),& x\in\mathcal{C}_1\\
\kappa_2 (|a_2|^2 - |b_2|^2),& x\in\mathcal{C}_2
\end{cases}
\]
\pause
Particle incident from left: $b_2 = 0$ then $a_2 = t_1a_1$ and $b_1 = r_1a_1$
\[
J(x) = \begin{cases}
\kappa_1 |a_1|^2(1 - |r_1|^2), x\in\mathcal{C}_1\\
\kappa_2 |a_2|^2 |t_1|^2, x\in\mathcal{C}_2
\end{cases}
\]
\pause
\fcolorbox{pdlblue}{white}{
\begin{minipage}{3in}
\begin{tabular}{ll}
Reflection probability & $R_1 = |r_1|^2$\\
Transmission probability & $T_1 = (\kappa_2/\kappa_1) |t_1|^2$
\end{tabular}
\end{minipage}
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Resonance and Tunneling}
\psfrag{Transmission}{\tiny Transmission}
\psfrag{Momentum}{\tiny Momentum}
\onslide*{1-3}{Rectangular potential with height = $1/2$ and width $2\eps$\\[12pt]}
\onslide*{4-6}{Rectangular potential with height = $-1/2$ and width $8\eps$\\[12pt]}
\onslide*{1}{Step up}
\onslide*{2}{Step up + step down \textbf{independently}}
\onslide*{3}{Step up + step down \textbf{combined}}
\onslide*{4}{Step down}
\onslide*{5}{Step down + step up \textbf{independently}}
\onslide*{6}{Step down + step up \textbf{combined}}
\begin{center}
\onslide*{1}{\includegraphics[width=3in,height=2in]{pic1}}
\onslide*{2}{\includegraphics[width=3in,height=2in]{pic2}}
\onslide*{3}{\includegraphics[width=3in,height=2in]{pic3}}
\onslide*{4}{\includegraphics[width=3in,height=2in]{pic4}}
\onslide*{5}{\includegraphics[width=3in,height=2in]{pic5}}
\onslide*{6}{\includegraphics[width=3in,height=2in]{pic6}}
\end{center}
\end{slide}
\section[slide=false]{Liouville Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Semi-classical Liouville Equation}
\begin{minipage}{1.25in}
\begin{center}
\includegraphics[width=1in]{reflect_transmit}
\end{center}
\end{minipage}
\begin{minipage}{2.5in}
Bicharacteristics:
\begin{itemize}
\item Classical particle is either transmitted \textbf{or} reflected
\item Quantum particle is generally both transmitted \textbf{and} reflected
\end{itemize}
\end{minipage}
\begin{itemize}
\item Hamiltonian $p^2/2m - V(x)$ constant along characteristics
\item Particle density $f(x,p,t)$ carried along bicharacteristics
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Finite Difference Scheme}
\onslide*{1}{
\ceqnbox{pdlblue}{f_t + v f_x - V_x f_v = 0}
Grid points at $(x_i,v_j)$.
Barrier at $x_{Z+1/2}$.
