Symmetric-split-operator techniques and finite-difference methods for the solution of classical and quantum evolution problems

Abstract

We show that the symmetrically split-operator technique provides a useful method to study evolution problems in classical and quantum mechanics. It is shown that it leads to a finite-difference scheme naturally preserving the symplectic nature of the problem in classical mechanics and the unitarity in quantum mechanics. We also prove its usefulness for the solution of Liouville and Fokker-Planck equations. The link with other difference schemes (Bender-Sharp, Moncrief, Vazquez) is finally discussed.