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I have solved the Time Independent Schrodinger Equation using the Numerov method and diagonalizing the Hamiltonian, in 1 - 3 dimensions. I suppose I could time-evolve it by multiplying every element in it by $e^{\frac{iEt}{\hbar}}$ to time evolve it (unless I cant). But I need an even more dynamical solution for what I need to do, and I need a time varying potential.

Is there some method where I can take an initial, known wave-function (like a free-particle solution) and evolve it in time while changing the potential? I have thought about using a Ritz solver, would that do the trick?

I would think the most practical solutions would be with the Schrodinger Equation, but if someone has a Quantum Field Theory solution that is also welcome.

Thank you!

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    $\begingroup$ If you have a time-evolving potential, then the energy levels are no longer constant and a particle that is in a pure state at the beginning will not remain in one. As a consequence, you can no longer just multiply the spatial states by the exponential factor you show: Time and space no longer separate this way. $\endgroup$ Commented Dec 30, 2022 at 22:50
  • $\begingroup$ @WolfgangBangerth Thank you for the response, agreed, that is why I am not sure what method to use. $\endgroup$
    – cgbsu
    Commented Jan 31, 2023 at 17:12

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