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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.

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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.

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  • $\begingroup$ 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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I would direct you to Numerical Recipes in C/C++ or any other language for more details. Even if you don't program in those languages the descriptions of the methods is valuable, they point out the issues with various solvers. Another good text is A first Course in the Numerical Analysis of Differential Equations. This type of equation has some challenges. The fact that it's first order in time and second in spatial derivatives leads to a very string stability criterion. This is discussed in NR. The standard RK4 would not do very well with this. Implicit solvers are designed for this problem and the canonical version is Crank-Nicolson method. In short, Schrodinger's equation is a Parabolic equation rather than a standard wave equation. NR specifically discusses Schrodinger as an example and has good references. Even without the time dependent potential stability issues arise.

If you are using MATLAB or Maple, or any other off the shelf high level programming language you should have access to these solvers. For MATLAB you need the ODE toolkit for implicit solvers but you can get a free version in SCILAB or Octave. Maple has them all. You still need to be cognizant of what you are doing as these are not self regulating. I have used MATLAB, SCILAB, and Maple solvers and they are very good but I can still make them fail. I've also written my own. While it may not be impossible to throw an RK4 or finite difference at the problem I'd be leary of this without experience. This is why the implicit solvers were invented.

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