# Runge-Kutta timestep in atomic units

I'm using 4th order RK to solve the schroedinger equation in atomic units. Say I want to simulate 400fs in intervals of h=10fs, then in atomic units this is h=413a.u and 400fs=16500a.u. 4RK involves repeated multiplication by the timestep, so with h=413 everything blows up. Could someone explain what I'm missing?

Edit: to be clear, I'm solving

\begin{align} \frac{d}{dt} \psi = -i H \psi \end{align} Where $\psi$ is a vector and $H$ a hermitian matrix.

• Runge Kutta methods are explicit schemes in time. There is a relationship between the time step and the spatial discretisation to fulfill in order to get rid of instabilities. I supose that the Hamiltonian operator has been discretised. – HBR Feb 1 '18 at 16:48
• Could you elaborate please? The system is a lattice with lattice constant 8 (so space is discrete) but the Hamiltonian is in a basis of Fock states. – Insert_Username Feb 1 '18 at 17:11
• Currently I am with mobile phone... and I am lazy to write here. Search on google about heat diffusion equation temporal scheme. In wikipedia you will find the restriction I was talking about. Regarding to the basis of Fock states... it is too specific to know what is that... – HBR Feb 1 '18 at 17:26