Simple examples of position-momentum correlated Gaussian free-particle wavepackets in one-dimension with the general form of the time-dependent spread in position
R. W. Robinett
We provide simple examples of closed-form Gaussian wavepacket solutions of the free-particle Schr\"{o}dinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely $(\Delta x_t)^2 = (\Delta x_0)^2 + At + (\Delta p_0)^2t^2/m^2$, where $A \equiv \left\langle (\hat{x}-\langle \hat{x} \rangle_0) (\hat{p}-\langle \hat{p} \rangle_0) + (\hat{p}-\langle \hat{p} \rangle_0) (\hat{x}-\langle \hat{x} \rangle_0) \right\rangle_0$ contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which $A<0$ so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the $x-p$ correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent $\Delta x_t$ and special correlated solutions, for the case of a particle subject to a uniform force. \vskip 1cm Keywords: wave packets, time-development, correlations, Gaussian
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\begin{document}

\preprint{ {\bf February 2005}}

\title{Simple examples of position-momentum correlated 
Gaussian free-particle wavepackets in one-dimension 
with the general form of the time-dependent spread in position}





\author{R. W. Robinett}
\email{rick@phys.psu.edu}
\affiliation{
Department of Physics\\
The Pennsylvania State University\\
University Park, PA 16802 USA \\
}



\author{M. A. Doncheski} \email{mad10@psu.edu}
\affiliation{%
Department of Physics\\
The Pennsylvania State University \\
Mont Alto, PA 17237 USA \\
}



\author{L. C. Bassett} \email{lcb36@cam.ac.uk}
\affiliation{%
Department of Applied Mathematics and Theoretical Physics (DAMTP) \\
University of Cambridge \\
Wilberforce Road, Cambridge, CB3~0WA, United Kingdom \\
}



\date{\today}


\begin{abstract}
We provide simple examples of closed-form Gaussian wavepacket solutions 
of  the free-particle Schr\"{o}dinger equation in one dimension
which exhibit the most general form of the time-dependent spread in position,
namely 
$(\Delta x_t)^2 = 
(\Delta x_0)^2 + At + (\Delta p_0)^2t^2/m^2$,
where $A \equiv 
\left\langle 
(\hat{x}-\langle \hat{x} \rangle_0)
(\hat{p}-\langle \hat{p} \rangle_0) 
+ 
(\hat{p}-\langle \hat{p} \rangle_0)
(\hat{x}-\langle \hat{x} \rangle_0)
\right\rangle_0$ 
contains information on the position-momentum correlation
structure of the initial wave packet. We exhibit straightforward
examples corresponding to squeezed states, as well as quasi-classical cases, 
for which $A<0$ so that the position spread can (at least initially) decrease
in time because of such correlations. We discuss how the  initial 
correlations in these examples can be dynamically
generated (at least conceptually) in various bound state systems. 
Finally, we focus on providing different ways 
of visualizing the $x-p$ correlations present in these cases, including the 
time-dependent distribution of kinetic energy and the use of the Wigner 
quasi-probability distribution. We discuss similar results, both for the 
time-dependent $\Delta x_t$ and special correlated solutions, for the case 
of a particle subject to a uniform force.

\vskip 1cm
Keywords: wave packets, time-development, correlations, Gaussian
\end{abstract}



\maketitle





\section{\label{sec_intro} Introduction}
\label{sec:introduction}

The study of time-dependent solutions of the one-dimensional
Schr\"{o}dinger equation is a frequent  topic in many
undergraduate textbooks on quantum mechanics. The problem of a Gaussian
or minimum-uncertainty wavepacket solution  for the case of a free particle
(defined more specifically below) is the most typical example cited, often 
being worked out in detail, or at least explored in problems \cite{texts}. 
The emphasis is often on the time-dependent position spread for such 
solutions, typically written in the forms
\begin{equation}
(\Delta x_t)^2 = 
(\Delta x_0)^2\left(1+\left(\frac{t}{t_0}\right)^2\right)
 =   (\Delta x_0)^2 + \frac{(\Delta p_0)^2 t^2}{m^2}
\label{not_general_case}
\end{equation}
where the spreading time or coherence time can be defined by $t_0 
\equiv m\Delta x_0/\Delta p_0$. Textbooks rightly point out the essentially
classical nature of much of this result, explained by the fact that 
the higher momentum components of the wave packet outpace the slower ones, 
giving a position-spread which eventually increases linearly with time as 
$\Delta x_t \approx \Delta v_0 t$,  where $\Delta v_0$ is identified with 
$\Delta p_0/m$.

The form of the expression for $\Delta x_t$ in Eqn.~(\ref{not_general_case})
is a special case of the most general possible form of the time-dependent
spatial width of a one-dimensional wave packet solution of the 
free-particle Schr\"{o}dinger equation which is well-known in the pedagogical
literature \cite{baird} - \cite{andrews}, but seemingly found in many fewer
textbooks \cite{merzbacher}. This general case can be written
in the form
\begin{equation}
(\Delta x_t)^2 = 
(\Delta x_0)^2 +
\left\langle 
(\hat{x}-\langle \hat{x} \rangle_0)
(\hat{p}-\langle \hat{p} \rangle_0) 
+ 
(\hat{p}-\langle \hat{p} \rangle_0)
(\hat{x}-\langle \hat{x} \rangle_0)
\right\rangle_0 \frac{t}{m}
+ \frac{(\Delta p_0^2) t^2}{m^2}
\label{general_case}
\end{equation}
where the coefficient of the term linear in $t$ measures a non-trivial 
correlation between the momentum- and position-dependence of the initial 
wave packet. 
While such correlations are initially not present in the standard Gaussian
wave packet example routinely used in textbook analyses, which therefore
gives rise to the simpler form in Eqn.~(\ref{not_general_case}), 
a non-vanishing $x-p$ correlation does develop for later times as has 
been discussed in at least
one well-known text \cite{bohm} and several pedagogical articles 
\cite{leblond}.

For wave packets which are constructed in such a way that large momentum 
components ($p > \langle \hat{p} \rangle_0$) are initially preferentially 
located in the `back' of the packet ($x < \langle \hat{x}\rangle_0$), 
the initial correlation can, in fact, be negative
leading to time-dependent wave packets which initially shrink in size,
while the long-time behavior of any 1D free particle wave packet is indeed 
always dominated
by the quadratic term in Eqn.~(\ref{general_case}), consistent with standard
semi-classical arguments. (We stress that we will consider here only 
localized wave packets which are square-integrable, for which the evaluation 
of $\Delta x_t$ and $\Delta p_t$ is possible, and not pure plane wave states 
nor the special non-spreading,  free-particle solutions discovered by Berry 
and Balazs \cite{berry}.)

For the standard Gaussian or minimum uncertainty wave packet used in most 
textbook examples, and in fact for any initial wave packet of the form 
$\psi(x,0) = R(x)\exp(ip_0(x-x_0)/\hbar)$ 
where $R(x)$ is a real function, this initial 
correlation vanishes and the more familiar special case of $\Delta x_t$
in Eqn.~(\ref{not_general_case}) results, leading many students to believe
that it  is the most general result possible. 
It is, however,  very straightforward to construct  initial quantum states 
consisting of simple Gaussian wave functions, such as squeezed states or 
linear combination of Gaussians, which have the required initial 
position-momentum correlations `built in', and which therefore exhibit 
the general form 
in  Eqn.~(\ref{general_case}), including examples where the position-space
wave packet can initially shrink in width. Since these examples can be 
analyzed with little or no more mathematical difficulty than the standard 
minimum-uncertainty cases commonly considered in textbooks \cite{texts}, 
we will focus on providing two such examples below. We will, however, also 
emphasize the utility of different ways of visualizing the time-dependent 
position-momentum correlations suggested by the form in 
Eqn.~(\ref{general_case}). 


The derivation of Eqn.~(\ref{general_case}) has been most often
discussed \cite{baird}, \cite{styer} using the evaluation of the 
time-dependence of expectation values described by 
\begin{equation}
\frac{d}{dt} \langle \hat{A} \rangle
= \frac{i}{\hbar} \left\langle [\hat{H},\hat{A}] \right\rangle
\label{time-development}
\end{equation}
using the free particle Hamiltonian, $\hat{H} = \hat{p}^2/2m$,
or related matrix methods \cite{nicola}; since we are interested only
in expectation values of operators ($\hat{A} = \hat{x}$ or $\hat{p}$) 
which are themselves
independent of time, there is no additional $\langle d\hat{A}/dt\rangle$ term
in Eqn.~(\ref{time-development}).  In the next section, we
derive the necessary time-dependent expectation values of powers of
position and momentum 
in a complementary way, using very general momentum-space ideas.
(Identical methods can then also be used to evaluate the general form 
of $\Delta x_t$ for the related case of uniform acceleration, which we
discuss in Appendix~\ref{sec:appendix}.)
Then in Sec.~\ref{sec:standard} we briefly review the special case of the
minimum-uncertainty Gaussian wave packet (to establish notation) focusing
on the introduction of useful tools to help visualize possible 
correlations between position and momentum in free particle wave
packets, especially the direct visualization of the real/imaginary
parts of $\psi(x,t)$, the time-dependent spatial distribution of kinetic 
energy, as well as the Wigner quasi-probability distribution. 
Then, in Sec.~\ref{sec:correlated}, we exhibit two cases of 
correlated wave packets with the general form of $\Delta x_t$
in Eqn.~(\ref{general_case}), which are
simple extensions of these  standard results. A similar
example, involving squeezed states, has been discussed in 
Ref.~\cite{ford}, 
but we will focus here on understanding the detailed
position-momentum correlations which give rise to the term linear in 
$t$ in Eqn.~(\ref{general_case}), especially using the techniques
outlined in Sec.~\ref{sec:standard} for their visualization.
Finally, we make some concluding remarks as well as noting
in an Appendix that very similar results (both for the general form of 
the time-dependent $\Delta x_t$  and for the exemplary cases studied
here) can be obtained for the Schr\"{o}dinger equation corresponding to
the case of constant acceleration.




\section{Time-dependent $\Delta x_t$ using momentum-space wavefunctions}
\label{sec:momentum_space}

While the general result for the free-particle $\Delta x_t$ is most 
often obtained using formal methods involving the time-dependence of 
expectation values as in Eqn.~(\ref{time-development}), 
one can also evaluate time-dependent powers of position and momentum 
for a free particle in terms of the
initial wave packet quite generally in terms of the momentum-space
description of the quantum state, namely $\phi(p,t)$, obtaining the
same results, in a manner which is nicely complementary to more standard 
analyses. Depending on the ordering of topics in a given quantum mechanics 
course syllabus, this discussion  might well be applicable and understandable 
earlier in the curriculum than the more formal method. 

In this approach, the most general momentum-space wave function 
which solves the free-particle time-dependent Schr\"{o}dinger equation 
\begin{equation}
\frac{p^2}{2m}\phi(p,t) = \hat{H} \phi(p,t) =  \hat{E} \phi(p,t)
= i\hbar \frac{\partial}{\partial t} \phi(p,t)
\, , 
\end{equation}
can be written in the form 
\begin{equation}
\phi(p,t) = \phi_{0}(p)\, e^{-ip^2t/2m\hbar}
\end{equation}
with $\phi(p,0) = \phi_{0}(p)$ being the initial state wavefunction.
The $t$-dependent expectation values for powers of momentum are trivial 
since
\begin{eqnarray}
\langle \hat{p} \rangle_t & = &  \int_{-\infty}^{+\infty}
\, p \, |\phi_{0}(p)|^2\,dp \equiv  \langle \hat{p} \rangle_0 
\label{p_1} \\
\langle \hat{p}^2 \rangle_t & = &  \int_{-\infty}^{+\infty}
\, p^2 \, |\phi_{0}(p)|^2\,dp \equiv  \langle \hat{p}^2 \rangle_0 
\label{p_2}
\end{eqnarray}
so that 
\begin{equation}
(\Delta p_t)^2 = \langle \hat{p}^2\rangle_t - \langle \hat{p}\rangle_t^2
= 
\langle \hat{p}^2\rangle_0 - \langle \hat{p}\rangle_0^2
= 
(\Delta p_0)^2
\end{equation}
as expected for a free-particle solution for which 
$|\phi(p,t)|^2 = |\phi_{0}(p)|^2$ is independent of time. 

In this representation, the position operator is given by the 
non-trivial form $\hat{x} = i\hbar (\partial/\partial p)$, and 
the time-dependent 
expectation value of position can be written as
\begin{eqnarray}
\langle \hat{x} \rangle_t & = &
\int_{-\infty}^{+\infty}
[\phi(p,t)]^{*}\, \hat{x}\, [\phi(p,t)]\,dp \nonumber \\
& = & 
\int_{-\infty}^{+\infty}
\left[\phi_{0}^{*}(p)\,e^{+ip^2t/2m\hbar}\right]
\,
\left(i\hbar \frac{\partial}{\partial p}\right)
\left[\phi_{0}(p)\,e^{-ip^2t/2m\hbar}\right]\,
dp \nonumber \\
& = & 
\int_{-\infty}^{+\infty}
[\phi_{0}^{*}(p)] \left(i\hbar \frac{\partial }{\partial p}\right)
[\phi_{0}(p)]\,dp
+ \frac{t}{m} \int_{-\infty}^{+\infty} \, p\, |\phi_{0}(p)|^2\,dp
\nonumber \\
& = & \langle \hat{x} \rangle_0 + \frac{t}{m} \langle \hat{p}\rangle_0
\label{x_1}
\end{eqnarray}
which is consistent with Ehrenfest's theorem for the essentially
classical behavior of $\langle \hat{x}\rangle_t$. 
The same formalism can
be used to evaluate $\langle \hat{x}^2\rangle_t$ and gives
\begin{equation}
\langle \hat{x}^2\rangle_t
 = 
\langle \hat{x}^2 \rangle_0
+ \frac{t}{m} \langle \hat{x} \hat{p} + \hat{p} \hat{x}\rangle_0
+ \langle \hat{p}^2\rangle_0 \frac{t^2}{m^2}
\label{x_2}
\end{equation}
where one can use the general representation-independent commutation
relation $[\hat{x},\hat{p}] = i\hbar$ to simplify the answer to this form.
The symmetric combination of position
and momentum operators, written here as $(\hat{x}\hat{p}+\hat{p}\hat{x})$, 
which is obviously Hermitian, guarantees that this expression is manifestly 
real.
(Discussions in textbooks on symmetrizing products of non-commuting operators
abound, but such results are seldom put into the context of being useful
or natural in specific calculations, as is apparent in their use here.)

