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:

$$\phi_{k}(\mbox{$\bf r$},t)=e^{\textstyle \frac{i}{\hbar} \, Q_{k}(t)}$$ (7.43)

with the quadratic form

$$Q_{k}(t) \equiv \left[ \mbox{$\bf r$}-\mbox{$\bf R$}_{k}(t)\right]^{T} \cdot \mbo... ... P$}_{k}(t) \cdot \left[ \mbox{$\bf r$}-\mbox{$\bf R$}_{k}(t)\right] + D_{k}(t)$$

$$\equiv \mbox{$\bf\xi$}_{k}^{T}(t) \cdot \mbox{${\bf A}$}_{k}(t) \cdot \m... ...\bf\xi$}_{k}(t) + \mbox{$\bf P$}_{k}(t) \cdot \mbox{$\bf\xi$}_{k}(t) + D_{k}(t)$$ (7.44)

$\mbox{$\bf R$}_{k}(t)$... center of the packet
$\mbox{${\bf A}$}_{k}(t)$ (matrix) ... shape, size and orientation. Simplest case: spherically symmetric packet $\Longrightarrow$ scalar $\mbox{${\bf A}$}_{k}$ scalar
in general: ellipsoidal ``cloud'' of size $\approx \sigma_{LJ}/10$
$\mbox{$\bf 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

$$\phi(x,t)=e^{\textstyle \frac{i}{\hbar}\, Q(t)}$$ (7.45)

where

$$Q(t) = A(t) [ x-X(t) ]^{2}+P(t) [x-X(t)] +D(t)$$

$$\equiv A(t) \xi ^{2}(t) +P(t) \xi (t) + D(t)$$ (7.46)

($A$ and $D$ are in general complex; $P$ is real.)
Expectation value of position:

$$\langle \phi \vert x\vert \phi \rangle \equiv \int dx\,x \,\phi^{*}(x,t) \phi(x,t) = X(t)$$ (7.47)

Expected momentum:

$$\langle \phi \vert -i\hbar \frac{\partial}{\partial x} \vert \phi \rangle = \dots = P(t)$$ (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)

$$\Psi(\mbox{$\bf r$},t)= \prod_{k=1}^{N} \phi_{k}(\mbox{$\bf r$},t)$$ (7.49)

Solve the Schroedinger equation

$$i\hbar\frac{\partial \Psi(\mbox{$\bf r$},t)}{\partial t}-\mbox{\rm H}\Psi(\mbox{$\bf r$},t)=0$$ (7.50)

by applying the minimum principle of Dirac, Frenkel, and McLachlan:

Temporal evolution of the parameters $\mbox{${\bf A}$}_{k}$, $\mbox{$\bf P$}_{k}$, and $D_{k}$ occurs such that the expression

$$I\left( \Psi , \frac{\partial \Psi}{\partial t}\right) \equi... ...{\partial t} -\mbox{\rm H}\Psi \right\vert^{2} d\mbox{$\bf r$}$$ (7.51)

will always be a minimum.

$\Longrightarrow$ Variational calculus, with the simplifying assumption that $\mbox{${\bf A}$}_{k}=A_{k} \mbox{${\bf I}$}$ (spherical Gaussian) leads to

$$\left( \dot{A} + \frac{2}{m} A^{2} \right) \langle \xi^{2} \rangl... ...} \rangle + \left[ - \frac{3 \hbar i}{m} A - \frac{P^{2}}{2m} + \dot{D} \right] = 0$$ (7.52)

$$\dot{P}_{\alpha} \langle \xi_{\alpha}^{2} \rangle + \langle \overline{U}\xi_{\alpha} \rangle = 0$$ (7.53)

$$\left( \dot{A} + \frac{2}{m} A^{2} \right) \langle (\xi^{2})^{2} ... ...ac{3 \hbar i}{m} A - \frac{P^{2}}{2m} + \dot{D} \right] \langle \xi^{2} \rangle = 0$$ (7.54)

where $\langle \dots \rangle$ ... expectation value, and

$$\overline{U}_{k} \equiv \sum_{l \neq k} \int U(r_{kl})\phi_{l}^{*} \phi_{l} \, d\mbox{$\bf r$}_{l}$$ (7.55)

is the potential created at $\mbox{$\bf 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 $Z$

$$c \equiv \langle (\xi^{2})^{2} \rangle - \langle \xi^{2} \rangle^{2}$$ (7.56)

$$d \equiv \langle \overline{U}\xi^{2} \rangle - \langle \overline{U} \rangle \langle \xi^{2} \rangle$$ (7.57)

and

$$A \equiv \frac{m}{2} \frac{\dot{Z}}{Z}$$ (7.58)

With $\dot{\mbox{$\bf R$}} \equiv \mbox{$\bf P$}/m$ we find for the position $\mbox{$\bf R$}$ and the shape parameter $Z$:

$$\ddot{R}_{\alpha} = - \frac{\langle \overline{U} \xi_{\alph... ...} \;\;\;\;\;\;\;\;\;\; \ddot{Z} = - \frac{2}{m} \frac{d}{c} Z$$ (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]