Franz J. Vesely > CompPhys Tutorial > Selected Applications > Wave Packet Dynamics  

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Part III: Ch. 7

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)


$ \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)


$ 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.

() 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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