Combining Eqns.~(\ref{x_1}) and (\ref{x_2}) then gives the most general
form for the time-dependent spread in position to be
\begin{eqnarray}
(\Delta x_t)^2 & = & \langle \hat{x}^2\rangle_t 
- \langle \hat{x}\rangle_t^2 \nonumber \\
& = & 
\left(\langle \hat{x}^2 \rangle_0
+ \frac{t}{m} \langle \hat{x} \hat{p} + \hat{p} \hat{x}\rangle_0
+ \langle \hat{p}^2\rangle_0 \frac{t^2}{m^2} \right)
\nonumber 
- \left(\langle \hat{x} \rangle_0 + \frac{t}{m} \langle \hat{p}\rangle_0\right)^2
\nonumber \\
& = & 
(\Delta x_0)^2 +
\left(
\langle \hat{x} \hat{p} + \hat{p} \hat{x} \rangle_0
- 2 
\langle \hat{x} \rangle_0 \langle \hat{p} \rangle_0
\right)
\frac{t}{m}
+ \frac{(\Delta p_0^2) t^2}{m^2} 
\nonumber \\
& = & 
(\Delta x_0)^2 +
\left\langle 
(\hat{x} - \langle \hat{x} \rangle_0)
(\hat{p} - \langle \hat{p} \rangle_0) 
+ 
(\hat{p} - \langle \hat{p} \rangle_0)
(\hat{x} - \langle \hat{x} \rangle_0)
\right\rangle_0 
\frac{t}{m}
+ \frac{(\Delta p_0^2) t^2}{m^2} 
\, . 
\end{eqnarray}
We have rewritten 
the term linear in $t$ in a form which stresses that it is a correlation 
between $x$ and $p$, similar in form to related classical quantities 
such as the covariance in probability and statistics. Recall that for
two classical quantities, $A$ and $B$, described by a joint probability 
distribution, the covariance is defined as
\begin{equation}
cov(A,B) = 
\left \langle
\left( A - \langle A \rangle \right)
\left( B - \langle B \rangle \right)
\right \rangle
= \langle AB \rangle - \langle A \rangle \langle B \rangle
\,. 
\end{equation}
As we will see in the next section, there is no initial correlation for 
the familiar minimum-uncertainty Gaussian wave packets. However, for simple 
variations on the standard example, as in Sec.~\ref{sec:correlated}, we will
find non-vanishing correlations, which we can visualize with the methods in
Sec.~\ref{sec:standard}.



We stress  that the notion of a time-dependent correlation between $x$ and 
$p$ at arbitrary times ($t>0)$ can be easily generalized from these results, 
and we can define a generalized covariance for these two variables 
\cite{merzbacher} -- \cite{leblond} 
(or any two operators, $\hat{A}, \hat{B}$) as
\begin{equation}
cov(\hat{x},\hat{p};t) \equiv \frac{1}{2} 
\left\langle 
(\hat{x} - \langle \hat{x} \rangle_t)
(\hat{p} - \langle \hat{p} \rangle_t) 
+ 
(\hat{p} - \langle \hat{p} \rangle_t)
(\hat{x} - \langle \hat{x} \rangle_t)
\right\rangle_t
\label{covariance}
\end{equation}
where the additional factor of $1/2$ accounts for the necessarily
symmetric combination which appears, compared to the classical
definition. One can then speak
of a time-dependent correlation coefficient defined by
\begin{equation}
\rho(x,p;t) \equiv
\frac{cov(x,p;t)}{\Delta x_t\cdot \Delta p_t}
\label{correlation_coefficient}
\end{equation}
in analogy with related quantities from statistics. This correlation
can be shown \cite{leblond} to satisfy the inequality
\begin{equation}
[\rho(x,p;t)]^2 \leq 1 - \left(\frac{|\langle [\hat{x},\hat{p}]\rangle|}{2\Delta x_t
\cdot \Delta p_t}\right)^2
= 1- \left(\frac{\hbar}{2\Delta x_t\cdot \Delta p_t}\right)^2
\end{equation}
which vanishes for the standard minimum-uncertainty Gaussian
at $t=0$, but which is non-zero for later times,  as we will see below.









\section{Standard minimum-uncertainty Gaussian wave packets}
\label{sec:standard}


The standard initial minimum-uncertainty Gaussian wave packet, which gives 
the familiar time-dependent spread in Eqn.~(\ref{not_general_case}), 
can be  written in generality as 
\begin{equation}
\phi_0(p) = \phi_{(G)}(p,0) = 
\sqrt{\frac{\alpha}{\sqrt{\pi}}}
\; e^{-\alpha^2(p-p_0)^2/2}
\; e^{-ipx_0/\hbar}
\label{initial_gaussian}
\end{equation}
where $x_0,p_0$ are used to characterize the arbitrary initial central 
position and momentum values respectively. This form gives 
\begin{equation}
\langle \hat{p} \rangle_{t} = p_0
\, , 
\qquad
\quad
\langle \hat{p}^2 \rangle_{t} = p_0^2 + \frac{1}{2\alpha^2}
\, ,
\qquad
\mbox{and}
\qquad
\Delta p_t = \Delta p_0 = \frac{1}{\alpha \sqrt{2}}
\label{momentum_results}
\end{equation}
which are, of course, consistent with the general results in 
Eqns.~(\ref{p_1}) and (\ref{p_2}).


The explicit form of the position-space wave function is given 
by Fourier transform as 
\begin{equation}
\psi_{(G)}(x,t) = \pre \sqrt{\frac{\alpha}{\sqrt{\pi}}}
\integral\, e^{ip(x-x_0)/\hbar}\,
e^{-\alpha^2 (p-p_0)^2/2}\,
e^{-ip^2t/2m\hbar}\,dp
\end{equation}
which can be evaluated in closed form (using the change of variables
$q \equiv p-p_0$ and standard integrals) to obtain
\begin{equation}
\psi_{(G)}(x,t) = \frac{1}{\sqrt{\sqrt{\pi} \alpha \hbar (1+it/t_0)}}
\,
e^{ip_0(x-x_0)/\hbar}
\, e^{-ip_0^2t/2m\hbar}
\,
e^{-(x-x_0-p_{0}t/m)^2/2(\alpha \hbar)^2(1+it/t_0)}
\label{free_particle_position_solution}
\end{equation}
where $t_0 \equiv m\hbar \alpha^2$ is the spreading time.  
This then gives 
\begin{equation}
|\psi_{(G)}(x,t)|^2 = \frac{1}{\sqrt{\pi}\beta_t}
\, e^{- [x-\overline{x}(t)]^2/\beta_t^2}
\end{equation}
where 
\begin{equation}
\overline{x}(t) \equiv x_0 + p_0t/m
\qquad
\mbox{and}
\qquad
\beta_t \equiv  \beta  \sqrt{1+(t/t_0)^2}
\qquad
\mbox{with}
\qquad
\beta \equiv \alpha \hbar
\,. 
\end{equation}
This gives
\begin{equation}
\langle \hat{x} \rangle_t = \overline{x}(t)
\quad
\qquad
\mbox{and}
\qquad
\quad
\langle \hat{x}^2 \rangle_t = [\overline{x}(t)]^2 + \frac{\beta_t^2}{2},
\end{equation}
so that
\begin{equation}
(\Delta x_t)^2 = 
\frac{\beta_t^2}{2}
= 
\frac{\beta^2}{2}
\left(1+\left(\frac{t}{t_0}\right)^2\right)
=  (\Delta x_0)^2 + (\Delta p_0 t/m)^2
\label{gaussian_result}
\end{equation}
which is the familiar textbook result, and for $t=0$ has the minimum
uncertainty product $\Delta x_0 \cdot \Delta p_0 = \hbar/2$. 

It is easy to confirm by direct calculation that there is no initial 
($t=0$) $x-p$ correlation ($cov(x,p;0)=0$) for this wavefunction, 
consistent with
the lack of a term linear in $t$ in Eqn.~(\ref{gaussian_result}). We 
emphasize that such correlations do indeed develop as the wavepacket 
evolves in time, which can be seen by examining the form of either the real or
imaginary parts of $\psi_{(G)}(x,t)$ as shown in Fig.~1 (where we specify
the model parameters used in that plot in the accompanying figure caption). 
We note that for times $t> 0$, the `front end' of the wave packet shown 
there is clearly more `wiggly' than the `back end' (simply count the nodes
on either side of $\langle x \rangle_t$.)
The time-dependent correlation function or covariance defined 
in Eqn.~(\ref{covariance}) and correlation coefficient
from Eqn.~(\ref{correlation_coefficient}) are easily calculated 
for this specific case to be
\begin{equation}
cov(x,p;t) = \frac{\hbar}{2} \left(\frac{t}{t_0}\right)
\qquad
\quad
\mbox{and}
\quad
\qquad
\rho(x,p;t) = \frac{t/t_0}{\sqrt{1+(t/t_0)^2}}
\label{standard_gaussian_correlations}
\end{equation}
which clearly expresses the increasingly positive correlation of
fast (slow) momentum components being preferentially in the leading 
(trailing) edge of the wave packet. We note that such correlations
have been discussed in Refs.~\cite{bohm} and \cite{leblond}.



This observation can also be described quantitatively by examining the 
distribution of kinetic energy of such a free-particle Gaussian wavepacket 
\cite{bassett}. 
In this approach, the standard expression for the kinetic energy is 
rewritten using  integration-by-parts in the form
\begin{equation}
\langle \hat{T}\rangle_{t}
 =  \frac{1}{2m}\langle \hat{p}^2\rangle_{t}
 =  -\frac{\hbar^2}{2m}
\integral dx \,\psi^*(x,t) \frac{\partial ^2 \psi(x,t)}{\partial x^2} 
 =  \frac{\hbar^2}{2m}\integral dx 
\left|\frac{\partial \psi(x,t)}{\partial x}\right|^2 
\end{equation} 
which can be used to define a {\it local kinetic energy density}, 
${\cal T}(x,t)$, via
\begin{equation}
{\cal T}(x,t) \equiv 
\frac{\hbar^2}{2m} 
\left|\frac{\partial \psi(x,t)}{\partial x} \right|^2
\qquad
\quad
\mbox{where}
\qquad
\quad
\langle \hat{T} \rangle_t =  \integral {\cal T}(x,t)\,dx 
\equiv T(t)
\, .
\label{kinetic_energy_distribution}
\end{equation}
As this notion is useful in systems other than for free particle states,
we allow for the possibility that the total kinetic energy varies with
time. 
Since this local density is clearly real and positive-definite, we can use it
to visualize the distribution of kinetic energy (or `wiggliness') 
in any time-dependent wavefunction. We can then define similar quantities
for the kinetic energy in the `front' and/or `back' halves of the wave
packet, using $\langle x\rangle_t$ as the measuring point, via
\begin{equation}
T^{(+)}(t) \equiv  \int_{\langle x \rangle_t}^{+\infty} {\cal T}(x,t)\,dx 
\qquad
\quad
\mbox{and}
\quad
\qquad
T^{(-)}(t) \equiv  \int^{\langle x \rangle_t}_{-\infty} {\cal T}(x,t)\,dx 
\label{half_kinetic_energies}
\, . 
\end{equation}

For the standard Gaussian wave packet in 
Eqn.~(\ref{free_particle_position_solution}), the local kinetic energy 
density is given by
\begin{equation}
{\cal T}_{(G)}(x,t) = \frac{1}{2m}
\left( p_0^2 + \left[\frac{2[x-\overline{x}(t)] p_0}{\alpha^2\hbar}\right]
\left[\frac{t/t_0}{(1+t^2/t_0^2)}\right]
+ \frac{[x-\overline{x}(t)]^2}{(\alpha^2 \hbar)^2 (1+t^2/t_0^2)}\right)
|\psi_{(G)}(x,t)|^2
\, . 
\label{gaussian_case}
\end{equation}
The expectation value of the kinetic energy is correctly given by
\begin{equation}
T_{(G)}(t) = \integral\, {\cal T}_{(G)}(x,t)\,dx = \frac{1}{2m} 
\left(p_0^2 + \frac{1}{2\alpha^2}\right)
\end{equation}
and receives non-zero contributions from only the first and last terms in 
brackets
in Eqn.~(\ref{gaussian_case}),  since the term linear in 
$[x-\overline{x}(t)]$ 
vanishes (when integrated over all space) for symmetry reasons.  The
individual values of $T^{(\pm)}_{(G)}(t)$ can also be calculated and 
are given by
\begin{equation}
T^{(\pm)}_{(G)}(t) 
= \frac{1}{2m}
\left(\frac{1}{2}\right)
\left( 
p_0^2 
\pm 
\left(\frac{2p_0}{\alpha \sqrt{\pi}} \right) \frac{t/t_0}{\sqrt{1+t^2/t_0^2}} 
+ \frac{1}{2\alpha^2}
\right)
\label{left_and_right_kinetic_energies}
\end{equation}
which are individually positive definite. The time-dependent fractions
of the total kinetic energy contained in the $(+)/(-)$ (right/left) halves 
of this standard wave packet are given by
\begin{equation}
R^{(\pm)}_{(G)}(t) \equiv 
\frac{T^{(\pm)}_{(G)}(t)}{T^{(+)}_{(G)}(t) + T^{(-)}_{(G)}(t)}
= \frac{1}{2} \pm 
 \left(\frac{2}{\sqrt{\pi}}\right)
\left( \frac{(p_0\alpha)}{(2(p_0\alpha)^2+1)}\right) 
\frac{t/t_0}{\sqrt{1+t^2/t_0^2}}
\label{define_r_function}
\,. 
\end{equation}
For the model parameters used in Fig.~1, for $t=2t_0$ this corresponds 
to $R^{(+)}/R^{(-)} = 56\%/44\%$, consistent with the small, but obvious,
difference in the kinetic energy distribution seen by `node counting'.



Finally, this growing correlation can be exhibited in yet another way, 
namely through the Wigner quasi-probability distribution, defined by
\begin{eqnarray}
P_{W}(x,p;t)
 & \equiv &
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\psi^{*}(x+y,t)\,\psi(x-y,t)\,e^{+2ipy/\hbar}\,dy \\
& = & 
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\phi^*(p+q,t)\, \phi(p-q,t)\, e^{-2ixq/\hbar}\,dq
\label{wigner_function}
\, . 
\end{eqnarray}
This distribution, first discussed by Wigner \cite{wigner}, 
and reviewed extensively in the research \cite{wigner_research}
and pedagogical \cite{wigner_pedagogical} literature (and even in the 
context of wave packet spreading \cite{wigner_lee}),
is as close as one can come to a quantum phase-space distribution,
and while not directly measurable, can still be profitably used to 
illustrate any $x-p$ correlations. 
For the standard minimum-uncertainty Gaussian wavefunctions defined by 
Eqns.~(\ref{initial_gaussian}) or (\ref{free_particle_position_solution}), 
one finds that \cite{kim_noz}
\begin{equation}
P_{W}(x,p;t) = \frac{1}{\hbar \pi}
\, e^{-(p-p_0)^2 \alpha^2}
\, e^{-(x-x_0-pt/m)^2/\beta^2}
= 
P_{W}(x-pt/m,p;0)
\,.
\label{explicit_wigner_function}
\end{equation}
Contour plots of $P_{W}(x,p;t)$ corresponding to the time-dependent
standard Gaussian wave packet for two different times ($t=0$ and
$t=2t_0$) are also shown at the bottom of Fig.~1, where the the 
elliptical contours with principal axes parallel to the $x,p$ 
axes for the $t=0$ case are indicative of the vanishing initial correlation, 
while the slanted contours at later times are consistent with the correlations
developing as described by Eqn.~(\ref{standard_gaussian_correlations}).
(We note that Bohm \cite{bohm} uses a similar illustration, but discusses it 
only in the context of classical phase space theory and Liouville's theorem.)
The visualization tools used in Fig.~1 (explicit plots of 
$Re[\psi(x,t)]$, and the Wigner function) and the distribution of
kinetic energy as encoded in Eqns.~(\ref{left_and_right_kinetic_energies})
or (\ref{define_r_function}), 
can then directly be used to examine the correlated wave packets we 
discuss in the next section.

