# Numerical solutions for time-dependent Hamiltonian

Currently I am facing the problem to solve numerically the following equation for a double well harmonic potential:

$$i\hbar \frac{\partial}{\partial t}\psi(x,t)= -\frac{\hbar}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t)+ V(x,t)\psi (x,t)$$

where

$$V(x,t)=\frac{m\omega_0^2}{2} \left[ \frac{a(t)}{x_0^2}x^4 -b(t)x^2 \right]$$

Where both $a(t)$ and $b(t)$ also depend sinusoidally on time. However I don't know where to start, I mean which numerical method should I use in order to solve this time of potential. Would the split step Fourier method work or which one should I use? I would appreciate any insight or recommendation that you could give me.

• Would Computational Science be a better home for this question? May 21, 2018 at 10:30
• May 21, 2018 at 16:02
• Thanks Emilio and Qmechanic for the remarks, I found the link very helpful. May 24, 2018 at 8:37
• You could make some progress by looking into the Floquet formalism. Nevertheless, I don't expect an analytical solution to be available. This could however guide you in a numerical approach. Sep 4, 2020 at 13:29