Numerical simulations of Landau-Zener transition

I tried to use a midpoint method and numerically solve the Schrödinger equation for the original Landau-Zener (LZ) problem: a 2X2 Hamiltonian $$\begin{pmatrix}\alpha t&\delta \\ \delta &-\alpha t\end{pmatrix}$$ with initial condition $$\psi=\begin{pmatrix}1\\0\end{pmatrix}$$ (the ground state) at some $$t=-1000$$, and say $$\alpha=0.01$$ and $$\delta=0.04$$. I took a huge amount of time slices ($$10^8$$) which gives a time step of ~$$10^{-5}$$.

My goal is to reach the exact value from the LZ formula, but no matter how small a time step I take, I always have an error of 0.1%, after averaging over the oscillations that rise in the asymptotic behaviour.

Has anyone encountered this problem?

• The problem is presumably numerical in origin, so you should post this question to the computational science SE instead and include your code. Aug 15 '19 at 10:25
• By the way: The preferred way here for mathematical notation is MathJax. Aug 15 '19 at 10:48