\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{novak@math.wisc.edu}
%\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}
