Further analysis of solutions to the time-independent wave packet equations for quantum dynamics: General initial wave packets

In this paper we reexamine, and analyze solutions to, the recently derived time-independent wave packet-Schrödinger (TIWS) and time-independent wave packet-Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time-dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrödinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninteracting to the interacting region. The method is illustrated by calculations for two 1-D scattering problems, namely the transmission of an electron through a single barrier and through a double barrier. The latter is especially challenging because of the occurrence of long-lived resonances with the electron trapped inside the double barrier. In addition, we show that the method can still make use of absorbing potentials to decrease grid size, and we show how the formalism can be used to treat inelastic and reactive scattering, and radial scattering variables, thereby constituting a general approach to time-independent wave packet quantum scattering.