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You're right. It seems NDsolve is clumsy with the SE and your initial condition. It seems a big part of the problem is the tiny numbers being used due to Planck's constan and the initial function you have. Setting hbar to 1, a Gaussian initial wave...
Thank you Humberto! I tried to use DSolve but its output is the command itself (I don't have boundary conditions), thats why I'm trying to solve the system in the Fourier's dominium.