Numerially integrate the radial schrodinger equation

[wdlang]

i now need to integrate the time-dependent Schrödinger equation in 2D

the potential is rotationally invariant and so is the initial wave function

thus the symmetry of the initial wave function will be preserved in time

Instead of a 2D equation, i now only need to integrate a 1d equation about the radial part of the wave function.

However, in doing so i encounter some difficulties. There are terms like 1/r \partial_r in the equation.

near the origin, this term will diverge!

I guess this is a well-solved problem. Could anyone give me some help?

 

[alxm]

(Why'd this get moved here, of all places? It's a math problem. Numerical methods, really.)

I guess this is a well-solved problem. Could anyone give me some help?

Well, more of an ill-conditioned problem, really. :)
I'm not at all a numerics expert, but the simplest (but surely not most efficient) numerical method that comes to mind would be to calculate the integral from h to infinity and shrink h until you're within convergence limits. You could also use Richardson extrapolation to improve on that.

 

[Modey3]

wdlang,

The 1/r potential term is best handled by recognizing that the value from the analytical solution at r = 0 for the hydrogen atom is 1. You should try to define a cut-off radius, which inside this radius the wavefuntion takes on the above value. Also, be careful of your radius step-size: use a small enough step size so your solution doesn't blow up.

 

[wdlang]

wdlang,

The 1/r potential term is best handled by recognizing that the value from the analytical solution at r = 0 for the hydrogen atom is 1. You should try to define a cut-off radius, which inside this radius the wavefuntion takes on the above value. Also, be careful of your radius step-size: use a small enough step size so your solution doesn't blow up.

Thanks a lot!

but I do not mean the 1/r potential

i mean the lapalace operator in polar coordinates, 2D

you will find a term 1/r(d/dr)

This term diverge near the origin.

I do not know how to handle it. Moreover, i am interested in the time-dependent S equation, not the time-independent S equation.

 

[confinement]

I recommend you take the unknown wavefunction psi(r) and replace it with u(r)/r. Substitute psi = u / r into your equation and you will hopefully find a less ill-conditioned problem for u.

 

[wdlang]

I recommend you take the unknown wavefunction psi(r) and replace it with u(r)/r. Substitute psi = u / r into your equation and you will hopefully find a less ill-conditioned problem for u.

Yes, but that works for 3D

I am concerned with the 2D case.