# 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? – Qmechanic May 21 '18 at 10:30
• – Emilio Pisanty May 21 '18 at 16:02
• Thanks Emilio and Qmechanic for the remarks, I found the link very helpful. – Oliver May 24 '18 at 8:37

## 1 Answer

I'd say it strongly depends on what you're looking to measure from the calculation. If you're looking at the system's evolution for a time window that isn't incredibly long, Runge-Kutta should be fine (4th order is usually a go to. ) For the spatial derivatives, good ol' finite differences should be sufficient.

• Pray expand on how you think a symplectic integrator would be useful in solving the Schrodinger equation. If you have reference on symplectic schemes used for this purpose I'd be very much interested in reading them. – ZeroTheHero May 21 '18 at 13:30
• @ZeroTheHero I'm mistaken. Editing the response – Captain Morgan May 21 '18 at 15:28