On the numerical solution of the time-dependent Schrödinger equation (Q1893769)

The microscopic description of many-body systems like atoms or nuclei is based on a many-body Hamiltonian. The related wave functions are given by Slater determinants for fermions. In the case of time dependent processes like atomic or nuclear collisions, nuclear fission or fusion, however, the situation is too complex due to the great number of degrees of freedom. Therefore, in most cases, a collective coordinate is introduced according to the essential physical properties of the considered system. This procedure leads to a macroscopic model with one degree of freedom, which is governed by an effective one-body Schrödinger equation including a time-dependent potential in the considered examples. The well-known coordinate representation reads \[ i\hbar {\partial \over \partial t} \phi({\mathbf r}, t) = \left[ - {\hbar \over 2M} \nabla^2 + V({\mathbf r},t) \right] \phi({\mathbf r}, t). \tag{1} \] In almost all practical cases Eq. (1) has to be evaluated numerically. The standard technique essentially consists of the following two steps: first apply a suitable scheme for the space discretization and then perform the time integration. The algorithm presented here is only related to the second step: The time integration in the case of an explicitly time dependent potential \(V({\mathbf r}, t)\). Our proposed convenient modification upgrades the standard method to a much more efficient version while negligibly increasing the computing time.