As a final reminder about the quantum mechanical ``engineering''
of model one-dimensional wavepackets, we recall that since an initial
$\phi_{0}(p)$ is related to the time-dependent $\psi(x,t)$ for
free-particle solutions via
\begin{equation}
\psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}}\,
\int_{-\infty}^{+\infty}\,
\left[\phi_{0}(p)\,e^{-ip^2t/2m\hbar}\right]\,e^{ipx/\hbar}\,dp
\end{equation}
then the simple modification 
\begin{equation}
\tilde{\phi}_{0}(p) = \phi_{0}(p)\, e^{-ipa/\hbar}\,e^{ip^2\tau/2m\hbar}
\label{change_phi}
\end{equation}
leads to the related position-space wavefunction satisfying
\begin{equation}
\tilde{\psi}(x,t)  = \frac{1}{\sqrt{2\pi\hbar}}\,
\int_{-\infty}^{+\infty}\,
\left[\left(\phi_{0}(p)\, e^{-ipa/\hbar}\,e^{ip^2\tau/2m\hbar}\right)\,e^{-ip^2t/2m\hbar}\right]\,e^{ipx/\hbar}\,dp 
 =  \psi(x-a,t-\tau)
\label{change_psi}
\end{equation}
so that simple shifts in coordinate and time labels are possible, 
and squeezed states often make use of similar connections.

\section{Correlated Gaussian wave packets}
\label{sec:correlated}

\subsection{Squeezed states}
\label{subsec:squeezed}

One of the simplest modifications of a standard minimum-uncertainty
Gaussian initial state which induces non-trivial initial correlations
between position and momentum is given by
\begin{equation}
\phi_{(S)}(p,0) = 
\sqrt{\frac{\alpha}{\sqrt{\pi}}}
\; e^{-\alpha^2(p-p_0)^2(1+iC)/2}
\;   e^{-ipx_0/\hbar}
\label{initial_squeezed}
\,. 
\end{equation}
(A similar version of a squeezed state, but with $\psi(x,0)$ modified,
has been discussed in Ref.~\cite{ford}.) Because the additional $C$ term
is a simple phase, the modulus of $\phi(p,t)$ is unchanged so that
the expectation values of momentum, $\langle \hat{p}\rangle_0$ and
$\langle \hat{p}^2 \rangle_0$, and the momentum-spread, are still given 
by Eqn.~(\ref{momentum_results}) as for the standard Gaussian example.
However, there is now an obvious coupling between the usual `smooth'
$\exp(-\alpha^2(p-p_0)^2/2)$ term which describes the peak momentum values
and the `oscillatory' $\exp(-ipx_0/\hbar)$ terms which dictates the
spatial location and spread, governed by the presence of the new $C$ term, 
which leads to a non-zero initial $x-p$ correlation.


The time-dependent position-space wavefunction is obtained via Fourier
transform with literally no more work than for the standard Gaussian and
one finds
\begin{equation}
\psi_{(S)}(x,t) = \frac{1}{\sqrt{\sqrt{\pi} \beta (1+i[C+t/t_0])}}
\,
e^{ip_0(x-x_0)/\hbar}
\, e^{-ip_0^2t/2m\hbar}
\,
e^{-(x-x_0-p_{0}t/m)^2/2\beta^2(1+i[C+t/t_0])}
\label{squeezed_position}
\end{equation}
giving 
\begin{equation}
|\psi_{(S)}(x,t)|^2
= \frac{1}{\sqrt{\pi}b(t)}
\, e^{-[x-\overline{x}(t)]^2/b^2(t)}
\qquad
\mbox{where}
\qquad
b(t) \equiv \beta \sqrt{1+(C+t/t_{0})^2}
\,. 
\end{equation}
Thus, the initial state in Eqn.~(\ref{initial_squeezed}) gives the same 
time-dependent Gaussian behavior as the standard case, still peaked at 
$x=\overline{x}(t)$, but with a spatial width shifted in time from 
$t \rightarrow t + Ct_0$. This can be understood from the results in 
Eqns.~(\ref{change_phi}) and (\ref{change_psi}) where  the new
$C$-dependent terms in Eqn.~(\ref{initial_squeezed}) give rise to 
effective $a$ and $\tau$ shifts given by 
\begin{equation}
a = -C\alpha^2 \hbar p_0
\qquad
\quad
\mbox{and}
\qquad
\quad
\tau = - C\alpha^2m\hbar = - Ct_0
\, . 
\end{equation}
The $\tau$ shift then affects the time-dependent width, $b(t)$, but the 
combined $a,\tau$ shifts undo each other in the argument of the Gaussian
exponential because they are highly correlated due to the form in
Eqn.~(\ref{initial_squeezed}).

The time-dependent position expectation values are then
\begin{equation}
\langle \hat{x}\rangle_t = \overline{x}(t)
\qquad
\quad
\mbox{and}
\qquad
\quad
\langle \hat{x}^2 \rangle_t = [\overline{x}(t)]^2 + \frac{[b(t)]^2}{2},
\end{equation}
so that
\begin{eqnarray}
(\Delta x_t)^2  =  \frac{[b(t)]^2}{2} 
& = &  
\frac{\beta^2}{2} \left(1 + (C+t/t_0)^2\right) \nonumber \\
& =& 
\frac{\beta^2}{2}(1+C^2) + C\beta^2\frac{t}{t_0} + \frac{\beta^2 t^2}{2t_0^2}
\nonumber \\
& = &  (\Delta x_0)^2 + At + \frac{(\Delta p_0)^2 t^2}{m^2}
\label{squeezed_spread}
\end{eqnarray}
which has a non-vanishing linear term if $C\neq 0$. The initial width
of this packet is larger than for the minimal uncertainty solution
by a factor of $\sqrt{1+C^2}$, but has the same quadratic time-dependence
since $\Delta p_0$ is the same.

One can confirm by direct calculation that $\phi_{(S)}(p,0)$ and 
$\psi_{(S)}(x,0)$ both do 
 have an initial non-vanishing
correlation leading to this form and this is also clear from plots of the
initial wave packet as shown in Fig.~2. We plot there an example with the
same model parameters as in Fig.~1, but with $C=-2$ which leads to an
anti-correlation (since $C<0$) with higher momentum components (more wiggles) 
in the `back edge' of the initial packet.  This gives an intuitive
expectation for a wave packet which
initially shrinks in time, consistent with the result in 
Eqn.~(\ref{squeezed_spread}), and with the plot shown in Fig.~2 for
$t=2t_0$. The parameters were chosen such that for this time the initial
correlation has become `undone', leading to something like the standard
Gaussian initial state, from which point it spreads in a manner which is
more familiar.  The initial correlation is achieved, however, 
at the cost of increasing the initial uncertainty principle product
by a factor of $\sqrt{1+C^2}$. 
The complete time-dependent correlation coefficient
from Eqn.~(\ref{correlation_coefficient}) is 
\begin{equation}
\rho(x,p;t) = \frac{(C+t/t_0)}{\sqrt{1+(C+t/t_0)^2}}
\end{equation}
corresponding in this case to a roughly $90\%$ initial correlation. 
The required initial correlation is also clearly evident
from the Wigner quasi-probability distribution for this case, where we
find 
\begin{equation}
P_{W}(x,p;t) = \frac{1}{\hbar \pi}
\, e^{-(p-p_0)^2 \alpha^2}
\, e^{-(x-x_0-pt/m - C(p-p_0)t_0/m)^2/\beta^2}
\,.
\end{equation}
In this case, the initial correlation for $C<0$ shown in Fig.~2 is
consistent with the desired anti-correlation, since the slope of the
elliptical contours is negative.


In a very similar manner, the expressions for the kinetic energy
density distribution from Eqn.~(\ref{define_r_function}) are simply 
shifted to
\begin{equation}
R^{(\pm)}_{(S)}(t) \equiv \frac{T^{(\pm)}_{(S)}(t)}{T^{(+)}_{(S)}(t) + T^{(-)}_{(S)}(t)}
= \frac{1}{2} \pm 
 \left(\frac{2}{\sqrt{\pi}}\right)
\left( \frac{(p_0\alpha)}{(2(p_0\alpha)^2+1)}\right) 
\frac{(C+t/t_0)}{\sqrt{1+(C+t/t_0)^2}}
\end{equation}
so that for $C<0$, there is an initial asymmetry in the front/back kinetic
energy distribution, with more `wiggles' in the trailing half of the
packet. For the $C=-2$ case in Fig.~2, the initial ($t=0$)
front/back asymmetry is $R^{(+)}/R^{(-)} = 44\%/56\%$.

We note that while a number of quantities (time-dependent spread in position,
correlation coefficient, kinetic energy distribution) are simply obtained 
by the $t \rightarrow t + Ct_0$ shift, other important metrics, such as the
autocorrelation function \cite{bassett_2}, $A(t)$, retain basically 
the same form.

One can imagine generating initial Gaussian states with non-zero 
correlations of the type in Eqn.~(\ref{initial_squeezed}), motivated 
by results obtained by the use of  modern atom trapping techniques, 
such as in Ref.~\cite{meekhof}. In a number of such experiments, 
harmonically bound ions are cooled to essentially their ground state,
after which changes in the external binding potential can generate
various {\it nonclassical motional states} such as coherent states
(by sudden shifts in the central location of the binding potential
\cite{heinzen}) and squeezed states (by changing the strength of the
harmonic binding force, i.e., the spring constant). The subsequent 
time-development of Gaussian packets in such states can then lead to
the desired correlated states, at which point the external binding
potential can be suddenly removed, with free-particle propagation
thereafter. 

As an example, the initial state in a harmonic oscillator potential
of the form $V(x) = m\omega^2x^2/2$ given by
\begin{equation}
\psi(x,0) = \frac{1}{\sqrt{\beta \sqrt{\pi}}}
\, e^{ip_0x/\hbar}\,e^{-x^2/2\beta^2}
\end{equation}
evolves in time as \cite{bassett}
\begin{equation}
\psi(x,t) = 
\exp
\left[
\frac{im\omega x^2 \cos(\omega t)}{2\hbar \sin(\omega t)}
\right]
\frac{1}{\sqrt{A(t) \sqrt{\pi}}}
\exp
\left[ 
-\frac{i m \omega \beta}{2\hbar \sin(\omega t)}
\frac{(x-x_s(t))^2}{A(t)}
\right]
\label{position_space_sho_solution}
\end{equation}
where
\begin{equation}
A(t) \equiv \beta \cos(\omega t) + i \left(\frac{\hbar}{m \omega \beta}
\right) \sin(\omega t)
\qquad
\mbox{and}
\qquad
x_s(t) \equiv \frac{p_0 \sin(\omega t)}{m \omega}
\, .
\end{equation}
The time-dependent expectation values are then
\begin{equation}
\langle x\rangle_t = x_s(t)
\, ,
\qquad
\Delta x_t = \frac{|A(t)|}{\sqrt{2}}
\, ,
\qquad
\mbox{and}
\qquad
\langle p \rangle_t = p_0\cos(\omega t)
\end{equation}
and it is then easy to show that the time-dependent correlation of this
state is given by
\begin{equation}
cov(x,p;t) = \frac{m\omega \sin(\omega t)\cos(\omega t)}{2}
\left[
\left(
\frac{\hbar}{m\omega \beta}\right)^2 - \beta^2
\right]
\,.
\end{equation}
For the special case of coherent states, where $\beta = \sqrt{\hbar/m\omega}$,
the correlations vanish identically for all times (as does the asymmetry in 
kinetic energy \cite{bassett}), while for more general solutions, removing 
the potential at times other than integral multiples of $\tau/2$ (where 
$\tau$ is the classical period) would yield an initially correlated Gaussian.







\subsection{Linear combinations of Gaussian solutions}
\label{subsec:linear_combination}

One of the simplest examples of correlated position-momentum behavior of
a system, leading to an initial shrinking of a spatial width, can be
classically modelled by two 1D non-interacting particles, with the faster 
particle placed initially behind the slower one. A quantum mechanical 
solution of the free-particle 
Schr\"{o}dinger equation involving simple Gaussian forms which mimics this quasi-classical behavior, and for which all expectation values and correlations
can be evaluated in simple closed form, consists of a linear combination
of two minimal-uncertainty Gaussian solutions of the form
\begin{equation}
\psi_{2}(x,t) = N\left[
\cos(\theta) \psi_{(G)}^{(A)}(x,t)
+ 
\sin(\theta) \psi_{(G)}^{(B)}(x,t)
\right]
\label{two_gaussians}
\end{equation}
where $A,B$ correspond to two different sets of initial position and 
momentum parameters, namely $(x_A,p_A)$ and $(x_B,p_B)$, $\theta$ describes
the relative weight of each component, and $N$ is an overall normalization;
we assume for simplicity that each component Gaussian has the same initial
width, $\beta$.
Since each $\psi_{(G)}(x,t)$ is separately normalized, the value of $N$
can be easily evaluated using standard Gaussian integrals with the result
that
\begin{equation}
N^{-2} = 
1 
+
\sin(2\theta)
\;
e^{-(x_A-x_B)^2/4\beta^2
- (p_A-p_B)^2\beta^2/4\hbar^2}
\cos[(x_B-x_A)(p_B+p_A)/2\hbar]
\end{equation}
so that if the two initial Gaussians are far apart in phase space, namely if 
\begin{equation}
\frac{(x_A-x_B)^2}{4\beta^2}
+ 
\frac{(p_A-p_B)^2\beta^2}{4\hbar^2}
>> 1
\, , 
\end{equation}
the normalization factor $N$  can be effectively set to unity, and 
all cross-terms in the evaluation of expectation values can also 
be neglected. 

