Splitting methods for the Schrödinger equation

The following text is a summary of the same-titled talk given within the frame of the seminar ’AKNUM: Seminar aus Numerik’ at the Vienna University of Technology. We address time splitting methods for the Schrödinger equation. In particular, we concentrate on the Strang splitting and prove second order convergence in the case of a bounded potential and given bounds on the (iterated) commutators. In addition, we briefly describe the Split-Step Fourier method, which allows an efficient computation of the solution on an equispaced spatial grid. 1 Theoretical background We consider the time-dependent Schrödinger equation (TDSE) i~ψ̇ = − ~ 2m ∆ψ + Vψ, (1) where ψ = ψ(x, t) is a wave function (state) with postulations • |ψ(., t)|2 is a probability distribution for the particle position, i.e. ∫ Ω |ψ(x, t)|2 dx gives the probabilty that the particle is located in Ω, • The initial state ψ(x, t0) determines later states (Causality), • Superposition: linear combinations of states are again states. For our mathematical considerations we consider the scaled version of (1), i.e. the scaling x← x √ 2m/~ leads to Definition 1 (TDSE). iψ̇ = − 2∆ψ + Vψ =: Hψ, (2) where V is a potential and H the Hamiltonian. It is easily verified that the following quantities are preserved by the TDSE (2). • Energy is real: 〈Hψ, ψ〉 H s.a. = 〈ψ,Hψ〉 = 〈Hψ, ψ〉 (3) ∗[email protected], [email protected]