Hi all, I'm trying to compute the solutions to a general case for a Schroedinger equation with a radial potential but I'm stuck on a rather small detail that I'm not sure about. It's well known that I can perform the change of variables to spherical coordinates and express the radial part of the wavefunction as:(adsbygoogle = window.adsbygoogle || []).push({});

## \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right)\chi(r) = E\chi(r)##

with

##\chi(r) = rR(r)##

and ##R(r)## radial part of the solution. Now this works wonderfully for me because the only thing I'm interested in is the radial density, so basically ##\chi^2##, which I think means I don't even have to worry too much with issues of low precision for small ##r##. I already have a Numerov solver for the 1D equation so I thought I'd apply that here. I compute the solutions for a potential ##V_{rot}(r) = V(r) + \frac{\hbar^2l(l+1)}{2mr^2}## imposing as conditions that ##\chi(r) = 0## both at zero and infinity (actually some high but finite r value). What I'm a bit perplexed about though is the degeneracy and the role of ##l##. My understanding of the problem is that I have to compute these solutions for various values of ##l##, and each of them is going to provide a number of states, all with degeneracy ##2l+1##, which then I can put together and sort by energy. I also assume I can apply a cutoff on ##l## (for example only compute solutions for ##l\leq5##) since high ##l## also brings high energy and I only care for the low energy solutions. Is this the right way to proceed? It sounds a bit iffy because it uses different quantum numbers from the typical hydrogen atom solution, so I can't tell how that would work out exactly. Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Computing solutions to the radial Schroedinger equation?

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Computing solutions radial |
---|

I Dimensions of Angular and Radial Nodes |

**Physics Forums | Science Articles, Homework Help, Discussion**