% -*- LaTeX -*-
\chapter{Analysis of results}

\section{Return probability plots}

A convenient way to plot the return probabilities is to gather
together the return probabilities that share the same $t$ and $\x_0$,
but whose momentum is oriented in different directions.  If we made a
polar plot of those probabilities, the directions with high
probabilities would stand out from the rest, forming ``spikes''
pointed in the directions in which the return is most likely.  Instead
of making just one plot, we can draw around each point $\x_0$ a polar
plot of return probabilities for wave packets originating from that
point, which will simultaneously display all of the return probability
data for a specific $\E$ and $t$.

The primary advantage of doing this is that the maxima of the polar
plots give the entire figure a sense of directional continuity similar
to the effect of the arrows of a quiver plot.  This visual grouping
corresponds to a group of wave packets whose return probabilities for
the specified propagation duration are high, and whose centers
approximately lie on a curved line.  (For an example, see \fig{fam4}.)
Our primary interest, motivated by discussion of
chapter~\ref{chp:return}, is to examine how closely these groupings
correspond to the classical periodic orbits for the values of $t$ when
the the return probabilities reach their local maxima.

\section{Return probabilities of the Delfi potential}

After writing software to implement the computational model described
in the preceding chapters, I computed a set of values of return
probabilities for the Delfi potential at the energy $\E=0.3$, and the
values of the other parameters chosen as described in
section~\ref{sec:parameters}.  All of the results are plotted in
Appendix~\ref{app:plots}.  Let us examine the features of those plots
and compare them with the data obtained for the classical problem.

\subsection{Family 1}

\begin{figure}[b]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_3_8.eps}
\end{center}
\caption{Return probabilities for $t=3.8$}
\label{fig:fam1}
\end{figure}
At $t=0$, the values of the return probabilities are actually all
equal to 1, since the packet hasn't yet evolved at all.  As the
packet is evolving, they rapidly become very small.  For times below
$t=3.0$, packets haven't been able to propagate far enough to get back
near their original position, so the return probabilities stay close
to zero.
After $t=3.0$, however, some of the probabilities begin to increase,
reaching a local maximum for $t=3.8$.  The data displayed in
\fig{fam1} is fairly easy to interpret.  All of the maxima of return
probabilities are exactly vertical, and their values are highest at
$\x=0$.  This suggests that any wave packet with $E=0.3$ and
$x\approx0$ that is initially moving in the vertical direction will
continue to oscillate in the vertical direction almost unchanged,
spreading out relatively slowly, until the spreading makes it
indistinguishable.

An examination of the classical behavior of the Delfi potential
indicates that one of the families of the periodic orbits, identified
as Family 1 in \cite{delfi}, does in fact consist of vertical
oscillations with $x=0$.  (The trajectories of Family 1 do stay
vertical for any energy, since
$\left. \partial V(x,y)\over\partial x \right|_{x=0} = 0$
for all $y$.)  The \Et\ plot shows that the family is unstable for
$E=0.3$, and confirms that the period is close to the return time of
the quantum model.  The trajectory for $E=0.3$ is very easy to find,
since it is known to be vertical, and its period is found to be
$\tau=3.865$, which is fairly close to the return time of the
associated maximum of return probability.

% 3.8656877754

\subsection{Family 2}

As the return time is increased above $t=3.8$, the return
probabilities decrease again, practically vanishing at $t=4.5$.  The
plots start showing visible data again slightly below $t=6$.  The
return probabilities increase more slowly than previously, and the
shape exhibited by the return probability plots changes noticeably,
which should be remembered for later analysis.

\begin{figure}[p]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_7_4.eps}
\end{center}
\caption{Return probabilities for $t=7.4$}
\label{fig:fam2}
\end{figure}
\begin{figure}[p]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/over_7_4.eps}
\end{center}
\caption{A plot of the classical Family~2 trajectory at $E=0.3$
         superimposed on the return probabilities for $t=7.4$}
\label{fig:fam2,traj}
\end{figure}
% 7.4509

The first clear maximum of the return probabilities occurs for
$t=7.4$.  According to the \Et\ plot, this corresponds to Family 2,
which starts as small horizontal oscillations at low energies.  Since
its trajectories aren't trivial like those of Family 1, finding its
trajectory for $E=0.3$ is somewhat harder.  One possible solution
involves the monodromy matrix method, which is how the \Et\ plot was
generated \cite{delfi}.  A somewhat more brute-force approach can be
made using Poincar\'e plots.  Since Family 2 is unstable, it cannot be
easily located on a Poincar\'e plot for $E=0.3$, but it is easy to
find it at a lower energy where it is stable and follow its movement
as the energy changes.  The trajectory for $E=0.3$, whose period is
$\tau=7.451$, is displayed in \fig{fam2,traj} along with the return
probabilities.

In examining \fig{fam2,traj}, one finds that the return probabilities
match the periodic orbit reasonably well in the left part of the
potential: the maxima lie close to the trajectory, and the direction
of the probabilities is almost tangential to the curve.  On the right,
however, the curve and the maxima end up visibly apart.

