New method in time-dependent quantum scattering theory: Integrating the wave function in the interaction picture

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Abstract
A new approach for solving the time-dependent wave function in quantum scattering problem is presented. The conventional wave packet method, which directly solves the time-dependent Schrödinger equation, normally requires a large number of grid points since the Schrödinger picture wave function both travels and spreads in time. Also, since the Schrödinger picture wave function oscillates in time with frequency ω=E/ℏ, a very small time increment is required to integrate the Schrödinger equation, especially for high energy collisions. The new method presented in this paper transforms the Schrödinger picture wave function into the interaction picture and carries out the integration in it. The new approach is superior to conventional one in that (1) a smaller numerical grid is required due to the localized nature of the interaction picture wave function, since it is not a traveling wave and does not spread appreciably in coordinate space, and thus behaves like a bound state wave function. (2) The interaction picture wave function varies slowly with time and is essentially independent of energy, permitting the use of a large time increment in the numerical integration. Because of these two features in this new approach, we are able to integrate the time dependent wave function once and obtain an accurate S matrix over a wide range of energy efficiently.