K
φ°
τ/h^{2}
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 ∂_{t}Ψ = H Ψ = -ΔΨ ,
Ψ(r, t=0) = exp( ikr - (r -
r_{o })^{2}/ a^{2}) 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)Ψ_{x}^{t+1}
= -2iβ Ψ_{x,y}^{t} +
(Ψ_{x+1,y}^{t} + Ψ_{x-1,y}^{t} +
Ψ_{x,y+1}^{t} + Ψ_{x,y-1}^{t} +
4Ψ_{x,y}^{t}) +
(Ψ_{x+1,y}^{t+1} +
Ψ_{x-1,y}^{t+1} +
Ψ_{x,y+1}^{t+1} + Ψ_{x,y-1}^{t+1}),
β = h^{2}/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.