7.3 Wave Packet Dynamics (WPD)
- Earliest attempt at a dynamical semiclassical simulation
for medium light particles such as Neon atoms
- Suggested by Konrad Singer [SINGER 86], based
on ideas by Heller et al. [HELLER 75,HELLER 76]
- Later development: see [HUBER 88,KOLAR 89,HERRERO 95,MARTINEZ 97].
Describe the smeared-out position of the atomic center of particle
$k$ by a Gaussian:
$
\phi_{k}( r ,t)=e^{\textstyle \frac{\textstyle i}{\textstyle \hbar} \, Q_{k}(t)}
$
(7.43)
with the quadratic form
$ \begin{eqnarray}
Q_{k}(t) & \equiv & \left[ r - R _{k}(t)\right]^{T} \cdot
{ A} _{k}(t) \cdot \left[ r - R _{k}(t)\right] + P _{k}(t)
\cdot \left[ r - R _{k}(t)\right] + D_{k}(t)
\\
& \equiv & \xi _{k}^{T}(t) \cdot { A} _{k}(t)
\cdot \xi _{k}(t) + P _{k}(t) \cdot \xi _{k}(t) + D_{k}(t)
\end{eqnarray} $
(7.44)
$ R _{k}(t)$... center of the packet
${A}_{k}(t)$ (matrix) ... shape, size and orientation.
Simplest case: spherically symmetric packet
$\Longrightarrow$ ${A}_{k}$ scalar; in general: ellipsoidal
"cloud" of size $\approx \sigma_{LJ}/10$
$ P_{k}(t)$ ... momentum of the wave packet
$D_{k}(t)$ ... phase factor (normalization)
One-dimensional case:
Each particle (i. e. wave packet) is described by
$
\phi(x,t)=e^{\textstyle \frac{\textstyle i}{\textstyle \hbar}\, Q(t)}
$
(7.45)
where
$
\begin{eqnarray}
Q(t)&=&A(t) [ x-X(t) ]^{2}+P(t) [x-X(t)] +D(t)
\\
& \equiv & A(t) \xi ^{2}(t) +P(t) \xi (t) + D(t)
\end{eqnarray}
$
(7.46)
($A$ and $D$ are in general complex; $P$ is real.)
Expectation value of position:
$
\langle \phi |x| \phi \rangle \equiv \int dx\,x \,\phi^{*}(x,t)
\phi(x,t) = X(t)
$
(7.47)
Expected momentum:
$
\langle \phi | -i\hbar \frac{\textstyle \partial}{\textstyle \partial x} | \phi \rangle = \dots =
P(t)
$
(7.48)
Assumption of a Gaussian shape: made for mathematical convenience.
Subject to a harmonic potential, a Gaussian w. p. will remain
Gaussian.
$\Longrightarrow$ Good approximation for any continuous potential.
$N$ wave packets: product ansatz (exchange neglected)
$
\Psi( r ,t)= \prod_{k=1}^{N} \phi_{k}( r ,t)
$
(7.49)
Solve the Schroedinger equation
$
i\hbar\frac{\textstyle \partial \Psi( r ,t)}{\textstyle \partial t}
-{\rm H}\Psi(r ,t)=0
$
(7.50)
by applying the minimum principle of Dirac, Frenkel, and McLachlan:
Temporal evolution of the parameters
$A_{k}$, $P_{k}$, and $D_{k}$ occurs such that the expression
$
I\left( \Psi , \frac{\textstyle \partial \Psi}{\textstyle \partial t}\right)
\equiv \int \dots \int \left| i \hbar \frac{\textstyle \partial \Psi}{\textstyle \partial t}
-{\rm H}\Psi \right|^{2} d r
$
(7.51)
will always be a minimum.
$\Longrightarrow$
Variational calculus, with the simplifying assumption that
${A}_{k}=A_{k} {I}$ (spherical Gaussian) leads to
$
\begin{eqnarray}
\left( \dot{A} + \frac{\textstyle 2}{\textstyle m} A^{2} \right) \langle \xi^{2} \rangle
+ \langle \overline{U} \rangle + \left[ - \frac{\textstyle 3 \hbar i}{\textstyle m} A
- \frac{\textstyle P^{2}}{\textstyle 2m} + \dot{D} \right] &=& 0\\
\dot{P}_{\alpha} \langle \xi_{\alpha}^{2} \rangle +
\langle \overline{U}\xi_{\alpha} \rangle &=& 0\\
\left( \dot{A} + \frac{\textstyle 2}{\textstyle m} A^{2} \right) \langle (\xi^{2})^{2} \rangle
+ \langle \overline{U} \xi^{2} \rangle + \left[ - \frac{\textstyle 3 \hbar i}{\textstyle m} A
- \frac{\textstyle P^{2}}{\textstyle 2m} + \dot{D} \right] \langle \xi^{2} \rangle &=& 0
\end{eqnarray} $
(7.52-7.54)
where
$\langle \dots \rangle$ ... expectation value, and
$
\overline{U}_{k} \equiv \sum \limits_{l \neq k} \int U(r_{kl})\phi_{l}^{*} \phi_{l}
\, d r _{l}
$
(7.55)
is the potential created at
$r_{k}$ by the "smeared out" particles
$l$.
Singer et al.: approximate the given pair potential $U(r)$
by a sum of Gaussian functions $\Longrightarrow$
right-hand side of
7.55 is a sum of simple definite integrals.
Introduce auxiliary variables
$c$, $d$, and $Z$
$
\begin{eqnarray}
c & \equiv & \langle (\xi^{2})^{2} \rangle - \langle \xi^{2} \rangle^{2} \\
d & \equiv & \langle \overline{U}\xi^{2} \rangle - \langle \overline{U} \rangle
\langle \xi^{2} \rangle
\end{eqnarray}
$
(7.56-7.57)
and
$
A \equiv \frac{\textstyle m}{\textstyle 2} \frac{\textstyle \dot{Z}}{\textstyle Z}
$
(7.58)
With $\dot{R} \equiv P/m$ we find for the position $R$
and the shape parameter $Z$:
$
\ddot{R}_{\alpha} =
- \frac{\textstyle \langle \overline{U} \xi_{\alpha}\rangle}{\textstyle m\langle \xi_{\alpha}^{2}
\rangle}
\;\;\;\;\;\;\;\;\;\;
\ddot{Z} = - \frac{\textstyle 2}{\textstyle m}
\frac{\textstyle d}{\textstyle c} Z
$
(7.59)
$\Longrightarrow$
Solve by any integration method, such as the
Størmer-Verlet algorithm.
Applications:
()
Liquid and gaseous neon [SINGER 86].
Basic thermodynamic properties in good agreement with experiment;
pair correlation function smeared out at its peaks (more than predicted).
Kinetic energy of the wave packets: given by the curvature of
$\phi_{k}$, i. e. by the shape parameter $A_{k}$:
temperature always too high if $A_{k}$ is allowed to vary between
individual wave packets; better agreement with
experiment by the "semi-frozen" approximation (all
$A_{k}$ equal, changing in unison under the influence of a force
that is averaged over all particles).
Recent applications: see [KNAUP 99]
More about the method:
see [HUBER 88,KOLAR 89,HERRERO 95,MARTINEZ 97]
vesely 2006
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