In this limit, the various initial expectation values required for the
evaluation of the time-dependent spread in Eqn.~(\ref{general_case}) are 
given by 
\begin{eqnarray}
\langle \hat{x} \rangle_0 & = & \cos^2(\theta) x_A + \sin^2(\theta) x_B
\\
\langle \hat{x}^2 \rangle_0 & = &
\cos^2(\theta) \left(x_A^2 + \frac{\beta^2}{2}\right)
+ 
\sin^2(\theta) \left(x_B^2 + \frac{\beta^2}{2}\right)
- \left[\cos^2(\theta) x_A + \sin^2(\theta) x_B\right]^2
\end{eqnarray}
so that
\begin{equation}
(\Delta x_0)^2 = 
[\sin(2\theta)]^2 \left(\frac{x_A-x_B}{2}\right)^2
+ \frac{\beta^2}{2}
\end{equation}
with a similar result for the momentum-spread, namely
\begin{equation}
(\Delta p_0)^2 = 
[\sin(2\theta)]^2 \left(\frac{p_A-p_B}{2}\right)^2
+ \frac{\hbar^2}{2\beta^2}\,.
\end{equation}
The necessary initial correlation is given by
\begin{equation}
\langle \hat{x}\hat{p} + \hat{p}\hat{x} \rangle_0 - 
2\langle \hat{x}\rangle_0 \langle \hat{p} \rangle_0
= 
2 [\sin(2\theta)]^2 \left[\frac{(x_A-x_B)(p_A-p_B)}{4}\right]
\end{equation}
so that the time-dependent spread in position is given by
\begin{equation}
(\Delta x_t)^2 =
[\sin(2\theta)]^2 
\left[
\left(\frac{x_A-x_B}{2}\right)
+ \left(\frac{p_A-p_B}{2}\right)\frac{t}{m}
\right]^2
+
\frac{\beta^2}{2}
+
\frac{\hbar^2 t^2}{2m^2\beta^2}
\,. 
\end{equation}
In the limit we're considering, namely when $|x_A-x_B| >> \beta$
and/or $|p_A-p_B| >> \hbar/\beta$, the time-dependent width can be dominated
by the quasi-classical value dictated by two well-separated `lumps' of
probability, and if $(x_A-x_B)$ and $(p_A-p_B)$ have opposite signs, then this
large position spread can initially decrease in time because of the
initial correlations. This example, while not as `quantum mechanical' as
that in Sec.~\ref{subsec:squeezed}, does clearly and simply  exhibit the 
position-momentum correlations necessary for the presence of the $A$ term 
in Eqn.~(\ref{squeezed_spread}), with the `fast one in the 
back, and the slow one in the front'.

One can imagine producing linear combinations of isolated, but highly 
correlated, Gaussian wave packets at very different points in phase space, 
by invoking the dynamical time-evolution of bound state wave packets which 
leads to the phenomenon of wave packet revivals, especially
fractional revivals \cite{revivals}. For the idealized case of the
infinite square well potential \cite{aronstein}, 
at $t=T_{rev}/4$ (where $T_{rev}$ is
the full revival time), an initially localized wave packet is 'split'
into two smaller copies of the original packet, located at opposite
ends of phase space \cite{belloni}, of the form in Eqn.~(\ref{two_gaussians}).
If, in this model system, the infinite wall boundaries are suddenly
removed at such a point in time, 
we then have the case considered in this section.



\section{Conclusion and discussion}
\label{sec:conclusion}


The study of the time-dependence of the spatial width of wave packets
in model systems can produce many interesting results, a number of which
are quasi-classical in origin, while some are explicitly quantum mechanical.
Time-dependent wave packet solutions of the Schr\"{o}dinger equation for
the harmonic oscillator are easily shown to exhibit intricate correlated
expansion/contraction of widths in position- and momentum-space 
\cite{saxon} and modern experiments \cite{meekhof}, \cite{heinzen} 
can probe a wide variety of such states. 
Even the behavior of otherwise free Gaussian wavepackets 
interacting with (or `bouncing from')  an infinite wall 
\cite{doncheski_1}, \cite{dodonov}, \cite{doncheski_2}
can lead to wave packets which temporarily shrink in size. 



While the fact that free-particle wavepackets can also exhibit 
initial shrinking of their spatial width is well-known in the
physics pedagogical literature, it is perhaps not appreciated enough 
in the context of introductory quantum mechanics courses because of the 
seeming lack of simple, mathematically tractable, and intuitively
visualizable examples, and we have provided two such simple cases here. 
We have also emphasized the usefulness of several tools for the detailed 
analysis of the structure of quantum states as they evolve, namely the direct
visualization of the real/imaginary part of the spatial wavefunction, the
time-dependent spatial distribution of the kinetic energy (how the
`wiggliness' changes in time), and the Wigner quasi-probability
distribution all of which provide insight into
the correlated $x-p$ structure of quantum states.





\appendix
\section{Time-dependent $\Delta x_t$ for the case of uniform acceleration}
\label{sec:appendix}


Many of the results discussed here  for the time-dependent widths of
localized quantum wavepackets for the free-particle case can be carried
over to the situation of a particle undergoing uniform acceleration, 
governed by the Hamiltonian $\hat{H} = \hat{p}^2/2m -Fx$, corresponding
to a constant force, $+F$, to the right. General expressions for the 
time-dependent values of expectation values of powers of both $x$ and $p$ 
can be obtained using Eqn.~(\ref{time-development}) (following the method of 
Styer \cite{styer} for example) and one obtains the results
\begin{equation}
\langle \hat{p} \rangle_t  =  Ft + \langle p \rangle_0 
\qquad
\quad
\mbox{and}
\quad
\qquad
\langle \hat{p}^2 \rangle_t  =  F^2t^2 + 2F\langle \hat{p}\rangle_0 t +
\langle \hat{p}^2 \rangle_0
\label{acc_p_1}
\end{equation}
which imply that 
$(\Delta p_t)^2 = (\Delta p_0)^2$. 
For position we have the corresponding results
\begin{eqnarray}
\langle \hat{x} \rangle_t 
& = & \frac{Ft^2}{2m} + \frac{\langle \hat{p} \rangle_0t}{m}
+ \langle \hat{x} \rangle_0 
\label{acc_x_1} \\
\langle \hat{x}^2 \rangle_t
& = &
  \frac{F^2t^4}{4m^2} 
+ \frac{F\langle \hat{p}\rangle_0 t^3}{m^2}
+ \frac{\langle \hat{p}^2\rangle_0 t^2}{m^2}
+ \frac{F\langle \hat{x}\rangle_0 t^2}{m}
+ \frac{\langle \hat{x}\hat{p} + \hat{p}\hat{x} \rangle_0t}{m}
+ \langle \hat{x}^2 \rangle _0 
\label{acc_x_2}
\end{eqnarray}
which actually combine to give the same expression for $(\Delta x_t)^2$ as in 
Eqn.~(\ref{general_case}). Thus, the model systems we have discussed
here can also be used as examples of correlated wave packets in the
related accelerating particle case.

The expressions above can also be derived using an approach similar to that
followed in Sec.~\ref{sec:momentum_space}, namely by using the most
general form for the time-dependent momentum-space wavefunction, $\phi(p,t)$.
In that case, the time-dependent Schr\"{o}dinger equation in $p$-space is 
written in the form
\begin{equation}
\frac{p^2}{2m} \phi(p,t)
+ F \left[i\hbar \frac{\partial}{\partial p}\right]
\phi(p,t) 
= i \hbar \frac{\partial}{\partial t} \phi(p,t)
\end{equation}
which has a general solution
\begin{equation}
\phi(p,t) = \phi_{0}(p-Ft)\, e^{i[(p-Ft)^3-p^3]/6mF\hbar}
\end{equation}
where $\phi_{0}(p) = \phi(p,0)$ is still the initial wavefunction.
The $t$-dependent expectation values in Eqn.~(\ref{acc_p_1}), 
(\ref{acc_x_1}) and (\ref{acc_x_2}) can then be obtained as in 
Sec.~\ref{sec:momentum_space} in terms of the $t=0$ results, just as
for Eqns.~(\ref{p_1}), (\ref{p_2}), (\ref{x_1}), and
(\ref{x_2}).







 \newpage


\begin{flushleft}
{\Large {\bf 
Figure Captions}}
\end{flushleft}
\vskip 0.5cm
 
\begin{itemize}
%	
\item[Fig.\thinspace 1.]  The time-development of a standard Gaussian
wavepacket, $\psi_{(G)}(x,t)$, described by 
Eqn.~(\ref{free_particle_position_solution}),  with the 
modulus (solid) and real part (dashed) shown for $t=0$ and $t=2t_0$
on the top plot. Contour plots of the Wigner function, from 
Eqn.~(\ref{explicit_wigner_function}), for the same two times are 
shown at the bottom,
corresponding to contours which are $70\%$, $30\%$, and $10\%$ of the
peak or central value. The model parameters used in this plot are 
$\hbar = m = 1$ and $\beta = 2$, which give $t_0 \equiv m\beta^2/\hbar= 4$, 
along with $p_0 = 4$ and $x_0 = -4$ for the initial packet.
%
%
\item[Fig.\thinspace 2.]  Same as Fig.~1, but for the correlated
squeezed state in Eqn.~(\ref{initial_squeezed}) or (\ref{squeezed_position}),
with the model parameters used in Fig.~1. For this case, we use $C=-2$, so 
that for $t=2t_0$ the initial correlations are `undone'.
%
%
\end{itemize}

\newpage
 
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M. Belloni,
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%
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%


%
\end{thebibliography}
%
%

\newpage

\begin{figure}[hbt]
\begin{center}
\epsfig{file=spread_1.eps,width=0.8\linewidth,angle=270}
\caption{}
\end{center}
\end{figure}


\newpage


\begin{figure}[hbt]
\begin{center}
\epsfig{file=spread_2.eps,width=0.8\linewidth,angle=270}
\caption{}
\end{center}
\end{figure}



\end{document}

The study of time-dependent solutions of the Schr\"{o}dinger equation
for initial wave packets of Gaussian form has a long history, dating
back to the earliest days of the development of quantum theory.
Schr\"{o}dinger himself \cite{schrodinger} and others
\cite{kennard} - \cite{debroglie} used such solutions to discuss the
connections between the quantum and classical formulations of mechanics
and found explicit wave packet solutions to many of the standard
problems of classical mechanics, including the free-particle, case
of uniform acceleration, the harmonic oscillator, and the particle in
a uniform magnetic field. These examples first appeared in a number of 
textbooks \cite{kemble} - \cite{rojansky} less than a decade later.



In contrast, many treatments of introductory quantum mechanics  today focus 
almost exclusively on time-independent, energy eigenstate (or stationary 
state) solutions
of the Schr\"odinger equation, although a large number of texts do 
include the explicit construction of a time-dependent Gaussian wave packet
solution for the free-particle case. This easily obtained, analytic
result (seemingly first obtained by Darwin \cite{darwin}) 
illustrates, in a closed form solution, many of the notions 
central to wave packet spreading and the essentially classical 
time-development of the expectation value as 
$\langle x \rangle_t = \langle p\rangle_0 t/m + \langle
x \rangle_0$. In many modern software packages, these solutions are also 
visualized, helping
to illustrate the correlated  time-development of the real and imaginary
parts, and especially their phase relationships, often using a popular
color-coding scheme \cite{styer}, \cite{thaller}, \cite{belloni}.

An example of a static image representing such time-development is
shown in Fig.~1. One can easily note the non-symmetric pattern of 
the `wiggliness' apparent for $t>0$ in both the real (dotted) and 
imaginary (dashed) parts of 
$\psi(x,t)$, which, as we will argue below,  can be used as a qualitative 
measure of the local kinetic energy. The faster (higher momentum, and hence
`wigglier') components  are more obviously associated with the `front end' 
of the spreading wave packet, while the `back end' exhibits much
less rapid spatial variation, consistent with the slower (low momentum)
components trailing behind. 

In this note, we attempt to extend and amplify upon 
this type of intuitive  observation
and try to answer such questions as {\it Where is the kinetic energy
localized in a wave packet?} To this end, we begin with 
 the familiar observation that the expectation value of the 
kinetic energy operator in any time-dependent
position-space state, $\psi(x,t)$,  can be written, using a simple 
integration-by-parts (or IBP) argument, in the form
\begin{eqnarray}
\langle \hat{T}\rangle 
= \frac{1}{2m}\langle \hat{p}^2\rangle 
& = & -\frac{\hbar^2}{2m}
\integral dx \,\psi^*(x,t) \frac{\partial ^2 \psi(x,t)}{\partial x^2} \nonumber \\
& \stackrel{IBP}{=} & 
-\frac{\hbar^2}{2m}
\left(\psi^*(x,t) \frac{\partial \psi(x,t)}{\partial x}\right)^{+\infty}_{-\infty}      \\
& & \;\;\;\;\;\;\;
+  \frac{\hbar^2}{2m}
\integral dx \,
\frac{\partial \psi^*(x,t)}{\partial x}
\, 
\frac{\partial \psi(x,t)}{\partial x} \nonumber 
\end{eqnarray}
or
\begin{equation}
\langle \hat{T} \rangle   =  \frac{\hbar^2}{2m}\integral dx 
\left|\frac{\partial \psi(x,t)}{\partial x}\right|^2 
\end{equation} 
where the `boundary terms' (involving $\psi$ and $\partial \psi/\partial x$ 
at $\pm \infty$) are assumed to vanish for any appropriately localized 
solution.

This global identification suggests that we define a 
local
{\it kinetic energy density}, ${\cal T}(x,t)$, via
\begin{equation}
{\cal T}(x,t) \equiv 
\frac{\hbar^2}{2m} 
\left|\frac{\partial \psi(x,t)}{\partial x} \right|^2
\;\;\; \;\;\;\;\;
\mbox{where}
\;\;\; \;\;\;\;\;
\langle \hat{T} \rangle_t =  \integral {\cal T}(x,t)\,dx 
\equiv T(t)
\, .
\label{kinetic_energy_distribution}
\end{equation}
This quantity, which is clearly locally real and positive-definite,
can then be used to quantify the distribution of kinetic energy, and
how it changes with time, for any time-dependent solution of the
Schr\"odinger equation. For Gaussian solutions, we will be able to
perform many of the desired integrals in closed form, leading to 
explicit analytic results for such related quantities as
\begin{equation}
T^{(+)}(t) \equiv  \int_{\langle x \rangle_t}^{+\infty} {\cal T}(x,t)\,dx 
\qquad
\mbox{and}
\qquad
T^{(-)}(t) \equiv  \int^{\langle x \rangle_t}_{-\infty} {\cal T}(x,t)\,dx 
\label{half_kinetic_energies}
\end{equation}
where the expectation value $\langle x \rangle_t$ serves to define the 
`center' of the wave
packet. These quantities can be respectively associated with the kinetic 
energy in the `right' (or `front', at least for packets moving generally
to the right) half and the `left' (or `back') half of the wave packet.

