How to solve Schrödinger equation numerically with time dependent potential How to solve the Schrödinger equation with time dependent potential in 1D or 3D (if it is easier):
$$i\hbar\dfrac{\partial \Psi}{\partial t}(x,t)=\left(-\dfrac{\hbar}{2m}\nabla^2-\frac{e^2}{x+\alpha}-exE(t)\right)\Psi(x,t)$$
where $E(t) = E_0 \exp(-t/\tau^2)sin(\omega_0 t)$ is a Gaussian pulse in time, $\alpha$ is a constant and $e$ is a constant (the electron charge). $\Psi(x,0)$ is hydrogen ground state.
What would it mean to find the solution in a self consistent manner?

this is not an easy topic to discus, since it would take some writing-up.. you can find in Google the detailed methods. $1D$ is simpler. You can use the split operator time-propagator, where the $x$ and $d/dx$-alike operators are treated differently. When applying the derivative operators, usually one makes use of the Fourier Transform, for which very good software packages are available as open source (FFTW - the Fastest Fourier Transform in the West). Not sure there is any meaning to "consistent manner" in this context . You need to specify the initial wave-function though.
