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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.
One of the often used numerical schemes for solving the time-dependent
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 +
Ψx,y+1t+1 + Ψx,y-1t+1),
β = h2/τ 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.