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Is it possible to use the Numerov method to solve the Time Dependent Schrodinger Equation ($\frac{i\partial\Psi(x, y, z, t)}{\partial t} = \nabla^2\Psi(x, y, z, t) + \Psi(x, y, z, t)V(x, y, z, t)$) with a time varying potential $V(x, y, z, t)$? If so, how?

Thanks!

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  • $\begingroup$ How do you apply a method for second-order equations to a first-order equation? $\endgroup$ Commented Jan 31, 2023 at 17:57
  • $\begingroup$ @LutzLehmann It is a second order equation on the spatial axis isn't it? $\endgroup$
    – cgbsu
    Commented Jan 31, 2023 at 17:59
  • $\begingroup$ Yes, but that is the Laplace operator, not a simple second derivative $\endgroup$ Commented Feb 1, 2023 at 13:10
  • $\begingroup$ @LutzLehmann I'm not sure I understand how that can change the nature of the problem, can you please explain? We could simplify to the 1 + 1 dimensional case with $\psi(x, t)$ and $V(x, t)$, right? Then it would simplify to be a simple second derivative on the spatial axis? $\endgroup$
    – cgbsu
    Commented Feb 4, 2023 at 20:05
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    $\begingroup$ @LutzLehmann I think the only difference I see with the wikipedia entry is that I am use to seeing it in a matrix form, I was following this paper when I did it in 3D. Could well be in Crank-Nicholson, I have been looking into it and it seems similar too what I have done already just in time. Thats part of what I am unsure of. I have a question involving that method as well. $\endgroup$
    – cgbsu
    Commented Feb 4, 2023 at 21:14

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