\[
\partial_t f_{ij} + v_j \cdot \partial_x f_{ij} - \partial_x V_i \cdot \partial_v f_{ij} = 0
\]
where
$
\partial_x f_{ij} = (f_{i+1/2,j} - f_{i-1/2,j})/\Delta x
$
\psfrag{x_0}{\scriptsize$x_{i-1}$}
\psfrag{x_1}{\scriptsize$x_{i}$}
\psfrag{x_2}{\scriptsize$x_{i+1}$}
\psfrag{f_0-}[l][r]{\scriptsize$f_{i-1/2}$}
\psfrag{f_0+}[c]{}
\psfrag{f_1-}[l][r]{\scriptsize$f_{i+1/2}$}
\psfrag{f_1+}[c]{}
\begin{center}
\includegraphics[width = 3in]{upwind}
\end{center}
Stability requires upwinding to approximate $f_{i\pm1/2}$\\
}
\onslide*{2}{
\psfrag{x_0}{\scriptsize$x_{i-1}$}
\psfrag{x_1}{\scriptsize$x_{i}$}
\psfrag{x_2}{\scriptsize$x_{i+1}$}
\psfrag{f_0-}{\scriptsize$f^-_{i-1/2}$}
\psfrag{f_0+}{\scriptsize$f^+_{i-1/2}$}
\psfrag{f_1-}{\scriptsize$f^-_{i+1/2}$}
\psfrag{f_1+}{\scriptsize$f^+_{i+1/2}$}
\begin{center}
\includegraphics[width = 3in]{upwind}
\end{center}
Where $f_{i\pm1/2}$ is continuous, $f_{i\pm1/2}^{-} = f_{i\pm1/2}^{+}$
\begin{align*}
\partial_x f_{ij} &= \frac{f^{-}_{i+1/2,j} - f^{-}_{i-1/2,j}}{\Delta x} \quad\text{if}\quad v_j>0\\
\partial_x f_{ij} &= \frac{f^{+}_{i+1/2,j} - f^{+}_{i-1/2,j}}{\Delta x} \quad\text{if}\quad v_j<0
\end{align*}
}
\end{slide}
\begin{slide}{Barrier Interface}
At the quantum barrier $x_{Z+1/2}$, we need to incorporate information from two bicharacteristics.\\[12pt]
Barrier interface condition
\begin{align*}
f^+_{Z+1/2,j} &= R_{-j}f^+_{Z+1/2,-j} + T_{-j}f^-_{Z+1/2,w(j)} \\
f^-_{Z+1/2,j} &= R_{-j}f^-_{Z+1/2,-j} + T_{-j}f^+_{Z+1/2,w(j)}.
\end{align*}
\onslide*{1}{
\begin{center}
\psfrag{rj}[r][r]{\scriptsize$j$}
\psfrag{r-j}[r][r]{\scriptsize$-j$}
\psfrag{rR-j}[r][r]{\scriptsize$R_{-j}$}
\psfrag{rT-j}[r][r]{\scriptsize$T_{-j}$}
\psfrag{r-w-j}[r][r]{\scriptsize$w_{j}$}
\psfrag{lj}[l]{\scriptsize$j$}
\psfrag{l-j}[l]{\scriptsize$-j$}
\psfrag{lR-j}[l]{\scriptsize$R_{-j}$}
\psfrag{lT-j}[l]{\scriptsize$T_{-j}$}
\psfrag{l-w-j}[l]{\scriptsize$w_{j}$}
\includegraphics[width = 3in]{barrier_interface}
\end{center}
}
\onslide*{2}{
We use the approximation
\[
T_{-j}f^+_{Z+1/2,w(j)} = \frac{1}{v_j \Delta v} \int_{w(v_{j-1/2})}^{w(v_{j+1/2})} T(v) v f^- \,dv
\]
where we use Hamiltonian to determine $w$
\[
w(v_{\pm|j|)} = \pm \sqrt{v_j^2 \pm 2(V_{Z+1/2}^+ - V_{Z+1/2}^-)}
\]
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{2nd order method}
Piecewise linear:
\begin{align*}
f_{i-1/2,j}^{+} &= f_{i,j} - \tfrac12 \(1- \lambda_j\) \Delta x \sigma^x_{ij} \\
f_{i+1/2,j}^{-} &= f_{i,j} + \tfrac12 \(1- \lambda_j\) \Delta x \sigma^x_{ij}
\end{align*} with the slope $\sigma^x_{ij}$ calculated using the Van~Leer slope limiter
\[
\sigma^x_{ij} = \( \frac{f_{ij} - f_{i-1,j}}{\Delta x}\)
\phi \( \frac{ f_{i+1,j} - f_{ij} }{f_{ij} - f_{i-1,j} } \)
\WHERE
\phi(\theta) = \frac{\theta + |\theta|}{1+|\theta|}
\]
and the Courant number $\lambda_j = |v_j| \Delta t / \Delta x$\\[12pt]
\begin{flushleft}
\fcolorbox{pdlblue}{white}{
%\begin{minipage}{4in}
%Since this calculation uses values on both sides of the barrier, the slope needs to be modified at the %barrier\emph{!}
%\end{minipage}
We can't do this directly across at the barrier\emph{!}
}
\end{flushleft}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Ghost fluid}
Across the barrier, we need to reconstruct ``unmixed'' flux. \\[12pt]
For $j>0$,
\begin{align*}
f_{Z+1,-w(-j)} &= T_j \tilde f_{Z+1,j} + R_j \tilde f_{Z,w(-j)} \\
f_{Z,-j} &= R_j \tilde f_{Z+1,j} + T_j \tilde f_{Z,w(-j)}
\end{align*}
with a similar system for $j<0$.