In what follows, we will present several explicit closed-form
examples of the
calculation of $T^{(\pm)}(t)$ for Gaussian wave packet solutions, 
starting in Sec.~\ref{sec_free_particle} with free-particle wave packets
for a standard Gaussian momentum distribution, while in 
Sec.~\ref{sec_uniform_acceleration} we illustrate 
similar results for Gaussian solutions to the problem of a particle 
undergoing uniform acceleration, working initially in momentum-space.
In Sec.~IV  we use standard propagator techniques to examine Gaussian wave 
packet solutions for the harmonic oscillator problem in this context,
while in Sec.~V we extend these results to the case of an
`inverted oscillator', corresponding to a  particle in unstable equilibrium.


\section{\label{sec_free_particle} Free-particle Gaussian wave packets}


The time-dependent Schr\"odinger equation for the one-dimensional free
particle case can be written, and easily solved, in either position- or
momentum-space in the equivalent forms
\begin{equation}
- \frac{\hbar^2}{2m} \frac{\partial ^2\psi(x,t)}{\partial x^2}
= i \hbar \frac{\partial \psi(x,t)}{\partial t}
\qquad
\quad
\mbox{or}
\qquad
\quad
\frac{p^2}{2m} \phi(p,t) = i \hbar \frac{\partial \phi(p,t)}{\partial t}
\, .
\end{equation}
Since we will find the momentum-space approach more useful in 
Sec.~\ref{sec_uniform_acceleration}
for the case of uniform acceleration, we will also use that approach here
and write
\begin{equation}
\phi(p,t) = \phi_{0}(p) e^{-ip^2t/2m\hbar}
\end{equation}
where $\phi(p,0) = \phi_{0}(p)$ is the initial momentum distribution.
Using this very general form, and the appropriate operator form of
$\hat{x} = i\hbar (\partial/\partial p)$, we recall that
\begin{eqnarray}
\langle \hat{x} \rangle_t & = & \int_{-\infty}^{+\infty}
\phi^*(p,t) \, \left(i\hbar \frac{\partial}{\partial p}\right)
\phi(p,t) \,dp \nonumber \\
& = &
\int_{-\infty}^{+\infty} \phi_{0}^*(p) \hat{x} \phi_{0}(p)\,dp
+ \frac{t}{m} \int_{-\infty}^{+\infty} p |\phi_{0}(p)|^2\,dp \\
& = & \langle \hat{x} \rangle_0 + \frac{\langle p\rangle_0 t}{m}
\, .
\nonumber
\end{eqnarray}
The position-space solution can be written, of course,
using the Fourier transform as
\begin{equation}
\psi(x,t) = \pre \integral e^{ipx/\hbar} \, \phi(p,t)\,dp
\, .
\label{fourier_transform}
\end{equation}
The standard initial Gaussian momentum-space distribution,
 which gives arbitrary initial momentum ($p_0$) 
and position ($x_0$) values,
 can be written in the form
\begin{equation}
\phi(p,0) = \phi_0(p) = 
\sqrt{\frac{\alpha}{\sqrt{\pi}}}
\, e^{-\alpha^2(p-p_0)^2/2}
\, e^{-ipx_0/\hbar}
\label{initial_gaussian}
\end{equation}
which gives
\begin{equation}
\langle p \rangle_{t} = p_0
\, , 
\qquad
\quad
\langle p^2 \rangle_{t} = p_0^2 + \frac{1}{2\alpha^2}
\, ,
\qquad
\mbox{and}
\qquad
\Delta p_t = \Delta p_0 = \frac{1}{\alpha \sqrt{2}}
\, .
\end{equation}
The explicit form of the position-space wave function is given by
the Gaussian integral
\begin{equation}
\psi(x,t) = \pre \sqrt{\frac{\alpha}{\sqrt{\pi}}}
\integral\, e^{ip(x-x_0)/\hbar}\,
e^{-\alpha^2 (p-p_0)^2/2}\,
e^{-ip^2t/2m\hbar}\,dp
\end{equation}
which can be evaluated in closed form (using the change of variables
$q \equiv p-p_0$ and standard integrals) to obtain
\begin{equation}
\psi(x,t) = \frac{1}{\sqrt{\sqrt{\pi} \alpha \hbar (1+it/t_0)}}
\,
e^{ip_0(x-x_0)/\hbar}
\, e^{-ip_0^2t/2m\hbar}
\,
e^{-(x-x_0-p_{0}t/m)^2/2(\alpha \hbar)^2(1+it/t_0)}
\label{free_particle_position_solution}
\end{equation}
where $t_0 \equiv m\hbar \alpha^2$. (This result is sometimes 
attributed to Darwin \cite{darwin}.)
The corresponding probability density is easily shown to be
\begin{equation}
P(x,t) = |\psi(x,t)|^2 = \frac{1}{\sqrt{\pi} \beta_t}
e^{-(x-x(t))^2/\beta_t^2}
\end{equation}
where
\begin{equation}
x(t) \equiv x_0 + p_0t/m
\qquad
\mbox{and}
\qquad
\beta_t \equiv \alpha \hbar \sqrt{1+ t^2/t_0^2}
\end{equation}
and the time-dependent expectation values of position are 
\begin{equation}
\langle x \rangle_t = x(t) = x_0 + p_0t/m 
\, ,
\qquad
\quad
\langle x^2 \rangle_t = (x(t))^2 + \frac{\beta^2_t}{2}
\, ,
\qquad
\mbox{and}
\quad
\Delta x_t = \frac{\beta_t}{\sqrt{2}}
\, ,
\end{equation}
all of which are familiar results.

Turning now to the kinetic energy distribution defined in 
Eqn.~(\ref{kinetic_energy_distribution}), we find that the required
spatial derivative is given by
\begin{equation}
\frac{\partial \psi(x,t)}{\partial x}
= \left(\frac{ip_0}{\hbar} - \frac{ (x-x(t))}{(\alpha \hbar)^2 (1+it/t_0)}
\right) \psi(x,t)
\, .
\end{equation}
The kinetic energy density can therefore be written in the form
\begin{equation}
{\cal T}(x,t) = \frac{1}{2m}
\left( p_0^2 + \left[\frac{2(x-x(t)) p_0}{\alpha^2\hbar}\right]
\left[\frac{t/t_0}{(1+t^2/t_0^2)}\right]
+ \frac{(x-x(t))^2}{(\alpha^2 \hbar)^2 (1+t^2/t_0^2)}\right)
|\psi(x,t)|^2
\, . 
\label{gaussian_case}
\end{equation}
The expectation value of the kinetic energy is correctly given by
\begin{equation}
T(t) = \integral\, {\cal T}(x,t)\,dx = \frac{1}{2m} 
\left(p_0^2 + \frac{1}{2\alpha^2}\right)
\end{equation}
and receives non-zero contributions from the first and last terms in brackets
in Eqn.~(\ref{gaussian_case}),  since the middle term vanishes (when
integrated over all space) for symmetry reasons. On the other hand,
the individual values of $T^{(\pm)}(t)$ in 
Eqn.~(\ref{half_kinetic_energies}) can also be evaluated giving
\begin{equation}
T^{(\pm)}(t) 
= \frac{1}{2m}
\left(\frac{1}{2}\right)
\left( 
p_0^2 
\pm 
\left(\frac{2p_0}{\alpha \sqrt{\pi}} \right) \frac{t/t_0}{\sqrt{1+t^2/t_0^2}} 
+ \frac{1}{2\alpha^2}
\right)
\label{left_and_right_kinetic_energies}
\end{equation}
both of which are easily seen to be positive definite, as they must,
due to the non-negativity of ${\cal T}(x,t)$. The time-dependent fractions
of the total kinetic energy contained in the $(+)/(-)$ (right/left) halves 
of this archetypical wave packet are given by
\begin{equation}
R^{(\pm)}(t) \equiv \frac{T^{(\pm)}(t)}{T^{(+)}(t) + T^{(-)}(t)}
= \frac{1}{2} \pm 
 \left(\frac{2}{\sqrt{\pi}}\right)
\left( \frac{(p_0\alpha)}{(2(p_0\alpha)^2+1)}\right) 
\frac{t/t_0}{\sqrt{1+t^2/t_0^2}}
\label{define_r_function}
\end{equation}
which will clearly increase/decrease monotonically as 
$t/t_0 \rightarrow \infty$.
The limiting values are then
\begin{equation}
R^{(\pm)}(t/t_0\rightarrow \infty) = \frac{1}{2} \pm 
\left(\frac{2}{\sqrt{\pi}}\right) \left(\frac{(p_0\alpha)}{2(p_0\alpha)^2+1}
\right)
\end{equation}
which have the  extremal values
\begin{equation}
R^{(+)}_{max}(t/t_0 \rightarrow \infty) 
 =  \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \approx 0.9
\qquad
\mbox{and}
\qquad
R^{(-)}_{min}(t/t_0 \rightarrow \infty) 
 =  \frac{1}{2} - \frac{1}{\sqrt{2\pi}} \approx 0.1
\label{maximum_minimum_values}
\end{equation}
which occur when
\begin{equation}
p_0 \alpha = \frac{1}{\sqrt{2}}
\qquad
\mbox{or}
\qquad
p_0 = \frac{1}{\alpha \sqrt{2}} = \Delta p_0
\label{extreme_value}
\, . 
\end{equation}
Thus, as much as $90\%$ of the total kinetic energy of this Gaussian
wave packet solution can be in the `front half' of the wave at long times
(i.e. those for which $t >> t_0$.)

To illustrate this effect, we plot in the left column of Fig.~2,  
the modulus ($|\psi(x,t)|$, solid),
and real (dotted) and imaginary (dashed) parts of a typical solution
corresponding to $p_0 = 0$ (top), 
$p_0 = \Delta p_0$ (the extremal value, middle), and
$p_0 = 4\Delta p_0$ (bottom) for long times ($t = 10t_0$).
The values are plotted in terms of the variable
$x(t) = x_0 + p_0t/m$ for easier comparison. For the $p_0=0$ case, it is
clear that the larger momentum components (in magnitude) spread faster,
but uniformly, in the opposite $+x$ and $-x$ directions, giving 
equal `wiggliness' on each 
side, while for large values of $p_0$, the total kinetic energy is clearly
increased (many more `wiggles' everywhere),  
but the amount on each side of the
expectation value peak at $\langle x \rangle_t$ is  roughly the same.
For the extremal value of $p_0 = \Delta p_0$, there is the clearest
distinction between the `front' and `back' halves,  as the magnitudes
of the momentum components in the `back' half are at a minimum, resulting
in the greatest separation between the kinetic energy in the two halves. 



We can also compare the distribution of probability, described as usual
by $P(x,t) \equiv |\psi(x,t)|^2$, to how the kinetic energy is localized.
The kinetic energy density, ${\cal T}(x,t)$, can be scaled to the total
(and possibly time-dependent) value of $T(t)$ via 
\begin{equation}
S(x,t) \equiv \frac{{\cal T}(x,t)}{T(t)}
\qquad
\mbox{which satisfies}
\qquad
\integral\, S(x,t)\,dx = 1
\label{scaled_kinetic_energy_density}
\end{equation}
and so is normalized in the same way as the probability density.
We plot both $P(x,t)$ (solid curve) and $S(x,t)$ (dot-dashed curve) 
in the right hand column of Fig.~2 for the same three cases considered above
and we note how the shape of $S(x,t)$ is correlated with the obvious 
`wiggliness'
shown in the real and imaginary parts of $\psi(x,t)$.











\section{\label{sec_uniform_acceleration} Uniform acceleration}

The problem of a particle under the influence of a constant
force is a staple in  classical mechanics,  and was considered early
in the history of quantum mechanics \cite{kemble}, but is less often 
discussed in introductory treatments of the subject,  especially in terms 
of time-dependent solutions. For that reason, we briefly review the most
straightforward momentum-space approach to this problem. In this case,
where the potential is given by $V(x) = -Fx$, we can write the
time-dependent Schr\"odinger equation in momentum-space as
\begin{equation}
\frac{p^2}{2m}\phi(p,t) - 
F\cdot \left[i\hbar \frac{\partial}{\partial p}\right] \phi(p,t)
= i\hbar \frac{\partial \phi(p,t)}{\partial t}
\label{pspace}
\end{equation}
or 
\begin{equation}
i\hbar\left(F\frac{\partial \phi(p,t)}{\partial p} 
+ \frac{\partial \phi(p,t)}{\partial t}\right) = \frac{p^2}{2m}\phi(p,t)
\, . 
\label{newp}
\end{equation}
We note that the simple combination of derivatives guarantees that a
function of the form $\Phi(p-Ft)$ will make the left-hand side vanish,  so
we assume a solution of the form $\phi(p,t) = \Phi(p-Ft)\tilde{\phi}(p)$,
with $\Phi(p)$ arbitrary and $\tilde{\phi}(p)$ to be determined.
Using this form, Eqn.~(\ref{newp}) reduces to
\begin{equation}
\frac{\partial \tilde{\phi}(p)}{\partial p} = 
-\frac{i p^2}{2m\hbar F}\tilde{\phi}(p)
\end{equation}
with the solution
\begin{equation}
\tilde{\phi}(p) = e^{-ip^3/6mF\hbar}
\, . 
\end{equation}
We can then write the general solution as
\begin{equation}
\phi(p,t) = \Phi(p\!-\!Ft)e^{-ip^3/6mF\hbar}
\end{equation}
or, using the arbitrariness of $\Phi(p)$, as
\begin{equation}
\phi(p,t) = \phi_0(p\!-\!Ft) e^{i((p-Ft)^3-p^3)/6mF\hbar )}
\label{pacc}
\end{equation}
where now $\phi_0(p)$ is some initial momentum distribution since
$\phi(p,0) = \phi_0(p)$. Note that because the $p^3$ terms cancel
in the exponential, we will be able to explicitly integrate
Gaussian type initial momentum-space waveforms.

For any general initial $\phi_{0}(p)$ we now have the time-dependent
expectation values
\begin{eqnarray}
\langle p \rangle_t &= & \langle p \rangle_0 + Ft \\
\langle p^2 \rangle_t &= & \langle p^2 \rangle_0 + 
2\langle p \rangle_0 Ft +(Ft)^2 \\
\langle \hat{x} \rangle_t & = &  \langle \hat{x} \rangle_0
+ \frac{\langle p\rangle_0 t}{m} + \frac{Ft^2}{2m} 
\end{eqnarray}
which also give the expectation value
\begin{equation}
\left \langle \frac{p^2}{2m} + V(x) \right\rangle = 
\frac{\langle p^2 \rangle_0}{2m}
- F\langle \hat{x} \rangle_0
\end{equation}
which, in turn, also  agrees with a similar calculation of 
$\langle \hat{E} \rangle_t$
using $\hat{E} = i\hbar (\partial/\partial t)$,  all of which are consistent
with a particle undergoing uniform acceleration.

