% -*- LaTeX -*-
\chapter{Limitations of a discrete model}

\section{A formal problem: aliasing}

As I mentioned in section~\ref{txt:cyclic}, the indices of $\Phi_{m_1\ldots
m_D}$ are cyclic, so in the discrete representation the space of momenta
has the topology of a torus rather than that of a Euclidean space.  What
causes this behavior and what are its effects?

On the purely formal level, the cyclic nature of the momenta stems
from the fact that the exponentials of equation~\eq{dft,p} are
periodic.  However, there is a more fundamental reason having to do
with the fact that $\Psi_{l_1\ldots l_D}$ is discretely sampled.  Let
us consider a wave~function of the form $\Psi(\x) = e^{i k_0 x_n}$ (for
some fixed index $n$).  This is a momentum eigenfunction with the
momentum pointed in the direction of the $n$th coordinate axis.  Its
discrete counterpart, along the lines of equation~\eq{sample,x}, is
\begin{equation}
\Psi_{\ldots l_n\ldots} = e^{i \; k_0 \; l_n \delta_n} ,
\end{equation}
where all indices other than $l_n$ are immaterial.  If the wave number
$k_0$ is increased by $2\pi\over\delta_n$, the discrete values become
\begin{equation}
\Psi'_{\ldots l_n\ldots} = e^{i (k_0 + \frac{2\pi}{\delta_n}) l_n \delta_n}
	= e^{i (k_0 \; l_n \delta_n + 2\pi l_n)}
	= e^{i \; k_0 \; l_n \delta_n} .
\end{equation}
There is no change in the sampled values of $\Psi$!  If the phase of $\Psi$
changes by some amount between two successive points, the phase of $\Psi'$
will change by that amount plus one full period, and since we have no way
of examining the data between the successive points, the extra period will
not change the value of the wave~function.  This makes it clear that
discretization inherently removes the distinction between momenta whose
components differ by ${2\pi\over\delta_i}\hbar$.  All of such momenta will
contribute to the same value of $\Psi_{l_1\ldots l_D}$ and become
indistinguishable from each other, which is known as {\it aliasing.}

This is a serious issue, since those differing momenta contribute very
differently to the wave~function propagation.  There are two basic ways of
resolving the conflict: one can ensure that only a single one of the set of
indistinguishable momenta can be available to the system, or one can
abandon the conventional kinetic-energy operator $\K = \hat\p^2/2m$ and
adopt one that is consistent with the periodic boundary conditions
mentioned above.

Choosing a different kinetic-energy operator is an intriguing
proposition, and it would perhaps be particularly interesting to
consider an ``infinitely mirrored'' system for which both coordinates
and momenta have periodic boundary conditions.  However, the drawback
is that this kind of periodic boundary conditions isn't commonly
studied in examinations of chaotic systems in classical mechanics.
This means that any serious interpretation of such a
quantum-mechanical system would require a full-fledged examination of
the behavior of the corresponding periodic classical system, which
would be at least as involving as the quantum-mechanical analysis
itself.  Since my work had been progressing in a different direction,
diverting my attention to a study of periodic Hamiltonians would have
required more time than I had available.

For a non-periodic $\K$, each value that can be taken by an index of
$\Phi_{m_1\ldots m_D}$ must be made to represent only one of the possible
corresponding values of the momentum.  The simplest way to achieve this is
to require $\Phi(\k)$ to be approximately zero everywhere outside some
specific range of momenta.  If the ratio $\delta_i/\hbar$ is then adjusted
to satisfy
\begin{equation}
{\delta_i\over\hbar} < { 2\pi \over (p_i)_{\text{max}} - (p_i)_{\text{min}} } ,
\label{eq:delta<}
\end{equation}
then any two accessible values of $p_i$ can only differ by less than
${2\pi\over\delta_i}\hbar$, which means that no aliasing will occur.