A part of the reason for this is optical: at the points for which a
small perpendicular element crosses the main one, the main element is
darker and therefore looks longer.  However, the region where the
return probabilities are large certainly does extend much further to
the right than the classical trajectory would suggest.
Can we explain what causes this?

\begin{figure}[b]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/over_7_4_with9.eps}
\end{center}
\caption{A plot of the classical Family~9 trajectory at $E=0.3$
         superimposed on the return probabilities for $t=7.4$
         (along with Family~2).}
\label{fig:fam2+9}
\end{figure}

\begin{figure}[p]
\vbox{\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_0.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_420.eps}%
}\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_500.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_580.eps}%
}\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_660.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-2/prop_740.eps}%
}}%
\caption{Time-evolution starting from the ``narrow'' bifurcation of Family 9}
The points between $t=0$ and $t=4.2$, when the packet moves to the
left, turns and returns on the ``wide'' branch, are not shown.
\label{fig:evol9,2}
\end{figure}
\begin{figure}[p]
\vbox{\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_0.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_420.eps}%
}\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_500.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_580.eps}%
}\hbox{%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_660.eps}%
\epsfxsize=.5\textwidth \epsfbox{plots/fam9-1/prop_740.eps}%
}}%
\caption{Time-evolution starting from the ``wide'' bifurcation of Family 9}
The points between $t=0$ and $t=4.2$, when the packet moves to the
left, turns and returns on the ``narrow'' branch, are not shown.
\label{fig:evol9,1}
\end{figure}

A look at the \Et\ plot provides a possible answer: in the region of
interest, Family 2 has a period doubling, Family 9.  At $E=0.3$, the
period of Family 9 is still close to twice the period of Family 2.  If
the the two doubled ``halves'' of Family 9 lie close enough to each
other, a packet which is initially on one of them could overlap its
propagated counterpart which is, half a period later, following the
other ``half''.

In order to verify if this guess is correct, we must actually find
Family 9, which can be done in an analogous way as with Family 2.  The
period of the orbit, which is also unstable, is $\tau=14.877$, and the
trajectory itself is displayed in \fig{fam2+9}.

The geometry of Family 9 does explain the spread of the maxima.  The
explanation is not as complete as one might wish, since the narrower
bifurcated part of the family seems to influence return probabilities
more than the wider one.  In order to explain this, we need to examine
the evolution of the wave~functions that correspond to classical
Family 9.

Packets that start in the ``narrow'' branch of the bifurcation
propagate as would be expected, so that at $t=7.4$ they are localized
at approximately the corresponding point in the ``wide'' branch
(\fig{evol9,2}).  However, the packets starting in the ``narrow''
branch initially spread much faster.  At $t\approx6.6$, when the
packet is being reflected from the side of the potential, it is almost
uniformly spread over a long segment.  As a result, after the turning
point, interference can be observed in the wave~function.  In this
case, the values of the wave~function close to its original location
increase dramatically, so that the return probability for $t=7.4$ is
found much higher than it would otherwise be.

This demonstrates that, while the correspondence between return
probabilities and the classical periodic orbits is usually very good,
there are cases where quantal phenomena---in this case, spreading and
interference---affect the return probabilities in a very visible way,
creating marked differences between cases which would be expected to
be similar if one were to rely on solely intuition about classical mechanics.

\subsection{Family 1 revisited}

As the return probabilities associated with Family 2 decrease for
$t>7.4$, the evidence of two other families begins to appear.
\begin{figure}[t]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_7_7.eps}
\end{center}
\caption{Return probabilities for $t=7.7$}
\label{fig:fam1,x2}
\end{figure}
The first of them is a recurrence of Family 1 (after two full orbits),
which peaks at $t=7.7$.  This is, as expected, very close to {\it
twice\/} the period found classically for Family 1, $2\tau=7.731$.
The vertical elements of the plot (\fig{fam1,x2}) once again follow
the shape of the classical trajectory; the curved ones are a part of
an unrelated family.

\subsection{Family 4}

The curved flow visible in the background of \fig{fam1,x2} reaches its
own maximum at $t=8.0$.  This maximum can be identified with Family 4,
which is a period doubling of Family 1, but which at this energy has a
very different shape.

\begin{figure}[ph]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_8.eps}
\end{center}
\caption{Return probabilities for $t=8.0$}
\label{fig:fam4}
\end{figure}

Unlike the other families, Family 4 is stable for $E=0.3$, and is
therefore easy to locate on a Poincar\'e plot.  The trajectory has a
period of $\tau=8.030$, and matches the return probability data
extremely well.

\begin{figure}[hbtp]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/over_8.eps}
\end{center}
\caption{A plot of the classical Family~4 trajectory at $E=0.3$
         superimposed on the return probabilities for $t=8.0$}
\label{fig:fam4,traj}
\end{figure}

\break
\subsection{Family 1, once again}

After $t=8$, the return probabilities once again decline to near-zero
values.  The only remaining feature that emerges before $t=12$, where
my calculation ended, was the third appearance of Family 1.  Once
again, the maxima of the return probabilities are aligned vertically
and grouped around $\x=0$.  The value of the return probabilities has
decreased since the second appearance of this family, just as it has
decreased between the first and second ones; since the wave~packet is
spreading, the overlap decreases with each orbit.  However, the third
maximum is still easily noticeable, particularly so since no other
features exhibit themselves in this time interval.

