# Finite Coulomb solutions

## Main Question or Discussion Point

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 $$r^{l}$$ and thus the derivative to be $$l r^{l-1}$$ 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 [Broken]

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

$$-\frac{1}{2R}(3-\frac{r^{2}}{R^{2}})$$ where i've set Q = 1, and you again take the Lagurre solution you get
http://members.iinet.net.au/~housewrk/int.jpg [Broken]
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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