%\addtocounter{chapter}{1}

\chapter{Introduction}

\section{Motivation}

One of the particularly useful ways of studying classical chaotic
systems is the examination of their periodic orbits.  It seems
sensible to assume that an examination of a similar concept for
a quantum-mechanical chaotic system might yield equally illuminating
results.  The work that follows examines a possible way of achieving
this, and applies it to a particular example sized to be computationally
feasible.

First, I will discuss the time-evolution operator in its usual form,
and then demonstrate several approximations well-suited to be used in
a numerical model that uses a discretized wave~function.
This provides the background needed to define the quantum-mechanical
return probabilities of a wave~packet.  I will suggest a possible
relationship between those return probabilities and the periodic
orbits of an analogous classical system.

I will also discuss some of the limitations of the numerical model I
used, and mention ways of avoiding those limitations.  This numerical
model will then be applied to the study of a specific problem: the
behavior of a quantum-mechanical system based on the Delfi
potential~\cite{delfi} for the energy $E=0.3$.  Finally, the results
of this computation will be discussed.

\section{A few notes on notation}

I have attempted to express the ideas behind this text as clearly as I
could.  However, in order to avoid loading the text with repetitive
explanations of what the reader may already be familiar with, I
adopted some notational conventions that are in more or less common
use in the physics community.

Vector quantities are typeset in boldface, so $\x$ denotes a vector
while $x_i$ is a scalar.  An integral over a vector quantity is
assumed to be an integral over the entire set of values of that
quantity; thus, $\int d^D \x$ is just shorthand for
$\mkern-6mu
\int\limits_{-\infty\mkern12mu}^{\infty}
\mkern-12mu{\cdotp}{\cdotp}{\cdotp}\mkern-12mu
\int\limits_{-\infty\mkern12mu}^{\infty}
\mkern-4mu dx_1\ldots dx_D$.

Since some of the covered topics invite confusion between state
variables or functions and the corresponding operators, the operators
are identified by a circumflex (``hat''), for example:
$\hat A\ket|\alpha> = A\ket|\alpha>$.

I will primarily use the Dirac bra-ket notation because it seems to
clarify certain questions, such as whether a state is being considered
time-dependent or fixed.  Unless otherwise noted, all kets under
consideration will be viewed in the Schr\"{o}dinger ``picture'': the
basis vectors are held time-independent, and the ket corresponding to
an evolving state changes with time.

Particular ket symbols will be used for certain commonly used
eigenkets.  The energy eigenfunctions of a system are written as
$\ket|n>$, with $\H\ket|n> = E_n\ket|n>$.  The ket $\ket|\x>$ is an
eigenket of position with the eigenvalue $\x$.  (More verbosely,
$\ket|\x>$ is a simultaneous eigenket of all position operators $\hat
x_i$, and the respective eigenvalues of each of those operators are
the components of the vector $\x$.)  Likewise, $\ket|\p>$ is an
eigenket of momentum with the eigenvalue $\p$.

I will occasionally make use of the convention, originated by
Einstein, of implied summation over a repeated index variable that occurs on
only one side of the equation.  This helps with the readability of
long expressions with many sums, where writing out the summations
explicitly would be cumbersome.  However, since I prefer writing out
sums where it is possible to do so, the use of this convention could
go unnoticed and cause some confusion.  To prevent that, I will make a
note of its use in the text surrounding the equations.

Finally, matrix mechanics sometimes blurs the distinction between
infinite sums and integrals over a basis.  For instance, the
completeness requirement for a basis is usually written as
$\sum\limits_a \ket|a>\bra<a| = 1$,
but when applied to the $\ket|\x>$ basis, it really means to say
$\int \ket|\x>\bra<\x| d^D\x = 1$.
Those two forms are clearly discrete and continuous formulations of
the same idea, but the sum and the integral are in fact not equivalent
in the strict mathematical sense.
%\footnote{The weight of the
%contribution of each ket's term to the total is one in the case of the
%sum, and a differential quantity in the case of the integral.}
However, the relationship between the two is trivial, and it is
usually clear whether summation or integration should be applied for a
particular set of kets.  Consequently, I will often write a summation
over a set of basis vectors where I really mean ``summation or
integration, as appropriate,'' and I hope I will be forgiven for it.

This notation will also be used to extend the Einstein convention to
an implied {\it integral\/} over a set of basis vectors when
expressions with matching indices (like $\ket|\x_i>\bra<\x_i|$) occur
on one side of an equation.  As with the regular implied summation,
any use of this will be noted in the text.
