K φ° τ/h2 delay fps
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"Reflection" once more with dispersion now. Schrodinger equation on 256×256 square. |Ψ|2 for the travelling wave packet is plotted. Colors correspond to the wave function phase. Ψ = 0 at the square border.

## Schrodinger equation

One of the often used numerical schemes for solving the time-dependent Schrodinger equation
i ∂tΨ = H Ψ = -ΔΨ ,     Ψ(r, t=0) = exp( ikr - (r - ro )2/ a2)
is the implicit Crank-Nicolson scheme
(I + ½iτ H)Ψ t+1 = (I - ½iτ H)Ψ t
with time step τ. It provides a second-order time-approximation to the equation, conserves the norm of the approximate solution and it is always stable. We solve implicit equations
(4 - 2iβ)Ψxt+1 = -2iβ Ψx,yt + (Ψx+1,yt + Ψx-1,yt + Ψx,y+1t + Ψx,y-1t + 4Ψx,yt) +
(Ψx+1,yt+1 + Ψx-1,yt+1 + Ψx,y+1t+1 + Ψx,y-1t+1),     β = h2

iteratively starting with Ψ t+1 = Ψ t. In spite of the known advantages of this scheme in practice one needs to carry out computing with a sufficiently small τ to obtain the solution with a reasonable accuracy. E.g. this script makes 100 itterations on every time step.
 "2D" wave packet [EXT]

Simulations on GPU
updated 4 July 2011