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
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 (a_{s}E+b_{s}), , and , respectively. To get from P_{r}(f) back to P_{r}(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 P_{r}(E) = |df/dE| P_{r}(f) we find the useful relation
For the T-matrix formalism, the same analysis yields