Using the standard initial Gaussian momentum-space wavefunction in 
Eqn.~(\ref{initial_gaussian}) as the $\phi_0(p-Ft)$ in Eqn.~(\ref{pacc}), 
we can evaluate the position-space solution
using Eqn.~(\ref{fourier_transform}) to obtain
\begin{eqnarray}
\psi(x,t) & = & \left[e^{iFt(x_0-Ft^2/6m)/\hbar}\, e^{i(p_0+Ft)(x-x_0 - p_0t/2m)/\hbar}\right] \left(
\frac{1}{\sqrt{\sqrt{\pi}\alpha \hbar (1+it/t_0)}}
\right) \nonumber \\ 
& & \,\,\,
\times
\, e^{-(x-(x_0+p_0t/m+Ft^2/2m))^2/2(\alpha \hbar)^2(1+it/t_0)}
\, . 
\label{accelerating_solution}
\end{eqnarray}
The corresponding probability density is given by
\begin{equation}
P(x,t) = \frac{1}{\sqrt{\pi} \beta_t}
e^{-(x-\tilde{x}(t))^2/\beta_t^2}
\end{equation}
where 
\begin{equation}
\tilde{x}(t) \equiv x_0 + \frac{p_0t}{m} + \frac{Ft^2}{2m}
\end{equation}
and 
\begin{equation}
\qquad
\langle x \rangle_t = \tilde{x}(t)
\, ,
\qquad
\langle x^2 \rangle_t = (\tilde{x}(t))^2 + \frac{\beta_t^2}{2}
\, ,
\quad
\mbox{and}
\qquad
\Delta x_t = \frac{\beta_t}{\sqrt{2}} 
\end{equation}
so that this accelerating wave packet spreads in the same manner as the
free-particle Gaussian example. 
The calculation of the kinetic energy density proceeds exactly 
as in Sec.~\ref{sec_free_particle}, with
\begin{eqnarray}
{\cal T}(x,t) & = &  \frac{1}{2m}
\left( (p_0+Ft)^2
+ \frac{2(p_0+Ft)(x-\tilde{x}(t))}{\hbar \alpha^2}
\frac{t/t_0}{(1+t^2/t_0^2)} \right. \nonumber \\
& &
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left.
+ \frac{(x-\tilde{x}(t))^2}{(\alpha^2\hbar)^2 (1+t^2/t_0^2)}
\right)
|\psi(x,t)|^2
\end{eqnarray}
The corresponding $(+)$ and $(-)$ kinetic energies are then derived
in the same way as before and are given by 
\begin{equation}
T^{(\pm)}(t)
= \frac{1}{2m} \left(\frac{1}{2}\right)
\left( (p_0+Ft)^2 
\pm 
\left( \frac{2(p_0+Ft)}{\alpha\sqrt{\pi}}\right)
\left(\frac{t/t_0}{\sqrt{1+t^2/t_0^2}}\right)
+
\frac{1}{2\alpha^2}
\right)
\end{equation}
which is the same result as in Eqn.~(\ref{left_and_right_kinetic_energies}),
with $p_0 \rightarrow p_0 +Ft$. This similarity in form implies that the
maximum (minimum) values of $T^{(\pm)}(t)$ are once again given by
Eqn.~(\ref{maximum_minimum_values}) (provided that $t/t_0 >> 1$) 
which now  occurs when 
$|p_0 +Ft| = \Delta p_0$. In this more dynamic situation, if $p_0$
and $F$ have different signs (so that the motion includes one `back'
and one `forth' component), this situation can occur twice during
a single trajectory, with the roles of $R^{(+)}(t)$ and
$R^{(-)}(t)$ changing between the `back' and the `forth' traversal.









\section{\label{sec_sho} Simple harmonic oscillator wave packets}

The third model system which is easily shown to exhibit time-dependent
Gaussian wave packet solutions
 is also the most frequently studied of all classical
and quantum mechanical examples, namely the simple harmonic oscillator,
defined by the potential energy $V(x) = m \omega^2 x^2/2$. In this case,
the initial value problem is perhaps most easily solved, especially for
Gaussian wave packets, by propagator techniques \cite{feynman_1} --
\cite{holstein}. In this approach, one writes
\begin{equation}
\psi(x,t) = \int_{-\infty}^{+\infty} \,dx'\, \psi(x',0)\, K(x,x';t,0)
\label{propagator}
\end{equation}
where the propagator can be derived in a variety of ways \cite{holstein}
and can be written in the form
\begin{equation}
K(x,x';t,0) = \sqrt{\frac{m \omega}{2\pi i \hbar \sin(\omega t)}}
\exp
\left[
\frac{im \omega}{2\hbar \sin(\omega t)} ((x^2 + (x')^2)\cos(\omega t)
- 2x x')
\right]
\, .
\end{equation}
For the initial state wavefunction we will use position-space
version of Eqn.~(\ref{initial_gaussian}), but for notational and
visualization simplicity, we will specialize to the case of $x_0 = 0$, 
namely
\begin{equation}
\psi(x',0) = \frac{1}{\sqrt{\beta \sqrt{\pi}}} \, e^{ip_0x'/\hbar}\,
e^{-(x')^2/2\beta^2}
\label{initial_position_gaussian}
\end{equation}
where $\beta = \alpha \hbar$ and $\Delta x_0 = \beta/\sqrt{2}$. 
In this state,  the expectation value of the energy is
\begin{equation}
\langle \hat{E} \rangle_t = \langle \hat{E} \rangle_0 =
\frac{1}{2m} \left(p_0^2 + \frac{\hbar^2}{2\beta^2}\right)
+ \frac{m\omega^2\beta^2}{4}
\label{total_sho_energy}
\end{equation}
The integral in Eqn.~(\ref{propagator}) can be done in closed form for
the initial Gaussian in Eqn.~(\ref{initial_position_gaussian}) with the
result
\begin{equation}
\psi(x,t) =
\exp
\left[
\frac{im\omega x^2 \cos(\omega t)}{2\hbar \sin(\omega t)}
\right]
\frac{1}{\sqrt{A(t) \sqrt{\pi}}}
\exp
\left[ 
-\frac{i m \omega \beta}{2\hbar \sin(\omega t)}
\frac{(x-x_s(t))^2}{A(t)}
\right]
\label{position_space_sho_solution}
\end{equation}
where
\begin{equation}
A(t) \equiv \beta \cos(\omega t) + i \left(\frac{\hbar}{m \omega \beta}
\right) \sin(\omega t)
\qquad
\mbox{and}
\qquad
x_s(t) \equiv \frac{p_0 \sin(\omega t)}{m \omega}
\, .
\end{equation}
(We note that in the force-free limit, when  $\omega \rightarrow 0$,
one can show that this solution reduces to the free-particle 
form  in Eqn.~(\ref{free_particle_position_solution}), with $x_0 = 0$,
as expected.)
The time-dependent probability density is given by
\begin{equation}
P(x,t) = |\psi(x,t)|^2
= \frac{1}{\sqrt{\pi}|A(t)|} e^{-(x-x_s(t))^2/|A(t)|^2}
\end{equation}
where
\begin{equation}
|A(t)| = \sqrt{\beta^2 \cos^2(\omega t) + (\hbar/m\omega \beta)^2
\sin^2(\omega t)}
\end{equation}
and
\begin{equation}
\langle x\rangle_t = x_s(t)
\qquad
\mbox{and}
\qquad
\Delta x_t = \frac{|A(t)|}{\sqrt{2}}
\, .
\end{equation}
So, while the expectation value of the wave packet oscillates in a
way which mimics the classical result, the time-dependent spatial
width of the packet changes in time, consistent with very
general expectations for the oscillator case \cite{gottfried},
\cite{styer_2}.
This more general time-dependent Gaussian solution is described
in several textbooks \cite{rojansky}, \cite{saxon} and
special cases of it are often rediscovered \cite{pulsate},
\cite{pulsate_2}.

We note that in the special case of the minimum uncertainty wavepacket
where 
\begin{equation}
\beta^2 = \left(\frac{\hbar}{m\omega \beta}\right)^2
\qquad
\mbox{or}
\qquad
\beta = \sqrt{\frac{\hbar}{m\omega}} \equiv \beta_0
\end{equation}
the time-dependent width of the Gaussian simplifies to
\begin{equation}
\Delta x_t = \frac{\beta_0}{\sqrt{2}}
= \Delta x_0
\end{equation}
so the wave packet oscillates with no change in shape. This is
the special result seen more standardly \cite{schiff} -- \cite{roy}
in textbooks and pedagogical articles, and is similar to the
coherent-state like solution discussed by
Schr\"{o}dinger \cite{schrodinger} in a famous paper.


The time-dependent
expectation value of the potential energy is easily found to be
\begin{equation}
\langle V(x)\rangle_t
= \frac{1}{2} m\omega^2 \langle x^2 \rangle_t
= 
\frac{1}{2m}\left( p_0^2 + \frac{\hbar^2}{2\beta^2}\right) \sin^2(\omega t)
+ \frac{m \omega^2 \beta^2 \cos^2(\omega t)}{4}
\, . 
\end{equation}
The time-dependent (total) kinetic energy then follows directly from
this equation combined with  Eqn.~(\ref{total_sho_energy}) and is given by
\begin{equation}
T(t) = \frac{1}{2m}\langle p^2 \rangle_t
= \frac{1}{2m} \left( p_0^2 + \frac{\hbar^2}{2\beta^2}\right) \cos^2(\omega t)
+ 
\frac{m \omega^2 \beta^2 \sin^2(\omega t)}{4}
\, .
\end{equation}
In order to determine the {\it distribution} of kinetic energy, however, 
we evaluate ${\cal T}(x,t)$ using
\begin{equation}
\frac{\partial \psi(x,t)}{\partial x}
= 
\left(
\frac{im\omega}{\hbar \sin(\omega t)}
\right)
\left( 
x \cos(\omega t) - \frac{\beta (x-x_s(t))}{A(t)}
\right)
\psi(x,t)
\end{equation}
and we find that
\begin{equation}
T^{(\pm)}(t) 
= \frac{T(t)}{2}
\pm 
\frac{p_0 \omega \sin(\omega t) \cos^2(\omega t)}{2\sqrt{\pi} |A(t)|}
\left[
\frac{\beta_0^4}{\beta^2} - \beta^2
\right]
\label{sho_fraction}
\, .
\end{equation}
The asymmetry ($T^{(+)}(t) \neq T^{(-)}(t)$) in the kinetic energy
distribution vanishes at half-integral multiples of the classical period
$\tau \equiv 2\pi/\omega$, namely when $t = n\tau/2$, but also at the
classical turning points, {\it i.e.}, at odd multiples of $\tau/4$. 
There is also no asymmetry in the case when $p_0 = 0$ and the wave packet 
expands and contracts uniformly in both the $+x$ and $-x$ directions.
Finally, there is no asymmetry in the special case of the `fixed width' 
Gaussian, when $\beta = \beta_0$, and this property can perhaps help
explain some of the special features of that minimum-uncertainty state.

To better visualize the general result in Eqn.~(\ref{sho_fraction}), 
we plot in Figs.~3 and 4 representations
of both the wave function (modulus, real, and imaginary parts) as well
as the probability ($P(x,t)$) and (scaled) kinetic energy 
($S(x,t)$) distributions over the
first quarter period. We note that for wave packets initially moving
to the right ($p_0 >0$) as shown here, narrow packets, i.e., ones
with $\beta < \beta_0$, typically have  more kinetic energy in the 
`front' half of the packet (Fig.~3, middle panels), while initially
wider packets have the opposite behavior (Fig.~4) consistent with
Eqn.~(\ref{sho_fraction}).


While the time-dependence  of $R^{(\pm)}(t)$ 
(defined in Eqn.~(\ref{define_r_function})) is more
varied than for the simpler, non-periodic, cases considered so far, 
as a specific example, we can examine the distribution of kinetic energy at 
times such
that 
\begin{equation}
\cos(\omega T) = \sin(\omega T) = \frac{1}{\sqrt{2}}
\qquad
\mbox{for instance, when}
\qquad
T = \frac{\tau}{8}
\, . 
\end{equation}
In that case we find
\begin{equation}
R^{(\pm)}(T) =
\frac{1}{2}
\pm
\left(
\frac{2}{\sqrt{\pi}}
\right)
\left(\frac{ (p_0/m\omega)}{(2(p_0/m\omega)^2 + (\beta_0^4/\beta^2
+ \beta^2)}\right)
\left(
\frac{(\beta_0^4/\beta^2) - \beta^2)}
{\sqrt{(\beta_0^4/\beta^2) + \beta^2)}}
\right)
\end{equation}
This has extremal values of
\begin{equation}
R^{(\pm)}(T) =
\frac{1}{2}
\pm
\frac{1}{\sqrt{2\pi}}
\left[
\frac{\beta_0^4/\beta^2 - \beta^2}
     {\beta_0^4/\beta^2 + \beta^2}
\right]
\end{equation}
when
\begin{equation}
p_0 =  \sqrt{\frac{(\beta m\omega)^2}{2} + \frac{\hbar^2}{2\beta^2}}
\label{sho_extreme_value}
\end{equation}
which are obvious generalizations of Eqns.~(\ref{maximum_minimum_values})
and (\ref{extreme_value}). 