\begin{figure}[btph]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_11_6.eps}
\end{center}
\caption{Return probabilities for $t=11.6$}
\label{fig:fam1,x3}
\end{figure}

\subsection{A shadow of Family 2}

Let us now return our attention to the time interval between $t=6.5$
and $t=7.3$.  As the probabilities increase, the maxima arrange
themselves into a shallow horizontal shape, but at $t\approx7.2$, the
shape suddenly transforms to become much more similar to Family 4.
\begin{figure}[p]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/returns_7.eps}
\end{center}
\caption{Return probabilities for $t=7.0$}
\label{fig:shadow2}
\end{figure}
\begin{figure}[p]
\begin{center}
\leavevmode\epsfysize=\imagesizetoo\epsfbox{plots/ret/over_7.eps}
\end{center}
\caption{A plot of the classical Family~4 trajectory at $E=0.3943$
         superimposed on the return probabilities for $t=7.0$}
\label{fig:shadow2,traj}
\end{figure}
A careful examination shows that the probabilities of the original
shape do in fact exhibit a local maximum for $t=7.0$.  But where does
this maximum come from?  There are no other periodic trajectories to
be found with the appropriate period or the shape similar to the one
displayed in \fig{shadow2}.

Once again, the \Et\ plot provides a clue.  While it displays no
trajectories with period close to $7.0$ for $E=0.3$, it shows that
Family 2 reaches a minimum in energy for $\tau\approx7.0$, with the
energy of $E\approx0.4$.  Clearly, this energy would be inaccessible in
the classical model, but wave packets are composed of many energy
eigenstates and therefore could be found at a range of energies, just
as they could be found at a range of positions or momenta.

I tried to look for a
resemblance between the return probabilities and the classical
trajectory corresponding to the local energy minimum of Family 4,
which occurs for $E\approx0.3943$ and $\tau\approx6.948$.  The result,
shown in \fig{shadow2,traj}, shows a very strong similarity between
the two.

% <H> =          0.3221852549 (3.221853e-01 + 5.088042e-17 i)
% <(H-E0)^2> =   0.0057739266 (5.773927e-03 + 0.000000e+00 i)
% sigma_E    =   0.0759863578

We can verify that the spread in energy for a Gaussian wave packet
with $\E=0.3$ is large enough to cover the energy of the classical
trajectory, $E=0.3943$.  While we can't easily recover the probability
distribution in energy (which is a decomposition of the packet into
energy eigenfunctions), it is easy to calculate both the average of
the energy, $E_0 = \bra<\psi|\H\ket|\psi>$, and its variance,
$\sigma_E^2 = \bra<\psi|(\H-E_0)^2\ket|\psi>$, by numerically applying
$\H=\K+\V$.  For one of the test packets used in calculating the
return probabilities,\footnote{Specifically, the parameters of the
packet that was used were $E=0.3$, $x=0$, $y=0.18$,
$\alpha=\tan^{-1}(p_y/p_x)=30^\circ$.} the values obtained are $E_0=0.32$,
$\sigma_E^2=5.8\cdot10^{-3}$, and $\sigma_E=0.076$.  Thus, the minimum
of Family 4 is only one standard deviation away from the mean energy,
which means that our explanation makes sense.  It must be concluded
that the features observed in return probabilities can be caused by
classical behavior that is classically inaccessible to the wave packet
under consideration!

\section{Conclusions}

The material in this chapter demonstrates that, at least in the case
under consideration, there exists a strong correspondence between the
local maxima of the return probabilities of a single-particle
quantum-mechanical system and the periodic orbits of the analogous
classical system.  However, since wave packets do not possess a
single, definite value of the energy, the return probabilities can in a
smaller degree reflect the behavior of the classical system at a {\it different\/} energy.
The quantum-mechanical system also exhibits certain kinds of behavior
which are not present in its classical counterpart, such as packet
spreading, interference and tunneling.

Another point of note is that there is relatively little observed
difference between the influence of stable and unstable orbits.  The
areas with relatively large return probabilities are found to be
slightly narrower for the stable orbits (Family 4) than for unstable
ones, but the difference is slight.  The packets corresponding to
strongly unstable periodic orbits may have their probability shift to
other periodic orbits (Families 9 and 2), but this effect---if present
at all---wasn't found to be strong.  In general, the
quantum-mechanical system seems to be much less affected by stability.
(Unfortunately, this is unlikely to make quantum-mechanical models
useful for study of unstable families of classical orbits, because
good results can be obtained with much less computationally intensive
classical models, such as the monodromy method.)

In general, I found that using explicit propagation of discretized
wave~functions works reasonably well as a means of calculating return
probabilities for Gaussian wave packets.  While relatively
computationally expensive, it would probably be particularly useful
when computing the stationary states is particularly hard, or in the
cases where a fast hardware-based way of computing discrete Fourier
transforms is available.