By inverting this system of equations, we have the unmixed state downwind of the barrier
\begin{align*}
\tilde f_{Z+1,j} &= \frac{T_j f_{Z+1,-w(-j)} - R_j f_{Z,-j}}{T_j - R_j} \qquad\text{when }j>0\\
\tilde f_{Z,j} &= \frac{T_j f_{Z,-w(-j)} - R_j f_{Z+1,-j}}{T_j - R_j} \qquad\text{when }j<0
\end{align*}
\end{slide}
\section[slide=false]{Examples}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\def\classical{\rotatebox{90}{\footnotesize classical}{ }}
\def\quantum{\rotatebox{90}{\footnotesize quantum}{ }}
\begin{slide}{Step Potential}
$V(x) = 0$ if $x<0$ and $V(x) = -\tfrac12$ if $x>0$, $v_0 = \tfrac14$, $\eps = .005$
\begin{center}
\classical\includemovie[poster=Qstep.ps]{8.00cm}{3.0cm}{Cstep.mpg}
\end{center}
\begin{center}
\quantum\includemovie[poster=Qstep.ps]{8.00cm}{3.0cm}{Qstep.mpg}
\end{center}
\end{slide}
%%% slide
\begin{slide}{Eckart Potential}
$V(x) = -2\sech^2(4x/\eps)$ with $\eps = .005$
\begin{center}
\classical\includemovie[poster=QEckart.ps]{8.00cm}{3.0cm}{CEckart.mpg}
\end{center}
\begin{center}
\quantum\includemovie[poster=QEckart.ps]{8.00cm}{3.0cm}{QEckart.mpg}
\end{center}
\end{slide}
%%% slide
\begin{slide}{Tunneling Diode}
%$V(x) = -x + \chi_{[-\eps/2,\eps/2]}(x)$
$V(x) = -x + \text{Rect}[-\eps/2,\eps/2](x)$ with $\eps = .005$
\begin{center}
\classical\includemovie[poster=QTunnel.ps]{8.00cm}{3.0cm}{CTunnel.mpg}
\end{center}
\begin{center}
\quantum\includemovie[poster=QTunnel.ps]{8.00cm}{3.0cm}{QTunnel.mpg}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Rectangular Potential}
$V = \tfrac12$, width = $25\eps$, $v_0 = 0$, $\eps = 0.005$
\begin{center}
\quantum\includemovie[poster=QResonant.ps]{8.50cm}{3.1098cm}{QResonant.mpg}
\end{center}
\end{slide}
\section[slide=false]{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% slide
\begin{slide}{Research Directions}
Simplifying assumptions
\begin{itemize}
\item Particle moves instantaneously across the barrier
\item Barrier is sufficiently local
\item Particle has no phase information (no long range interaction)
\end{itemize}
Incorrect/inaccurate for
\begin{itemize}
\item Larger quantum structures
\item Smaller domains (nonvanishing $\eps$)
\item Periodic crystalline structures
\item Highly resonant barriers
\end{itemize}
Extension of model
\begin{itemize}
\item Introduce time delay
\item Introduce phase information
\item Reconstruct solution inside the quantum barrier
\end{itemize}
\end{slide}
%\begin{slide}{Conclusion}
%Relatively simple model
%\begin{itemize}
% \item Accurate for a large class of problems
% \item Arbitrary probability distribution\dots not just $\delta$-distributions
% \item Connections to several related problems
%\end{itemize}
%\end{slide}
\begin{slide}{Thank you}
\vspace{1in}
\begin{center}
Questions?
\end{center}
\end{slide}
\end{document}
答案2
元数据上写的是“Radical Eye Software”，我的 Google-Fu 不足以收集更多信息。