\section{\label{sec_unstable} `Inverted' oscillator wave packets for 
unstable equilibrium}

The final case we present for which time-dependent Gaussian wave packet 
solutions are easily obtained is a generalization of the harmonic oscillator
which has been described as the `inverted oscillator' \cite{nardone}
or the `unstable particle' \cite{robinett_2}, corresponding to
the classical motion of a particle at the top of potential hill, given
by 
\begin{equation}
\tilde{V}(x) \equiv -\frac{1}{2} m\tilde{\omega}^2 x^2
\, . 
\end{equation}
Many of the results obtained using propagator techniques can be easily
carried over to this problem with the simple identifications
\begin{equation}
\omega \rightarrow i \tilde{\omega}
\, ,
\qquad
\sin(\omega t) \rightarrow i \sinh(\tilde{\omega} t)
\, ,
\qquad
\mbox{and}
\qquad
\cos(\omega t) \rightarrow \cosh(\tilde{\omega} t)
\, . 
\end{equation}
For example, the position-space probability density corresponding
to the initial state in Eqn.~(\ref{initial_position_gaussian}) 
is given by
\begin{equation}
P(x,t) = |\psi(x,t)|^2
= \frac{1}{\sqrt{\pi}|B(t)|} \exp^{-(x-\tilde{x}_s(t))^2/|B(t)|^2}
\end{equation}
where
\begin{equation}
|B(t)| = \sqrt{\beta^2 \cosh^2(\tilde{\omega} t) + 
(\hbar/m\tilde{\omega} \beta)^2
\sinh^2(\tilde{\omega} t)}
\end{equation}
and
\begin{equation}
\langle x\rangle_t = \tilde{x}_s(t) = 
\frac{p_0 \sinh(\tilde{\omega}t)}{m\tilde{\omega}}
\qquad
\mbox{and}
\qquad
\Delta x_t = \frac{|B(t)|}{\sqrt{2}}
\end{equation}
and the probability density exhibits a `runaway' (exponential)
behavior. 
In this case, the long time behavior of the kinetic energy fractions
is dictated by the limits
\begin{equation}
\cosh(\tilde{\omega}t), \,\sinh(\tilde{\omega} t)
\longrightarrow 
e^{\tilde{\omega}t}/2
\end{equation}
which gives
\begin{equation}
R^{(\pm)}(t \rightarrow \infty)
= \frac{1}{2}
\pm
\left(\frac{2}{\sqrt{\pi}}\right)
\left[
\frac{(p_0/m\tilde{\omega})}
{ 2(p_0/m\tilde{\omega})^2 + (\beta_0^4/\beta^2 + \beta^2)}
\right]
\sqrt{\beta_0^4/\beta^2 + \beta^2}
\end{equation}
with the same extremal values as in Eqn.~(\ref{maximum_minimum_values}),
when $p_0$ satisfies the equivalent of Eqn.~(\ref{sho_extreme_value})
with $\omega \rightarrow \tilde{\omega}$.






\section{\label{sec_conclusions} Conclusions and discussion}

In this note we have provided a detailed analysis of Gaussian wave packet
solutions of the Schr\"odinger equation in several model systems, helping to
elucidate  some of the qualitative aspects of wave packet time-development
and spreading  often seen in standard visualizations, 
and their relationship to the distribution of
kinetic energy. We have focused on closed-form analytic results,  
all of which have been obtainable due to the special nature of the integrals 
which arise for
Gaussian wave packets. An obvious and interesting extension would be to
extend these results to free particle wave packets for arbitrary 
non-Gaussian initial momentum distributions, performing the required 
integrals in Eqn.~(\ref{fourier_transform}) numerically to see just
how general the results in Eqns.~(\ref{maximum_minimum_values})
and (\ref{extreme_value}) are. For example, are there other initial
wave packets and initial conditions for which the kinetic energy can
be even more localized that the $90\%/10\%$ fraction seen here for
Gaussians? Another straightforward generalization is to consider 
two-dimensional extensions of all four systems discussed here 
(then visualized as functions of powers of both $x$ and $y$) where closed 
form expressions are also possible, as well as the related case of the
charged particle (confined to a plane) subject to a uniform magnetic
field. 

 
Other studies of dynamical wave packet propagation have
discussed aspects of the time-development of $\psi(x,t)$ which
arise from effects related to the differing behavior of various
momentum-components. Examples have included discussions of 
the average speed of wave packet components
which are transmitted or reflected from a rectangular barrier
\cite{bramhall} (due to the energy dependence of probabilities of 
transmission and reflection) as well as the time-dependent shape of 
$|\phi(p,t)|^2$ for a wave packet hitting an infinite wall \cite{doncheski}
or similar `bouncing' systems \cite{bouncing_bob}.
The consideration of the distribution of kinetic energy distribution
in many other such time-dependent wave packet solutions might prove useful
in understanding their behavior. One such example certainly would be
Gaussian-like wave packet solutions in the infinite square well, where 
additional interesting features, such as exact wave packet revivals, are 
present.

Gaussian solutions such as those considered here can find use as examples 
of model systems in discussing other theoretical constructs, such as the 
Wigner quasi-probability distribution 
\cite{wigner} - \cite{belloni_doncheski_robinett}.
In that case, one defines $P_{W}(x,p;t)$ via
\begin{eqnarray}
P_{W}(x,p;t)
 & \equiv &
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\psi^{*}(x+y,t)\,\psi(x-y,t)\,e^{2ipy/\hbar}\,dy \\
& = & 
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\phi^*(p+q,t)\, \phi(p-q,t)\, e^{-2ixq/\hbar}\,dq
\end{eqnarray}
which can then be evaluated explicitly 
in closed form, in either position- or 
momentum-space, for all of the solutions discussed above. In an accompanying 
paper \cite{bassett} we study the same four examples considered here, 
in yet a different context.



\newpage
 
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\bibitem{cohen}
S. M. Cohen,
{\it Path integral for the quantum harmonic oscillator using
elementary methods},
Am. J. Phys. {\bf 66}, 537-540 (1998).
%
%
\bibitem{holstein}
B. R. Holstein,
{\it The harmonic oscillator propagator},
Am. J. Phys. {\bf 66}, 583-589 (1998).
%
%
\bibitem{gottfried}
K. Gottfried,
{\it Quantum Mechanics: Volume I Fundamentals}
(Benjamin, New York, 1966) pp~260-264.
%
%
\bibitem{styer_2} 
D. F. Styer, 
{\it The motion of wave packets through their expectation values and 
uncertainties},
 Am. J. Phys. {\bf 58}, 742-744 (1990).
%
%
\bibitem{pulsate}
A. S. de Castro and N. C. da Cruz,
{\it A pulsating Gaussian wave packet},
Eur. J. Phys. {\bf 20}, L19-L20 (1999).
%
%
\bibitem{pulsate_2}
W. Waldenstr\"{o}m and Razi K. Naqvi,
{\it A neglected aspect of the pulsating Gaussian wave packet},
Eur. J. Phys. {\bf 20}, L41-L43 (1999).
%
%
\bibitem{schiff}
L. I. Schiffer,
{\it Quantum Mechanics} (First Edition)
(McGraw Hill, New York, 1949) pp~67-69.
%
%
\bibitem{bohm}
D. Bohm,
{\it Quantum Theory}
(Prentice-Hall, Englewood Cliffs, 1951) pp~306-309.
%
%
\bibitem{fong}
P. Fong,
{\it Elementary Quantum Mechanics}
(Addison-Wesley, Reading, 1962) pp~88-91.
%
%
\bibitem{messiah}
A. Messiah,
{\it Quantum Mechanics: Volume I}
(North-Holland, Amsterdam, 1961) pp~446-447.
%
%
\bibitem{haar}
D. ter Haar,
{\it Selected Problems in Quantum Mechanics}
(Academic Press, New York, 1964) pp~14, 143-145.
%
%
\bibitem{french_cohen}
C. Cohen-Tannoudji,  B. Diu,  and F. Lalo\"{e}, 
{\it Quantum Mechanics: Volume I}
(Wiley, New York, 1977) pp~572-573.
%
%
\bibitem{roy}
S. Howard and S. K. Roy, 
{\it Minimum uncertainty states and their time evolution},
Am. J. Phys. {\bf 53}, 538-542 (1985).
%
%
\bibitem{robinett_2} 
R. W. Robinett,
{\it Quantum Mechanics: Classical Results, Modern Systems, and 
Visualized Examples} 
(Oxford University Press, New York, 1997) pp~208-209, 213-214.
%
%
\bibitem{bramhall}
M. H. Bramhall and B. M. Casper,
{\it Reflections on a wave packet approach to quantum mechanical
barrier penetration},
Am. J. Phys. {\bf 38}, 1136-1145 (1970).
%
%
\bibitem{doncheski}
M. A. Doncheski and R. W. Robinett,
{\it Anatomy of a `quantum bounce'},
Eur. J. Phys {\bf 20}, 29-37 (1999).
%
%
\bibitem{bouncing_bob}
L. de la Torre and F. Gori,
{\it The bouncing bob: quasi-classical states},
Eur. J. Phys. {\bf 24}, 253-259 (2003).
%
%
%%%%%%%%%%%%%% This is the original Wigner reference  %%%%%%%%%%%
\bibitem{wigner}
E. Wigner,
{\it On the quantum correction for thermodynamic equilibrium},
Phys. Rev. {\bf 40}, 749-759 (1932).
%
%
%%%%%%%%%  These are the review articles %%%%%%%%%%%%%%%%%%%%%%%
\bibitem{tatarskii}
V. I. Tatarski\u{i},
{\it The Wigner representation of quantum mechanics},
Sov. Phys. Usp. {\bf 26}, 311-327 (1983).
%
%
\bibitem{earlier_wigner_review}
N. L. Balaczs and B. K. Jennings,
{\it Wigner's function and other distribution functions in
mock phase space},
Phys. Rep. {\bf 105}, 347-391 (1984).
%
%
\bibitem{carruthers}
P. Carruthers and F. Zachariasen,
{\it Quantum collision theory with phase-space distributions},
Rev. Mod. Phys. {\bf 55}, 245-285 (1983).
%
%
\bibitem{wigner_review}
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,
{\it Distribution functions in physics: Fundamentals},
Phys. Rep.  {\bf 106}, 121-167 (1984).
%
%
\bibitem{tomagraphic}
J. Bertrand and P. Bertrand,
{\it A tomographic approach to Wigner's function},
Found. Phys. {\bf 17}, 397-405 (1987).
%
%
\bibitem{kim_wigner}
Y. S. Kim and E. P. Wigner,
{\it Canonical transformations in quantum mechanics},
Am. J. Phys. {\bf 58}, 439-448 (1990).
%
%
\bibitem{kim_noz}
Y. S. Kim and M. E. Noz,
{\it Phase space picture of quantum mechanics: Group theoretical
approach}, Lecture Notes in Physics Series, Vol. 40
(World Scientific, Singapore, 1990)
%
%
\bibitem{lee}
H.-W. Lee,
{\it Theory and application of the quantum phase-space distribution
functions},
Phys. Rep. {\bf 259}, 147-211 (1995).
%
%
\bibitem{weyl_review}
A. M. Ozorio de Almeida,
{\it The Weyl representation in classical and quantum mechanics},
Phys. Rep. {\bf 296}, 265-342 (1998)
%
%
\bibitem{belloni_doncheski_robinett}
M. Belloni, M. Doncheski, and R. W. Robinett,
{\it Wigner quasi-probability distribution for the infinite square
well: energy eigenstates and time-dependent wave packets}; 
e-Print arXiv: quant-ph/0312086.
%
%
\bibitem{bassett}
R. W. Robinett and L. C. Bassett,
{\it Analytic results for Gaussian wave packets in four model
systems: II. Autocorrelation functions},
submitted to Found. Phys.
%
%
\end{thebibliography}



 \newpage


\begin{flushleft}
{\Large {\bf 
Figure Captions}}
\end{flushleft}
\vskip 0.5cm
 
\begin{itemize}
\bibitem[Fig.\thinspace 1.]  Time-development of a free-particle Gaussian wave 
packet, illustrating the modulus ($|\psi(x,t)|$, solid), 
the real ($Re(\psi(x,t))$, dotted)
and imaginary ($Im(\psi(x,t))$, dashed) parts,  as functions of time.
%
%
\bibitem[Fig.\thinspace 2.] The figures in the left column show the modulus 
(solid),
and the real (dotted) and imaginary (dashed) parts of Gaussian
wave packets at long times ($t = 10t_0$) plotted versus $x_0 + p_0 t/m$,
illustrating the large fraction of kinetic energy (note the pattern
of `wiggliness') in the front of the wave for the $p_0 = \Delta p_0$
case, as in Eqn.~(\ref{maximum_minimum_values}). In the right column, 
we plot the probability density ($|\psi(x,t)|^2$, solid) and the (scaled)
kinetic energy density ($S(x,t) = {\cal T}(x,t)/T(t)$, dashed)
from Eqn.~(\ref{scaled_kinetic_energy_density}).
%
%
\bibitem[Fig.\thinspace 3.] The figures in the top row show the modulus (solid),
and the real (dotted) and imaginary (dashed) parts of Gaussian wave packet
solutions of the harmonic oscillator corresponding to $x_0 = 0$ and
$p_0 > 0$ with $\beta = \beta_0/2 < \beta_0$. Times over the first quarter
period are shown. In the bottom row, we plot the probability density ($|\psi(x,t)|^2$, solid) and the (scaled) kinetic energy density 
($S(x,t) = {\cal T}(x,t)/T(t)$, dashed) 
from Eqn.~(\ref{scaled_kinetic_energy_density}) at the corresponding times.
%
%
\bibitem[Fig.\thinspace 4.]  Same as Fig.~3, but for an initial Gaussian
wave packet solution of the harmonic oscillator with 
$\beta = 2\beta_0 > \beta_0$.
%
%
\end{itemize}
 
\newpage
 



\noindent
\hfill
\begin{figure}[hbt]
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\begin{minipage}{0.7\linewidth}
\epsfig{file=fig1.eps,width=\linewidth}
\caption{}
\end{minipage}
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\end{figure}
\hfill 


\newpage



\noindent
\hfill
\begin{figure}[hbt]
\, \hfill \,
\begin{minipage}{0.7\linewidth}
\epsfig{file=fig2.eps,width=\linewidth}
\caption{}
\end{minipage}
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\end{figure}
\hfill 


\newpage

 


\noindent
\hfill
\begin{figure}[hbt]
\, \hfill \,
\begin{minipage}{0.7\linewidth}
\epsfig{file=fig3.eps,width=\linewidth}
\caption{}
\end{minipage}
\, \hfill \,
\end{figure}
\hfill 


\newpage

 


\noindent
\hfill
\begin{figure}[hbt]
\, \hfill \,
\begin{minipage}{0.7\linewidth}
\epsfig{file=fig4.eps,width=\linewidth}
\caption{}
\end{minipage}
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\end{figure}
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\newpage

 
 
\end{document}
 
 
 
\begin{equation}
      f(x) = \left\{ \begin{array}{ll} 0 & \mbox{for $  $} \\ 1 &
               \mbox{for $-a<x<0$} \end{array} \right.
\end{equation}


\newcounter{temp}
\begin{list}
{(\roman{temp})}{\usecounter{temp}}
\item 
\item
\item
\end{list}





 




%
\bibitem{lai}
C. S. Lai,
{\it  Spreading of wave packets in the coordinate representation}, 
Am. J. Phys., {\bf 47}, 766-768 (1979).
%
%
\bibitem{almeida}
C. Almeida and A. Jabs,
{\it  Spreading of a relativistic wave packet}, 
Am. J. Phys., {\bf 52}, 921-925 (1984).
%


