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Flux analysis of real wave packets

If one has just = Re[ ], propagated using the time-dependent Schrödinger equation, one may use , and expand as with Eq. (1),

 = (15)

From this it follows that

 (16)

We require that the initial wave packet be such that , for the Evalues of interest, be zero, i.e., that the initial wave packet has no appreciable amplitude for energies -E. [This can be achieved if the energy scale is chosen such that only E > 0 corresponds to scattering states. Alternatively, as with the energies f(E) associated with the modified Schrödinger equation, no physical states correspond to -f(E).] We obtain

 (17)

which differs from Eq. (1), the complex wave packet result, by simply a factor of 2. We want to point out here that this is a general result. Therefore, we can use it to adapt any method mentioned in Sec. 2. For reaction probabilities using the flux method from Refs. [5,6], we get for real wave packets that Pr(E) is simply four times the right hand site of Eq. (13).

Consider now the modified Schrödinger equation and its corresponding real part , which also satisfies Eq. (14). One simply repeats the entire analysis above, but with , E, , and replaced by , f(E) = arccos (asE+bs), , and , respectively. To get from Pr(f) back to Pr(E), one has to realize that f'' normalized scattering functions are related to E normalized ones by . Thus, if the initial condition for the modified Schrödinger equation is the same as for the ordinary one, , then . Moreover, since we have the real part at discrete times (or iterations) , we can employ a Fourier series to evaluate the integrals in Eq. (11). Using this and the fact that Pr(E) = |df/dE| Pr(f) we find the useful relation

 Pr(E) = (18)

where and df/dE has been explicitly evaluated. (Note that has canceled out of the final expression.)

For the T-matrix formalism, the same analysis yields

 (19)

Next: Results for D + H Up: Flux and real wave Previous: Real wave packet propagation
Anthony J. H. M. Meijer
1998-06-19