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

Time-independent wave-packet forms of the Schr\"odinger 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 Schr\"odinger equation. The new equations retain the initial wave packet \ensuremath{\chi}(${\mathit{t}}_{0}$) 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 ${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\ensuremath{\Psi}}}^{(+)}$(E) of the standard time-independent Lippmann-Schwinger or Schr\"odinger equation is given and the method illustrated by a computational application.