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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...
If I understand what you're trying to do, you should note that FourierTransform[f[x,y,z],{x,y,z},{kx,ky,kz}] does return a function of kx,ky, and kz, even if it remains in that form. That is, running the following code[mcode]Ff[kx_, ky_, kz_] := ...