# How to solve the time-dependent Schroedinger equation numerically? [duplicate]

I want to solve the following Schroedinger equation: $$i\hbar\frac{\partial}{\partial t}\Psi(t,x)=\left[-\frac{\hbar}{2m}\partial_x^2+V(x)\right]\Psi(t,x),$$ where $\hbar$, $m$ are constants and $V(x)$ is a given potential. The initial condition is a Gaussian distribution around $0$, that is $$\Psi(0,x)=e^{-\frac{\pi}{2}x^2}.$$

I decompose the complex function $\Psi(t,x)$ into two real parts such that we have two coupled PDEs: $$\Psi(x,t)=\Psi_{R}(t,x)+i\Psi_M(t,x)$$ and give the initial conditions $$\Psi_R(0,x)=e^{-\frac{\pi}{2}x^2},\\ \Psi_M(0,x)=0.$$

But the problem is that, mathematica always tells me that an insufficient boundary conditions have been specified. What is going wrong? How will you implement the computation? Thank you very much for your help.