How Do Numerical and Analytical Solutions Compare in Finite Coulomb Problems?

[FunkyDwarf]

Hey guys,

I'm doing some numerical solutions to wave equations and i started by checking that my method using NDSolve in mathematica worked by comparing it to the analytic solutions DSolve would produce. Now in terms of numerically solving the schrodinger equation i have done the usual trick to integrate out from zero by assuming the solution at some small r0 to be [tex]r^{l}[/tex] and thus the derivative to be [tex]l r^{l-1}[/tex] which means NDSolve doesn't crap its pants at zero where the equation is singular.

Now, this all works fine when i do the normal coulomb potential of -1/r (assume attractive) as when compared with the analytic solution which is regular at the origin (lagurre polynomials) it gives the image below (in both images numeric solutions are purple):

http://members.iinet.net.au/~housewrk/ext.jpg

Where i have taken the magnitude as the waves are in fact pi out of phase, but in terms of 'measureables' you only measure the magnitudes anyway so for all intents and purposes these waves are infact in phase (also the analytic solution is complex). I should point out i don't really care about magnitudes as eventually i am simply hunting for a phase shift.

Anyway, if i do the same procedure with the interior potential for a uniform charged sphere

[tex]-\frac{1}{2R}(3-\frac{r^{2}}{R^{2}})[/tex] where I've set Q = 1, and you again take the Lagurre solution you get
http://members.iinet.net.au/~housewrk/int.jpg
which are clearly out of phase. The radius R of the sphere is 30 here.

Now, i know that i should automatically assume the Lagurre solution is the correct one but I'm not sure what to do as both (analytic) solutions seem to have trouble at the origin, but the probability integrand for both is bounded at the origin (its zero). The derivatives, however, are not always zero at the origin which is another issue.

I guess my question is: has anyone else done this? I can't find a whole lot on the finite coulomb problem :(

Cheers
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[eljose79]

Hello,

As a fellow scientist, I have also encountered similar issues when trying to numerically solve wave equations. Your approach of comparing numerical solutions to analytic solutions is a good way to check for accuracy and validity.

In terms of the Schrodinger equation, I have also used the same trick of assuming a solution at some small r0 to avoid singularities at the origin. However, I have not specifically worked with the finite Coulomb problem, so I am not familiar with any specific solutions for that potential.

From what I can see in your images, there seems to be a phase shift between the two solutions. This could be due to the different potentials you are using, as well as the fact that the Laguerre solution may not be the only valid solution for the finite Coulomb problem. It might be worth exploring other possible solutions or numerical methods to see if you can achieve a better match.

I suggest looking into literature on the finite Coulomb problem and seeing if there are any other known solutions or techniques that could help you in your work. It's always helpful to see what others have done in similar situations.

Best of luck with your research!