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Real wave packet propagation

The real wave packet approach applies either to the usual Schrödinger equation or to a modified Schrödinger equation, $
i \hbar \partial \chi ({{\bf x}}, t)/ \partial t = f( \hat{H}) \chi
({{\bf x}},t)$, where $f(\hat{H})$ replaces $\hat H$ and $\chi ({\bf x},t)$replaces $\psi ({\bf x},t)$[12]. Since only the real part is manipulated, computational requirements can be reduced relative to complex wave packet propagation. The modified Schrödinger equation, through judicious choice of f, results in greater savings because f can be chosen to simplify the propagation. Derivations of S matrix or reaction probability formulae are simpler if one refers to the usual Schrödinger equation. The results are then generalized to the modified Schrödinger equation without difficulty.

Consider the modified Schrödinger equation, and take $f(\hat{H})
=-\hbar/\tau \arccos (\hat{H}_s)$, where $\hat{H_s}=a_s \hat{H} +b_s$with as and bs chosen such that the maximum and minimum eigenvalues of $\hat{H_s}$ lie between -1 and 1. Ref. [12] showed that the real part $q_\chi({\bf x},t)$ = ${\rm Re}
\left[ \chi({\bf x},t)\right]$ can be propagated directly and satisfies


 \begin{displaymath}q_{\chi}({\bf x},t+\tau)=\hat{A}\left[-\hat{A}q_{\chi}({\bf x},t-\tau)
+ 2 \hat{H}_s q_{\chi}({\bf x},t)\right],
\end{displaymath} (14)

where $\tau$ is an arbitrary time step. We have also introduced an absorption operator $\hat{A}$ to remove wave packet amplitude as it approaches grid edges. (Eq. (14) is also the damped Chebyshev iteration introduced in Ref. [9].) While this real wave packet approach focuses on the real part of a wave packet $\chi$, it should be noted that the initial condition $\chi (t=0)$ can be complex, as discussed in Ref. [12] and explicitly implemented in Ref. [16].


next up previous
Next: Flux analysis of real Up: Flux and real wave Previous: Flux and real wave
Anthony J. H. M. Meijer
1998-06-19