So I used the split step method on the Schrodinger equation and have produced the following equation:
$\Psi(x,t+dt)=F^{-1} \left\{ e^{-i\frac{\hbar^2k^2}{2m}\frac{dt}{\hbar}}F\left\{e^{-iV(x)\frac{dt}{\hbar}}\Psi(x,t) \right\}\right\}$
Which when scaled to dimensionless the time evolution step can be written as:
$\Psi(x,t+dt)=F^{-1}\left\{e^{-ik^2dt}F\left\{e^{-iV(x)dt}\Psi(x,t)\right\}\right\}$
F above represents the Fourier transform operator acting on the equations. Now my problem is that I don't know how to implement the above in Fast Fourier Transform. For example: If I were to take the exponential factor with V(x), do I multiply -iV(x) by dt? Not sure I understand what is going on here....
Thanks