The astute reader has probably noticed that, just as the
transform~\eq{dft,p} makes the indices of $\Phi$ cyclic, the opposite
transform~\eq{dft,x} makes the indices of the resulting $\Psi$ cyclic.
This implies that $\Psi$ needs to be rescued from an analogous aliasing
problem.  Since the distance between the two adjacent points in the $i$th
dimension is $\delta_i$, the indistinguishable points will be separated by
multiples of $N_i \delta_i$.  (The same result can be obtained by a more
rigorous analysis parallel to the one performed for $\Phi$.)  Thus, if the
positions are restricted to a limited area, the condition
\begin{equation}
\delta_i > { (x_i)_{\text{max}} - (x_i)_{\text{min}} \over N_n }
\label{eq:delta>}
\end{equation}
ensures that any value of each of the indices corresponds to exactly one
possible position.

For any bound state in a conservative potential, the positions and momenta
are already restricted by the fact that the total energy of the system is
conserved.  The wave~function may tunnel into the classically forbidden
regions of either positions or momenta, but it will decay exponentially and
eventually become insignificantly small.  Thus, given a conservative system
whose energy states are all bound, and given some fixed value of its total
energy, we can find a set of ranges of positions and momenta outside of
which the position-representation and momentum-representation wave~functions
practically vanish.  The equations~\eq{delta<} and~\eq{delta>} then provide
the conditions required to avoid aliasing problems.

\section{A physical problem: good sampling}

The discussion in the previous section is based solely on the mathematical
properties of discrete Fourier transforms, and it merely insures the
self-consistency of the computational model.  It turns out that there are
additional physical requirements which must be considered, and which give
rise to constraints that are stronger than the conditions of the previous
section.

Given the physical nature of our model, positive momenta are equally
likely as the negative ones.  It is therefore most reasonable to
assume that the available range of momenta is centered around $\p=0$,
i.e. that $(p_i)_{\text{min}} = -(p_i)_{\text{max}}$.  Without any
loss of generality we can also assume that the available range of
coordinates is also centered around $\x=0$, since we are free to
choose the origin for the potential.

If a wave~function is localized in momentum, then its phase in
coordinate representation precesses with the wavelength approximately
equal to the de Broglie wavelength
\begin{equation}
\lambda = {2\pi\hbar \over p} .
\label{eq:deBroglie}
\end{equation}
If we compare this with condition~\eq{delta<}, we see that the
condition is equivalent to
\begin{equation}
\delta_i < { \pi\hbar \over p_{\text{max}} } = {\lambda_{\text{min}}\over2} ,
\label{eq:lambdahalf}
\end{equation}
i.e. the distance between the grid points is allowed to be as large as
half of the shortest possible wavelength.  Since the real and the
imaginary part of the wave~function are periodic with a period that is
approximately equal to the wavelength, this requirement is clearly not
strong enough, since taking only two points for each period of the
function is almost guaranteed to lose most of the meaningful data.  A
commonly used ``rule of thumb'' for this kind of a numerical model of
a wave~function is that the grid spacing must be no larger than $\lbar
= \lambda/2\pi$, which results in a stronger condition
\begin{equation}
{\delta_i\over\hbar} < { 1\over p_{\text{max}} } .
\label{eq:delta,p}
\end{equation}

Of course, since the state under consideration is also localized in
position, the same problem will occur for the wave~function in
momentum representation, which will precess with the ``wavelength''
\begin{equation}
\Lambda = {2\pi\hbar \over x}
\label{eq:deBroglie,p}
\end{equation}
where $\Lambda$ has units of momentum (not wave number).  An analogous
chain of reasoning leads to the condition
\begin{equation}
\delta_i > \pi { (x_i)_{\text{max}} - (x_i)_{\text{min}} \over N_n }.
\label{eq:delta,x}
\end{equation}

The same result can be achieved by other means, for example by
considering the quantization of the area in phase space bounded by any
classical trajectory.  This illustrates that it is a fundamental
requirement linked to the fact that the wave~function that resembles a
classical particle is localized in both position and momentum.

The two new conditions, \eq{delta,p} and~\eq{delta,x}, are both much
stronger than the conditions that come from the mathematical
properties of discrete Fourier transforms.  This means that their
restrictions apply within the restrictions of the discrete grid; in
either representation, large areas of the grid will be inaccessible to
the wave~function in order to insure adequate sampling in the other
representation.  In each dimension, less than $1/\pi$ of the range
covered by the grid is actually available.  This may at first seem
like a waste of good grid points, but it is impossible to avoid: it is
a fundamental limitation of discretization, in position and momentum
representations, of a wave~function which is localized in momentum and
position.
