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

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


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

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.


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.

  • $\begingroup$ 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. $\endgroup$ – ZeroTheHero May 21 '18 at 13:30
  • $\begingroup$ @ZeroTheHero I'm mistaken. Editing the response $\endgroup$ – Captain Morgan May 21 '18 at 15:28

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