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Solves Schrodinger equation for a Gaussian wave packet in the harmonic oscillator potential using the split-operator method Feit, Fleck and Steiger J. Comput Phys. 47, 412-433, (1982).

Algorithm

They show that the wave function can be expressed as:

ψ(x,t+Δ t)=exp(-iΔt V(x)/2) exp(-iΔt p/2m)exp(-iΔtV(x)/2) ψ(x,t)

The first operation on the wave function (right of last equation)

exp(-iΔtV(x)/2) ψ(x,t)= cos(ΔtV(x)/2)-i sin(ΔtV(x)/2)[Re (ψ(x,t) +i Im (ψ(x,t)]

since the operation is all in coordinate space.

Now take Fast Fourier Transform of last term and call the result ψ₁(x,t):

ψ₁(x,t)= F[exp(-iΔtV(x)/2) ψ(x,t)]

Since we are now in momentum space, we multiply the result by:

ψ₂(x,t)=[cos(Δt p²/2m)-i sin(Δt p²/2m)]ψ₁(x,t)

Now take inverse FFT to go back to coordinate space again and multiply it as:

ψ₃(x,t)=F^-1[exp(-iΔtV(x)/2)ψ₂(x,t)

The wavefunction was advanced in time by Δt, repeat the whole procedure to continue advancing in time steps.