Best way to solve Schrodinger's wave equation numerically.

[Slide rule]

I have been trying to research the best way to solve the Schrödinger wave equation numerically so that I can plot and animate it in Maple. I'd also like to animate as it is affected by a potential. I have been trying for weeks to do this and I don't feel any closer than when I started. I have looked at finite difference method but I get so far and don't know what to do next.

Any help would be greatly appreciated.

The sort of thing I'm looking for is like in this presentation on youtube, especially at 11s leading onto something like the animation at 13s.

http://m.youtube.com/watch?hl=en-GB&client=mv-google&gl=GB&v=Xj9PdeY64rA&fulldescription=1

Thanks you very much

 

[DrClaude]

Can you give more detail? What is the Hamiltonian? In how many dimensions?

And there is no link to the YouTube video.

 

[Slide rule]

Hi, sorry for not posting the link, I was very tired when making this post, definitely an oversight on my part. I will be able to post the link in just over an hour.

With regards to dimensions in would only be in 1 dimension along x. Regarding the hamiltonian I am trying to solve the equation as
i x hbar x diff(psi, t) = -(hbar^2)/2m x diff(psi, x$2) + V(x) x psi
where psi = psi(x, t).

Thank you for replying.

 

[DrClaude]

Since your potential is time independent, the fastest way to solve your problem is to first find the eigenfunctions of the Hamiltonian
$$H \phi_i = E_i \phi_i$$
To do this, discretize space and write the Hamiltonian as a matrix. As you said, you can use a finite difference approximation for the momentum operator.

Once you have the $\phi_i$, find the initial coefficients of your wave function in this basis,
$$\psi(x,t=0) = \sum_i c_i \phi_i(x)$$
by calculating
$$c_i = \int \phi_i^*(x) \psi(x,t=0) dx$$

Then, the wave function at any time time is simply given by
$$\psi(x,t) = \sum_i c_i \phi_i(x) \exp(-i E_i t / \hbar)$$
By advancing $t$ and refreshing the plot, you will get your animation. I have no idea how to do this in Maple

 

[Bill_K]

I have been trying to research the best way to solve the Schrödinger wave equation numerically so that I can plot and animate it in Maple. I'd also like to animate as it is affected by a potential. I have been trying for weeks to do this and I don't feel any closer than when I started. I have looked at finite difference method but I get so far and don't know what to do next.

Slide rule, Why not ask this question over on the Diff Eq forum.