\documentclass[12pt]{article}
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{0pt}
\setlength{\textheight}{9in}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
\title{Physics 381C Project Outline}
\author{Todd Timberlake}
\date{5 March 1997}
\pagestyle{empty}
\begin{document}
\maketitle
\begin{description}
\item[I] Introduction
	\begin{description}
	\item[A] Discussion of high-harmonic generation and its
    	relation to classical chaos.
	\item[B] The model:  sinusoidally driven triangle well.
    	\end{description}
\item[II] Numerical Computation of Radiation Spectra
    	\begin{description}
	\item[A] Compute the solution to the Time-Dependent
    	Schr\"{o}dinger's Equation.
		\begin{description}
		\item[1] This can be done by using a PDE solver and an
    	initial condition for the wavefunction \ldots
		\item[2] \ldots or by decomposing the solution into a
    	basis of energy eigenstates for the triangle well and solving
    	for the coefficients of that decomposition.  This second
    	method helps avoid any complications caused by the infinite
    	wall in the potential.
		\end{description}	
	\item[B] Once the solution is computed, the radiation
    	spectrum can be calculated by taking the Fourier transform of
    	$<\psi(x,t)|\ddot{x}|\psi(x,t)>$.
	\end{description}
\item[III] Periodic Orbit Theory
	\begin{description}
	\item[A] Locate periodic orbits of the classical driven
	triangle well.
		\begin{description}
		\item[1] This is essentially a shooting problem with
	two parameters ($x_{0}$ and $p_{0}$).
		\item[2] Only periodic orbits with periods
	$T = \frac{2\pi n}{\omega}$, where $\omega$ is the frequency
	of the driving force (classical electric field), need to be identified.
		\item[3] Periodic orbits need to found up to some
	specified maximum period ($T_{max} = \frac{2\pi n_{max}}{\omega}$).
		\end{description}
	\item[B] For each periodic orbit calculate the trace of the
	stability matrix.
	\item[C] For each periodic orbit calculate the classical
	action along that orbit.
	\item[D] This information can then be used to construct a
	finite-resolution density of states for the quasi-energies
	(Floquet eigenvalues) of the system.
	\end{description}
\item[IV] Semiclassical Calculation of Radiation Spectra
	\begin{description}
	\item[A] Decompose the initial quantum state into wave
	packets.
	\item[B] Propagate these wave packets using a semiclassical
	propagator (i.e. along classical trajectories).
	\item[C] When the wave packets return to the nucleus, calculate
	the probability for a transition back to the initial state and
	emission of a photon.
	\item[D] Use this information to compute a radiation spectrum
	for the system.
	\item[E] Note: this theory has not yet been fully worked out.
	\end{description}
\item[V] Classical Chaos
	\begin{description}
	\item[A] If time permits, strobe plots of the classical phase
	space can be computed to show the onset of chaos in the
	classical system.
	\item[B] The field strengths at which the classical phase
	space becomes mixed (part regular, part chaotic) and globally
	chaotic (no regular motion) should be identified.
	\end{description}
\item[VI] Conclusion
	\begin{description}
	\item[A] What do these results say about the connection
	between classical chaos and high-harmonic generation?
	\item[B] What do the results say about the problem of
	``quantum chaos''.
	\end{description}
\end{description}
\end{document}