# Energy levels of hydrogen in atomic units

1. Nov 14, 2008

### russdot

[solved] Energy levels of hydrogen in atomic units

1. The problem statement, all variables and given/known data
The effective potential for a hydrogen atom can be thought of as the actual potential plus the centrifugal repulsion, written as:
$$V_{eff} = -\frac{1}{r} + \frac{l (l+1)}{2r^{2}}$$
Remembering that you are working in atomic units, make a plot of V_eff(r) for l=0,1,2,3. Mark the energy levels for n=2,3,4. Explain why, for a given n, l cannot be larger than n-1.

The attempt at a solution
I have plotted the potentials for l=0,1,2,3 here: http://img385.imageshack.us/my.php?image=plotte7.gif

I used $$E=-\frac{1}{n^{2}}$$ for the energy levels in atomic units (is this correct?)
As far as explaining why l <= n-1, I thought that because the energy levels would intersect the curves for l <= n-1 and the potential energies with l>=n would be unbound so that l cannot be larger than n-1. However, this does not seem to be the case from the plot I've constructed. I hope the energies I've used are incorrect, because I can't seem to extract the answer out of my plot??
I realised the energy levels should be : $$E = -\frac{1}{2n^{2}}$$