Time-dependent wave-packet forms of Schrödinger and Lippmann-Schwinger equations

Time-independent wave-packet forms of the Schrödinger equation (TIWSE) and Lippmann-Schwinger equation (TIWLSB) have been derived by a partial time-to-energy Fourier transform of ${\mathit{L}}^2$ wave-packet solutions to the time-dependent Schrödinger equation. The new equations retain the initial wave packet χ(${\mathit{t}}_0$) as a ‘‘universal source’’ of scattered waves, which applies for all collision energies E contained in the initial wave packet. The relationship between the solution ${\mathrm{\Psi }}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\Psi }}^{\mathrm{}(+)\mathrm{}}$(E) of the standard time-independent Lippmann-Schwinger or Schrödinger equation is given and the method illustrated by a computational application.