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?

  • 1
    $\begingroup$ The problem is presumably numerical in origin, so you should post this question to the computational science SE instead and include your code. $\endgroup$
    – lemon
    Aug 15 '19 at 10:25
  • $\begingroup$ By the way: The preferred way here for mathematical notation is MathJax. $\endgroup$ Aug 15 '19 at 10:48

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