2d Schrodinger equation - numerical solutions

[diegzumillo]

Hi there,

I was expecting to find a "simulations forums" somewhere here, if there is a better place for this thread please let me know :)

OK, here's the problem: I'm trying to make a simulation with PYthon, at first with a square potential, for simpler potential/boundary conditions. But I'm new on this numerical stuff, so I'm studying about solving PDEs numerically and all, and this is the equation I have arrived:
(with the indices i,j and l associated with x,y and t respectively)
$$\Psi^{l+1}_{i,j}=\frac{i\Delta t \hbar}{m (\Delta x)^2}(\Psi^{l}_{i+1,j}+\Psi^{l}_{i-1,j}+\Psi^{l}_{i,j+1}+\Psi^{l}_{i,j-1}-4\Psi^{l}_{i,j})+\Psi^{l-1}_{i,j}+\frac{2\Delta t}{i\hbar}V^{l}_{i,j}\Psi^{l}_{i,j}$$
(it may not visualize correctly)

From this equation I see that it needs two previous steps to calculate a new one. So my initial conditions must fill the first two steps? Is this how I create a moving Gaussian, for example? by making the first two steps gaussians with a displacement between them? I'm a little lost here :P

 

[badel hyder]

please take this link
http://www.chemistry.emory.edu/faculty/bowman/old_classes/chem430sp99/num%20sol/NUM%20SOL.html

 

[diegzumillo]

I think I'm on the right track, his discrete equation is just like mine.

 

[The_Duck]

A state is completely specified by the wave function at one instant in time. A moving gaussian has a wave function like exp(-(x^2+y^2)/(2*w^2))*exp(i(ax+by)). The first term gives the shape; the second term gives the momentum.

Can I ask why you have a $$\Psi^{l-1}_{i,j}$$ instead of $$\Psi^{l}_{i,j}$$ ?