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## Main Question or Discussion Point

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:

## \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!

## \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!