%
\bibitem{jordan}
T. F. Jordan,
{\it  Spreading velocity of a relativistic wave packet}, 
Am. J. Phys., {\bf 54}, 182 (1986).
%
%
\bibitem{vandegrift}
G. Vandegrift,
{\it  The diffraction and spreading of a wave packet}, 
Am. J. Phys., {\bf 72} 404-407 (2004).
%





\begin{equation}
\phi(p,0) = \phi_0(p) = 
\sqrt{\frac{\alpha}{\sqrt{\pi}}}
\, e^{-\alpha^2(p-p_0)^2/2}
\, e^{-ipx_0/\hbar}
\label{initial_gaussian}
\end{equation}
which gives
\begin{equation}
\langle p \rangle_{t} = p_0
\, , 
\qquad
\quad
\langle p^2 \rangle_{t} = p_0^2 + \frac{1}{2\alpha^2}
\, ,
\qquad
\mbox{and}
\qquad
\Delta p_t = \Delta p_0 = \frac{1}{\alpha \sqrt{2}}
\, .
\end{equation}
The explicit form of the position-space wave function is given by
the Gaussian integral
\begin{equation}
\psi(x,t) = \pre \sqrt{\frac{\alpha}{\sqrt{\pi}}}
\integral\, e^{ip(x-x_0)/\hbar}\,
e^{-\alpha^2 (p-p_0)^2/2}\,
e^{-ip^2t/2m\hbar}\,dp
\end{equation}
which can be evaluated in closed form (using the change of variables
$q \equiv p-p_0$ and standard integrals) to obtain
\begin{equation}
\psi(x,t) = \frac{1}{\sqrt{\sqrt{\pi} \alpha \hbar (1+it/t_0)}}
\,
e^{ip_0(x-x_0)/\hbar}
\, e^{-ip_0^2t/2m\hbar}
\,
e^{-(x-x_0-p_{0}t/m)^2/2(\alpha \hbar)^2(1+it/t_0)}
\label{free_particle_position_solution}
\end{equation}
where $t_0 \equiv m\hbar \alpha^2$. 

\begin{equation}
\phi(p,0) = \phi_0(p) = 
\sqrt{\frac{\alpha}{\sqrt{\pi}}}
\, e^{-\alpha^2(p-p_0)^2/2}
\, e^{-ipx_0/\hbar}
\label{initial_gaussian}
\end{equation}
which gives
\begin{equation}
\langle p \rangle_{t} = p_0
\, , 
\qquad
\quad
\langle p^2 \rangle_{t} = p_0^2 + \frac{1}{2\alpha^2}
\, ,
\qquad
\mbox{and}
\qquad
\Delta p_t = \Delta p_0 = \frac{1}{\alpha \sqrt{2}}
\, .
\end{equation}
The explicit form of the position-space wave function is given by
the Gaussian integral
\begin{equation}
\psi(x,t) = \pre \sqrt{\frac{\alpha}{\sqrt{\pi}}}
\integral\, e^{ip(x-x_0)/\hbar}\,
e^{-\alpha^2 (p-p_0)^2/2}\,
e^{-ip^2t/2m\hbar}\,dp
\end{equation}
which can be evaluated in closed form (using the change of variables
$q \equiv p-p_0$ and standard integrals) to obtain
\begin{equation}
\psi(x,t) = \frac{1}{\sqrt{\sqrt{\pi} \alpha \hbar (1+it/t_0)}}
\,
e^{ip_0(x-x_0)/\hbar}
\, e^{-ip_0^2t/2m\hbar}
\,
e^{-(x-x_0-p_{0}t/m)^2/2(\alpha \hbar)^2(1+it/t_0)}
\label{free_particle_position_solution}
\end{equation}
where $t_0 \equiv m\hbar \alpha^2$. 







\begin{eqnarray}
P_{W}(x,p;t)
 & \equiv &
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\psi^{*}(x+y,t)\,\psi(x-y,t)\,e^{2ipy/\hbar}\,dy \\
& = & 
\frac{1}{\pi \hbar}
\int_{-\infty}^{+\infty}
\phi^*(p+q,t)\, \phi(p-q,t)\, e^{-2ixq/\hbar}\,dq
\label{wigner_function}
\end{eqnarray}

quant-ph0502097.extracted_bib
\begin{thebibliography}{99}
%
%
\bibitem{texts}
L. I. Schiff, 
{\it Quantum Mechanics}
(McGraw-Hill, New York, 1949) First Edition pp.57-59;
D. S. Saxon,
{\it Elementary Quantum Mechanics}
(McGraw-Hill, New York, 1968) pp.62-66;
C. Cohen-Tannoudji, B. Diu, and F. Lalo\"{e},
{\it Quantum Mechanics} Vol. 1
(Wiley, New York, 1977), pp.~61-66;
R. L. Liboff,
{\it Introductory Quantum Mechanics}
(Addison-Wesley, Reading, 1980, pp.~151-156;
R. A. Serway, C. J. Moses, and C. A. Moyer,
{\it Modern Physics}, 
(Saunders, Philadelphia, 1989), pp.~142-143;
M. A. Morrison,
{\it Understanding Quantum Mechanics}
(Prentice-Hall, Englewood Cliffs, 1990), pp.~206-213;
A. Goswami,
{\it Quantum Mechanics}
(Wm. C. Brown, Dubuque, 1992) pp.~37-43;
D. Griffiths,
{\it Introduction to Quantum Mechanics},
(Prentice-Hall, Englewood Cliffs, 1995), p. 50;
S. Gasiorowicz,
{\it Quantum Physics} 
(Wiley, New York, 1996) Second Edition, pp.~30-31;
R. W. Robinett,
{\it Quantum Mechanics: Classical Results, Modern Systems, and
Visualized Examples}
(Oxford, New York, 1997), pp.~57-60, 86-87, 90, 106.
%

%
\bibitem{baird}
L. C. Baird,
{\it  Moments of a wave packet}, 
Am. J. Phys. {\bf 40}, 327-329 (1972).
%
%
\bibitem{nicola}
M. Nicola,
{\it  Position uncertainty of  a free particle}, 
Am. J. Phys. {\bf 40}, 342 (1972).
%
%
\bibitem{bradford}
H. M. Bradford,
{\it  Propagation and spreading of a pulse or wave packet}, 
Am. J. Phys. {\bf 44}, 1058-1063 (1976).
%
%
\bibitem{woodsum}
H. C. Woodsum and K. R. Brownstein,
{\it  Tumbling motion of free three-dimensional wave packets}, 
Am. J. Phys. {\bf 45}, 667-670 (1977).
%
%
\bibitem{farina}
J. E. G. Farina,
{\it  Classical and quantum spreading of position probability}, 
Am. J. Phys. {\bf 45}, 1200-1202 (1977).
%
%
\bibitem{klein}
J. R. Klein,
{\it  Do free quantum mechanical wave packets always spread?},
Am. J. Phys. {\bf 48}, 1035-1037 (1980).
%
%
\bibitem{styer}
D. F. Styer,
{\it  The motion of wave packets through their expectation values
and uncertainties}, 
Am. J. Phys. {\bf 58}, 742-744 (1990).
%
%
\bibitem{andrews}
M. Andrews,
{\it  Invariant operators for quadratic Hamiltonians}, 
Am. J. Phys. {\bf 67}, 336-343 (1999).
%
%
\bibitem{merzbacher} E. Merzbacher, {\it Quantum Mechanics}
(Wiley and Sons, New York, 1961) pp~155, 165.
%
%
\bibitem{bohm} 
D. Bohm,  
{\it Quantum Theory} 
(Prentice-Hall, Englewood Cliffs, 1963) pp~200-205.
%
%
\bibitem{leblond} 
J.-M. L\'{e}vy-Leblond,
{\it Correlation of quantum properties and the generalized Heisenberg
inequalities}, 
Am. J. Phys. {\bf 54}, 135-136 (1986);
R. A. Campos, {\it Correlation coefficient for incompatible observables
of the quantum harmonic oscillator}, 
Am. J. Phys. {\bf 66}, 712-718 (1998).
%

%
\bibitem{berry}
M. V. Berry and N. L. Balazs,
{\it Non-spreading wave packets},
Am. J. Phys. {\bf 47}, 264-267 (1979); see also the very relevant Comment 
on this paper by D. M. Greenberger, Am. J. Phys. {\bf 48}, 256 (1980).
We note that the special free-particle but `accelerating' solutions involving
Airy packets found by  Berry and Balazs  are not square integrable, but
localized wave packets constructed from them exhibit the standard
result for $\langle x \rangle_t$ consistent with Ehrenfest's theorem
in Eqn.~(\ref{x_1}) and an expression for $\Delta x_t$ consistent with 
Eqn.~(\ref{general_case}).
%
%
\bibitem{ford}
G. W. Ford and R. F. O'Connell,
{\it  Wave packet spreading: temperature and squeezing effects with
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Am. J. Phys. {\bf 70}, 319-324 (2002).
%
%
\bibitem{bassett}
R. W. Robinett and L. C. Bassett,
{\it Analytic results for Gaussian wave packets in four model systems:
I. Visualization of the kinetic energy}, to appear in Found. Phys. Lett.
[arXiv:quant-ph/0408049].
%
%
\bibitem{wigner}
E. Wigner,
{\it On the quantum correction for thermodynamic equilibrium},
Phys. Rev. {\bf 40}, 749-759 (1932).
%
%
\bibitem{wigner_research}
V. I. Tatarski\u{i},
{\it The Wigner representation of quantum mechanics},
Sov. Phys. Usp. {\bf 26}, 311-327 (1983);
N. L. Balaczs and B. K. Jennings,
{\it Wigner's function and other distribution functions in
mock phase space},
Phys. Rep. {\bf 105}, 347-391 (1984);
P. Carruthers and F. Zachariasen,
{\it Quantum collision theory with phase-space distributions},
Rev. Mod. Phys. {\bf 55}, 245-285 (1983);
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,
{\it Distribution functions in physics: Fundamentals},
Phys. Rep.  {\bf 106}, 121-167 (1984);
J. Bertrand and P. Bertrand,
{\it A tomographic approach to Wigner's function},
Found. Phys. {\bf 17}, 397-405 (1987);
Y. S. Kim and M. E. Noz,
{\it Phase space picture of quantum mechanics: Group theoretical
approach}, Lecture Notes in Physics Series, Vol. 40
(World Scientific, Singapore, 1990);
H.-W. Lee,
{\it Theory and application of the quantum phase-space distribution
functions},
Phys. Rep. {\bf 259}, 147-211 (1995);
A. M. Ozorio de Almeida,
{\it The Weyl representation in classical and quantum mechanics},
Phys. Rep. {\bf 296}, 265-342 (1998).
%
%
\bibitem{wigner_pedagogical}
J. Snygg,
{\it Wave functions rotated in phase space},
Am. J. Phys. {\bf 45}, 58-60 (1977);
N. Mukunda,
{\it Wigner distribution for angle coordinates in quantum mechanics},
Am. J. Phys. {\bf 47}, 192-187 (1979);
S. Stenholm,
{\it The Wigner function: I. The physical interpretation},
Eur. J. Phys. {\bf 1}, 244-248 (1980);
G. Mourgues, J. C. Andrieux, and M. R. Feix,
{\it Solution of the Schroedinger equation for a system excited by
a time Dirac pulse of potential. An example of the connection with
the classical limit through a particular smoothing of the Wigner
function},
Eur. J. Phys. {\bf 5}, 112-118 (1984);
Y. S. Kim and E. P. Wigner,
{\it Canonical transformations in quantum mechanics},
Am. J. Phys. {\bf 58}, 439-448 (1990);
M. Casas, H. Krivine, and J. Martorell,
{\it On the Wigner transforms of some simple systems and their
semiclassical interpretations},
Eur. J. Phys. {\bf 12}, 105-111 (1991);
R. A. Campos,
{\it Correlation coefficient for incompatible observables of the
quantum harmonic oscillator},
Am. J. Phys. {\bf 66}, 712-718 (1998);
M. Belloni, M. A. Doncheski, and R. W. Robinett,
{\it Wigner quasi-probability distribution for the infinite square well:
Energy eigenstates and time-dependent wave packets},
Am. J. Phys. {\bf 72}, 1183-1192 (2004) 
%
%
\bibitem{wigner_lee}
H.-W. Lee,
{\it  Spreading of a free wave packet}, 
Am. J. Phys., {\bf 50}, 438-440 (1982).
%
%
\bibitem{kim_noz} 
See Y. S. Kim and M. E. Noz in Ref.~\cite{wigner_research}.
%
%
\bibitem{bassett_2}
R. W. Robinett and L. C. Bassett,
{\it Analytic results for Gaussian wave packets in four model systems:
II. Autocorrelation functions}, to appear in Found. Phys. Lett.
[arXiv:quant-ph/0408050].
%
%
\bibitem{meekhof} 
D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland,
{\it Generation of nonclassical motional states of a trapped atom},
Phys. Rev. Lett. {\bf 76} 1796-1799 (1996).
%
%
\bibitem{heinzen} D. J. Heinzen and D. J. Wineland,
{\it Quantum-limited cooling and detection of radio-frequency oscillations
by laser-cooled atoms},
Phys. Rev. {\bf A42}, 2977-2994 (1990).
%
%
\bibitem{saxon} See D. A. Saxon in Ref.~\cite{texts}.
%
%
\bibitem{revivals} For a recent review, see R. W. Robinett,
{\it Quantum wave packet revivals},
Phys. Rep. {\bf 392},  1-119 (2004).
%
%
\bibitem{aronstein} 
D. L. Aronstein and C. R. Stroud, Jr.,
{\it Fractional wave-function revivals in the infinite square
well},
Am. J. Phys. {\bf 55}, 4526-4537 (1997)
%
%
\bibitem{belloni} 
M. Belloni,
M. A. Doncheski,
and R. W. Robinett,
{\it Wigner quasi-probability distribution for the infinite well:
Energy eigenstates and time-dependent wave packets},
Am. J. Phys. {\bf 72}, 1183-1192 (2004) [arXiv:quant-ph/0312086].
%
%
\bibitem{doncheski_1}
M. A. Doncheski and R. W. Robinett,
{\it  Anatomy of a quantum `bounce'}, 
Eur. J. Phys,  {\bf 20}, 29-37 (1999)
[arXiv:quant-ph/0307010].
%
%
\bibitem{dodonov}
V. V. Dodonov and M. A. Andreata, 
{\it Deflection of quantum particles by impenetrable boundary},
Phys. Lett. A {\bf 275}, 173-181 (2000);
M. A. Andreata and V. V. Dodonov,
{\it The reflection of narrow slow quantum packets from mirrors},
J. Phys. A: Math. Gen. {\bf 35}, 8373-8392 (2002);
V. V. Dodonov and M. A. Andreata, 
{\it Quantum deflection of ultracold atoms from mirrors},
Laser Physics, {\bf 12}, 57-70 (2002).
%
%
\bibitem{doncheski_2}
M. Belloni, M. A. Doncheski, and R. W. Robinett,
{\it Exact results for `bouncing' Gaussian wave packets},
to appear in Phys. Scripta.
[arXiv:quant-ph/0408182];
%


%
\end{thebibliography}