Calculation of sharp-energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> matrices via time-dependent and time-independent wave packets

This paper explores momentum-space wave-packet methods to calculate sharp-energy $S$-matrix elements from a time-dependent description of collisions. Two aspects of the computational time-dependent (TD) scattering theory are examined and elucidated: (i) Extraction of the sharp-energy $S$ matrices from the asymptotic analysis of a numerically propagated wave packet, and ii) use of a time-to-energy transform of the numerical wave packet to obtain an energy-resolved time-independent wave packet which provides an alternative form of the asymptotic analysis. The second approach gives rise to a novel expression for the $S$ matrix in terms of time-independent wave packets, which may be useful in the time-independent context as well. In the computational implementation of the methods introduced, the discretization of the spatial variables turns the time-evolution problem of an initial precollision wave packet into a set of first-order differential equations in time. Propagation of the wave packet is then achieved by invoking the central-difference approximation for the time derivative. While time-propagation of the wave packet is carried out, the time-to-energy transform is concurrently accumulated using the trapezoidal rule. These procedures are tested on the $s$-wave scattering problem for a model interaction potential. $S$-matrix elements over a wide range of energies can be extracted from a single wave packet. These wave-packet results are in good agreement with the reference results obtained by solving the Lippmann-Schwinger integral equation one energy at a time for large number of energies.